\(\int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [410]

Optimal result

Integrand size = 24, antiderivative size = 249 \[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^3 d}+\frac {\left (3 a^2-6 a b+35 b^2\right ) \text {arctanh}(\sin (c+d x))}{8 (a-b)^3 d}-\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^3 d}+\frac {1}{16 (a-b) d (1-\sin (c+d x))^2}+\frac {3 a-11 b}{16 (a-b)^2 d (1-\sin (c+d x))}-\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac {3 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))} \]

[Out]

1/8*(3*a^2-6*a*b+35*b^2)*arctanh(sin(d*x+c))/(a-b)^3/d+1/16/(a-b)/d/(1-sin(d*x+c))^2+1/16*(3*a-11*b)/(a-b)^2/d
/(1-sin(d*x+c))-1/16/(a-b)/d/(1+sin(d*x+c))^2+1/16*(-3*a+11*b)/(a-b)^2/d/(1+sin(d*x+c))-1/2*b^(5/4)*arctanh(b^
(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)/d/(a^(1/2)-b^(1/2))^3+1/2*b^(5/4)*arctan(b^(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)
/d/(a^(1/2)+b^(1/2))^3

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3302, 1185, 213, 1181, 211, 214} \[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^3}-\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^3}+\frac {\left (3 a^2-6 a b+35 b^2\right ) \text {arctanh}(\sin (c+d x))}{8 d (a-b)^3}+\frac {3 a-11 b}{16 d (a-b)^2 (1-\sin (c+d x))}-\frac {3 a-11 b}{16 d (a-b)^2 (\sin (c+d x)+1)}+\frac {1}{16 d (a-b) (1-\sin (c+d x))^2}-\frac {1}{16 d (a-b) (\sin (c+d x)+1)^2} \]

[In]

Int[Sec[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

(b^(5/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^3*d) + ((3*a^2 - 6*a*b + 35*b^
2)*ArcTanh[Sin[c + d*x]])/(8*(a - b)^3*d) - (b^(5/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt
[a] - Sqrt[b])^3*d) + 1/(16*(a - b)*d*(1 - Sin[c + d*x])^2) + (3*a - 11*b)/(16*(a - b)^2*d*(1 - Sin[c + d*x]))
- 1/(16*(a - b)*d*(1 + Sin[c + d*x])^2) - (3*a - 11*b)/(16*(a - b)^2*d*(1 + Sin[c + d*x]))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 1181

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q))
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]

Rule 1185

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rule 3302

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a-b x^4\right )} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{8 (a-b) (-1+x)^3}+\frac {3 a-11 b}{16 (a-b)^2 (-1+x)^2}+\frac {1}{8 (a-b) (1+x)^3}+\frac {3 a-11 b}{16 (a-b)^2 (1+x)^2}+\frac {-3 a^2+6 a b-35 b^2}{8 (a-b)^3 \left (-1+x^2\right )}+\frac {b^2 \left (-3 a-b-(a+3 b) x^2\right )}{(a-b)^3 \left (a-b x^4\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {1}{16 (a-b) d (1-\sin (c+d x))^2}+\frac {3 a-11 b}{16 (a-b)^2 d (1-\sin (c+d x))}-\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac {3 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^2 \text {Subst}\left (\int \frac {-3 a-b+(-a-3 b) x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{(a-b)^3 d}-\frac {\left (3 a^2-6 a b+35 b^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sin (c+d x)\right )}{8 (a-b)^3 d} \\ & = \frac {\left (3 a^2-6 a b+35 b^2\right ) \text {arctanh}(\sin (c+d x))}{8 (a-b)^3 d}+\frac {1}{16 (a-b) d (1-\sin (c+d x))^2}+\frac {3 a-11 b}{16 (a-b)^2 d (1-\sin (c+d x))}-\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac {3 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))}-\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )^3 d}-\frac {b^2 \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^3 d} \\ & = \frac {b^{5/4} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^3 d}+\frac {\left (3 a^2-6 a b+35 b^2\right ) \text {arctanh}(\sin (c+d x))}{8 (a-b)^3 d}-\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^3 d}+\frac {1}{16 (a-b) d (1-\sin (c+d x))^2}+\frac {3 a-11 b}{16 (a-b)^2 d (1-\sin (c+d x))}-\frac {1}{16 (a-b) d (1+\sin (c+d x))^2}-\frac {3 a-11 b}{16 (a-b)^2 d (1+\sin (c+d x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.87 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.27 \[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {2 \left (3 a^2-6 a b+35 b^2\right ) \text {arctanh}(\sin (c+d x))}{(a-b)^3}+\frac {4 b^{5/4} \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^3}+\frac {4 i b^{5/4} \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^3}-\frac {4 i b^{5/4} \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^3}-\frac {4 b^{5/4} \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )}{a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^3}+\frac {1}{(a-b) (-1+\sin (c+d x))^2}+\frac {-3 a+11 b}{(a-b)^2 (-1+\sin (c+d x))}-\frac {1}{(a-b) (1+\sin (c+d x))^2}+\frac {-3 a+11 b}{(a-b)^2 (1+\sin (c+d x))}}{16 d} \]

[In]

Integrate[Sec[c + d*x]^5/(a - b*Sin[c + d*x]^4),x]

[Out]

((2*(3*a^2 - 6*a*b + 35*b^2)*ArcTanh[Sin[c + d*x]])/(a - b)^3 + (4*b^(5/4)*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]]
)/(a^(3/4)*(Sqrt[a] - Sqrt[b])^3) + ((4*I)*b^(5/4)*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] +
Sqrt[b])^3) - ((4*I)*b^(5/4)*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] + Sqrt[b])^3) - (4*b^(5/
4)*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^3) + 1/((a - b)*(-1 + Sin[c + d*x])^2) +
(-3*a + 11*b)/((a - b)^2*(-1 + Sin[c + d*x])) - 1/((a - b)*(1 + Sin[c + d*x])^2) + (-3*a + 11*b)/((a - b)^2*(1
+ Sin[c + d*x])))/(16*d)

Maple [A] (verified)

Time = 3.40 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a -11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-6 a b +35 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}+\frac {1}{2 \left (8 a -8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a -11 b}{16 \left (a -b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a^{2}+6 a b -35 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a -b \right )^{3}}+\frac {b^{2} \left (\frac {\left (-3 a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (-a -3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a -b \right )^{3}}}{d}\)	\(322\)
default	\(\frac {-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a -11 b}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-6 a b +35 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}+\frac {1}{2 \left (8 a -8 b \right ) \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a -11 b}{16 \left (a -b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a^{2}+6 a b -35 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a -b \right )^{3}}+\frac {b^{2} \left (\frac {\left (-3 a -b \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {\left (-a -3 b \right ) \left (2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\left (a -b \right )^{3}}}{d}\)	\(322\)
risch	\(\text {Expression too large to display}\)	\(1171\)

[In]

int(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2/(8*a-8*b)/(1+sin(d*x+c))^2-1/16*(3*a-11*b)/(a-b)^2/(1+sin(d*x+c))+1/16*(3*a^2-6*a*b+35*b^2)/(a-b)^3*
ln(1+sin(d*x+c))+1/2/(8*a-8*b)/(sin(d*x+c)-1)^2-1/16*(3*a-11*b)/(a-b)^2/(sin(d*x+c)-1)+1/16/(a-b)^3*(-3*a^2+6*
a*b-35*b^2)*ln(sin(d*x+c)-1)+b^2/(a-b)^3*(1/4*(-3*a-b)*(1/b*a)^(1/4)/a*(ln((sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x
+c)-(1/b*a)^(1/4)))+2*arctan(sin(d*x+c)/(1/b*a)^(1/4)))-1/4*(-a-3*b)/b/(1/b*a)^(1/4)*(2*arctan(sin(d*x+c)/(1/b
*a)^(1/4))-ln((sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1/b*a)^(1/4))))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3703 vs. \(2 (207) = 414\).

Time = 1.65 (sec) , antiderivative size = 3703, normalized size of antiderivative = 14.87 \[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/16*(4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 -
20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*
a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 92
4*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a
^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(1/2*(a^3*b^4 + 15*a^
2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5
- 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^1
0 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^
8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) - (3*a^5*b^3 + 46*a^4*b^4 + 60
*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^
3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 +
30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6
- 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15
*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) - 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a
^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqr
t((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*
a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9
+ 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*
b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 -
3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^
5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^1
2*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12
*a^4*b^11 + a^3*b^12)*d^4)) + (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 +
20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^
5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2
- 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*
b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b
^6)*d^2))) - 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*
b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8
+ 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^
5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7
- 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(-1/2*(a^3*b^4
+ 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*((a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*
a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 3
0*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 -
792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) - (3*a^5*b^3 + 46*a^4*b
^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 + (a^7 - 6*a^6*b + 15*a^5*b^2 - 20
*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^
2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*
a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6
*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2))) + 4*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sq
rt((6*a^2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*
d^2*sqrt((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*
b + 66*a^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a
^6*b^9 + 66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 -
6*a^2*b^5 + a*b^6)*d^2))*cos(d*x + c)^4*log(-1/2*(a^3*b^4 + 15*a^2*b^5 + 15*a*b^6 + b^7)*sin(d*x + c) + 1/2*(
(a^10 - 3*a^9*b - 3*a^8*b^2 + 25*a^7*b^3 - 45*a^6*b^4 + 39*a^5*b^5 - 17*a^4*b^6 + 3*a^3*b^7)*d^3*sqrt((a^6*b^5
+ 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a^13*b^2 -
220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 + 66*a^5*b
^10 - 12*a^4*b^11 + a^3*b^12)*d^4)) + (3*a^5*b^3 + 46*a^4*b^4 + 60*a^3*b^5 + 18*a^2*b^6 + a*b^7)*d)*sqrt((6*a^
2*b^3 + 20*a*b^4 + 6*b^5 - (a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*d^2*sqrt
((a^6*b^5 + 30*a^5*b^6 + 255*a^4*b^7 + 452*a^3*b^8 + 255*a^2*b^9 + 30*a*b^10 + b^11)/((a^15 - 12*a^14*b + 66*a
^13*b^2 - 220*a^12*b^3 + 495*a^11*b^4 - 792*a^10*b^5 + 924*a^9*b^6 - 792*a^8*b^7 + 495*a^7*b^8 - 220*a^6*b^9 +
66*a^5*b^10 - 12*a^4*b^11 + a^3*b^12)*d^4)))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b
^5 + a*b^6)*d^2))) - (3*a^2 - 6*a*b + 35*b^2)*cos(d*x + c)^4*log(sin(d*x + c) + 1) + (3*a^2 - 6*a*b + 35*b^2)*
cos(d*x + c)^4*log(-sin(d*x + c) + 1) - 2*((3*a^2 - 14*a*b + 11*b^2)*cos(d*x + c)^2 + 2*a^2 - 4*a*b + 2*b^2)*s
in(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(d*x + c)^4)

Sympy [F]

\[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]

[In]

integrate(sec(d*x+c)**5/(a-b*sin(d*x+c)**4),x)

[Out]

Integral(sec(c + d*x)**5/(a - b*sin(c + d*x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.46 \[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {4 \, b^{2} {\left (\frac {2 \, {\left (b {\left (3 \, \sqrt {a} - \sqrt {b}\right )} + a^{\frac {3}{2}} - 3 \, a \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b {\left (3 \, \sqrt {a} + \sqrt {b}\right )} + a^{\frac {3}{2}} + 3 \, a \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {2 \, {\left ({\left (3 \, a - 11 \, b\right )} \sin \left (d x + c\right )^{3} - {\left (5 \, a - 13 \, b\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{2} + a^{2} - 2 \, a b + b^{2}}}{16 \, d} \]

[In]

integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

1/16*(4*b^2*(2*(b*(3*sqrt(a) - sqrt(b)) + a^(3/2) - 3*a*sqrt(b))*arctan(sqrt(b)*sin(d*x + c)/sqrt(sqrt(a)*sqrt
(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + (b*(3*sqrt(a) + sqrt(b)) + a^(3/2) + 3*a*sqrt(b))*log((sqrt(b)
*sin(d*x + c) - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*sin(d*x + c) + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*s
qrt(b))*sqrt(b)))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + (3*a^2 - 6*a*b + 35*b^2)*log(sin(d*x + c) + 1)/(a^3 - 3*a^
2*b + 3*a*b^2 - b^3) - (3*a^2 - 6*a*b + 35*b^2)*log(sin(d*x + c) - 1)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 2*((3*
a - 11*b)*sin(d*x + c)^3 - (5*a - 13*b)*sin(d*x + c))/((a^2 - 2*a*b + b^2)*sin(d*x + c)^4 - 2*(a^2 - 2*a*b + b
^2)*sin(d*x + c)^2 + a^2 - 2*a*b + b^2))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (207) = 414\).

Time = 1.28 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.53 \[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {8 \, {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{4} b - 3 \, \sqrt {2} a^{3} b^{2} + 3 \, \sqrt {2} a^{2} b^{3} - \sqrt {2} a b^{4}} + \frac {8 \, {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} + \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{4} b - 3 \, \sqrt {2} a^{3} b^{2} + 3 \, \sqrt {2} a^{2} b^{3} - \sqrt {2} a b^{4}} - \frac {4 \, {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{\sqrt {2} a^{4} b - 3 \, \sqrt {2} a^{3} b^{2} + 3 \, \sqrt {2} a^{2} b^{3} - \sqrt {2} a b^{4}} + \frac {4 \, {\left (\left (-a b^{3}\right )^{\frac {3}{4}} {\left (a + 3 \, b\right )} - \left (-a b^{3}\right )^{\frac {1}{4}} {\left (3 \, a b^{2} + b^{3}\right )}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{\sqrt {2} a^{4} b - 3 \, \sqrt {2} a^{3} b^{2} + 3 \, \sqrt {2} a^{2} b^{3} - \sqrt {2} a b^{4}} - \frac {{\left (3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {{\left (3 \, a^{2} - 6 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {2 \, {\left (3 \, a \sin \left (d x + c\right )^{3} - 11 \, b \sin \left (d x + c\right )^{3} - 5 \, a \sin \left (d x + c\right ) + 13 \, b \sin \left (d x + c\right )\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]

[In]

integrate(sec(d*x+c)^5/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

-1/16*(8*((-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4)
+ 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) + 8*((
-a*b^3)^(3/4)*(a + 3*b) + (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) - 2*sin(d*
x + c))/(-a/b)^(1/4))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) - 4*((-a*b^3)^(3
/4)*(a + 3*b) - (-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(
-a/b))/(sqrt(2)*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) + 4*((-a*b^3)^(3/4)*(a + 3*b) -
(-a*b^3)^(1/4)*(3*a*b^2 + b^3))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(sqrt(2)
*a^4*b - 3*sqrt(2)*a^3*b^2 + 3*sqrt(2)*a^2*b^3 - sqrt(2)*a*b^4) - (3*a^2 - 6*a*b + 35*b^2)*log(abs(sin(d*x + c
) + 1))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + (3*a^2 - 6*a*b + 35*b^2)*log(abs(sin(d*x + c) - 1))/(a^3 - 3*a^2*b +
3*a*b^2 - b^3) + 2*(3*a*sin(d*x + c)^3 - 11*b*sin(d*x + c)^3 - 5*a*sin(d*x + c) + 13*b*sin(d*x + c))/((a^2 -
2*a*b + b^2)*(sin(d*x + c)^2 - 1)^2))/d

Mupad [B] (verification not implemented)

Time = 20.00 (sec) , antiderivative size = 12217, normalized size of antiderivative = 49.06 \[ \int \frac {\sec ^5(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^5*(a - b*sin(c + d*x)^4)),x)

[Out]

(atan(((((18064*a*b^13 + 256*b^14 + 119760*a^2*b^12 - 275888*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 1371
2*a^6*b^8 + 6768*a^7*b^7 - 720*a^8*b^6)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*
b^4 - 56*a^5*b^3 + 28*a^6*b^2)) - (((4096*a*b^15 + 12288*a^2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 24576
00*a^5*b^11 + 3440640*a^6*b^10 - 3182592*a^7*b^9 + 2002944*a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*
a^11*b^5 + 6144*a^12*b^4)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b
^3 + 28*a^6*b^2)) - (sin(c + d*x)*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^
4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5
*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*(8192*a^2*b^15 - 73728*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 +
737280*a^6*b^11 - 344064*a^7*b^10 - 344064*a^8*b^9 + 737280*a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 7372
8*a^12*b^5 + 8192*a^13*b^4))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^
5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a
*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^
6*b^3 + 15*a^7*b^2)))^(1/2) - (sin(c + d*x)*(256*b^15 - 50464*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 2
12736*a^5*b^10 - 111296*a^6*b^9 + 57088*a^7*b^8 - 20096*a^8*b^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a
^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2
) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(
1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*(-(a^3*(a^3*b^
5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3
*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) - (sin(c
+ d*x)*(6802*a*b^12 + 1257*b^13 - 857*a^2*b^11 + 892*a^3*b^10 + 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^
8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^
5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3
*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*1i - (((
18064*a*b^13 + 256*b^14 + 119760*a^2*b^12 - 275888*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 13712*a^6*b^8
+ 6768*a^7*b^7 - 720*a^8*b^6)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a
^5*b^3 + 28*a^6*b^2)) - (((4096*a*b^15 + 12288*a^2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 2457600*a^5*b^1
1 + 3440640*a^6*b^10 - 3182592*a^7*b^9 + 2002944*a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*a^11*b^5 +
6144*a^12*b^4)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^
6*b^2)) + (sin(c + d*x)*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15
*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*
a^6*b^3 + 15*a^7*b^2)))^(1/2)*(8192*a^2*b^15 - 73728*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 + 737280*a^6
*b^11 - 344064*a^7*b^10 - 344064*a^8*b^9 + 737280*a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 73728*a^12*b^5
+ 8192*a^13*b^4))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28
*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*
b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15
*a^7*b^2)))^(1/2) + (sin(c + d*x)*(256*b^15 - 50464*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 212736*a^5*
b^10 - 111296*a^6*b^9 + 57088*a^7*b^8 - 20096*a^8*b^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a^7*b - 8*a
*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a
^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*
(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*(-(a^3*(a^3*b^5)^(1/2) +
b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2
))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (sin(c + d*x)*(6
802*a*b^12 + 1257*b^13 - 857*a^2*b^11 + 892*a^3*b^10 + 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^8 - 8*a^7*
b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) +
b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2
))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*1i)/((((18064*a*b^
13 + 256*b^14 + 119760*a^2*b^12 - 275888*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 13712*a^6*b^8 + 6768*a^7
*b^7 - 720*a^8*b^6)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 2
8*a^6*b^2)) - (((4096*a*b^15 + 12288*a^2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 2457600*a^5*b^11 + 344064
0*a^6*b^10 - 3182592*a^7*b^9 + 2002944*a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*a^11*b^5 + 6144*a^12
*b^4)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) -
(sin(c + d*x)*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^
3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 +
15*a^7*b^2)))^(1/2)*(8192*a^2*b^15 - 73728*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 + 737280*a^6*b^11 - 34
4064*a^7*b^10 - 344064*a^8*b^9 + 737280*a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 73728*a^12*b^5 + 8192*a^
13*b^4))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2))
)*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2)
+ 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2))
)^(1/2) - (sin(c + d*x)*(256*b^15 - 50464*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 212736*a^5*b^10 - 111
296*a^6*b^9 + 57088*a^7*b^8 - 20096*a^8*b^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8
+ 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1
/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a
^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*
b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^
9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) - (sin(c + d*x)*(6802*a*b^12
+ 1257*b^13 - 857*a^2*b^11 + 892*a^3*b^10 + 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^8 - 8*a^7*b - 8*a*b^
7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*
b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^
9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (((18064*a*b^13 + 256*b^14
+ 119760*a^2*b^12 - 275888*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 13712*a^6*b^8 + 6768*a^7*b^7 - 720*a^
8*b^6)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) -
(((4096*a*b^15 + 12288*a^2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 2457600*a^5*b^11 + 3440640*a^6*b^10 -
3182592*a^7*b^9 + 2002944*a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*a^11*b^5 + 6144*a^12*b^4)/(64*(a^
8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) + (sin(c + d*x)
*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2)
+ 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))
^(1/2)*(8192*a^2*b^15 - 73728*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 + 737280*a^6*b^11 - 344064*a^7*b^10
- 344064*a^8*b^9 + 737280*a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 73728*a^12*b^5 + 8192*a^13*b^4))/(16*
(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3
*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(
a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (si
n(c + d*x)*(256*b^15 - 50464*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 212736*a^5*b^10 - 111296*a^6*b^9 +
57088*a^7*b^8 - 20096*a^8*b^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6
- 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b
^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^
6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) -
6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b +
a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (sin(c + d*x)*(6802*a*b^12 + 1257*b^13
- 857*a^2*b^11 + 892*a^3*b^10 + 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*
a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) -
6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b +
a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) - (1505*b^12 - 748*a*b^11 + 318*a^2*b^10
- 60*a^3*b^9 + 9*a^4*b^8)/(32*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b
^3 + 28*a^6*b^2))))*(-(a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) - 6*a^2*b^5 - 20*a^3*b^4 - 6*a^4*b^3 + 15*a*b
^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*
b^3 + 15*a^7*b^2)))^(1/2)*2i)/d - (log(sin(c + d*x) - 1)*((2*b^2)/(a - b)^3 + 3/(16*(a - b))))/d + (atan(((((1
8064*a*b^13 + 256*b^14 + 119760*a^2*b^12 - 275888*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 13712*a^6*b^8 +
6768*a^7*b^7 - 720*a^8*b^6)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^
5*b^3 + 28*a^6*b^2)) - (((4096*a*b^15 + 12288*a^2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 2457600*a^5*b^11
+ 3440640*a^6*b^10 - 3182592*a^7*b^9 + 2002944*a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*a^11*b^5 +
6144*a^12*b^4)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6
*b^2)) - (sin(c + d*x)*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a
*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^
6*b^3 + 15*a^7*b^2)))^(1/2)*(8192*a^2*b^15 - 73728*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 + 737280*a^6*b
^11 - 344064*a^7*b^10 - 344064*a^8*b^9 + 737280*a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 73728*a^12*b^5 +
8192*a^13*b^4))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a
^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5
)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^
7*b^2)))^(1/2) - (sin(c + d*x)*(256*b^15 - 50464*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 212736*a^5*b^1
0 - 111296*a^6*b^9 + 57088*a^7*b^8 - 20096*a^8*b^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a^7*b - 8*a*b^
7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b
^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9
- 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*((a^3*(a^3*b^5)^(1/2) + b^3*
(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(1
6*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) - (sin(c + d*x)*(6802*a
*b^12 + 1257*b^13 - 857*a^2*b^11 + 892*a^3*b^10 + 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^8 - 8*a^7*b - 8
*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(
a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16
*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*1i - (((18064*a*b^13 + 2
56*b^14 + 119760*a^2*b^12 - 275888*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 13712*a^6*b^8 + 6768*a^7*b^7 -
720*a^8*b^6)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*
b^2)) - (((4096*a*b^15 + 12288*a^2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 2457600*a^5*b^11 + 3440640*a^6*
b^10 - 3182592*a^7*b^9 + 2002944*a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*a^11*b^5 + 6144*a^12*b^4)/
(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) + (sin(c
+ d*x)*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^
(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*
b^2)))^(1/2)*(8192*a^2*b^15 - 73728*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 + 737280*a^6*b^11 - 344064*a^
7*b^10 - 344064*a^8*b^9 + 737280*a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 73728*a^12*b^5 + 8192*a^13*b^4)
)/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3
*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^
2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)
+ (sin(c + d*x)*(256*b^15 - 50464*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 212736*a^5*b^10 - 111296*a^6*
b^9 + 57088*a^7*b^8 - 20096*a^8*b^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^
2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a
^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^
3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2)
+ 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*
b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (sin(c + d*x)*(6802*a*b^12 + 1257*b^
13 - 857*a^2*b^11 + 892*a^3*b^10 + 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 +
28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2)
+ 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b
+ a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*1i)/((((18064*a*b^13 + 256*b^14 + 11976
0*a^2*b^12 - 275888*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 13712*a^6*b^8 + 6768*a^7*b^7 - 720*a^8*b^6)/(
64*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) - (((4096
*a*b^15 + 12288*a^2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 2457600*a^5*b^11 + 3440640*a^6*b^10 - 3182592*
a^7*b^9 + 2002944*a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*a^11*b^5 + 6144*a^12*b^4)/(64*(a^8 - 8*a^
7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) - (sin(c + d*x)*((a^3*(
a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*
b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*(8
192*a^2*b^15 - 73728*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 + 737280*a^6*b^11 - 344064*a^7*b^10 - 344064
*a^8*b^9 + 737280*a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 73728*a^12*b^5 + 8192*a^13*b^4))/(16*(a^8 - 8*
a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2
) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(
1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) - (sin(c + d*x)
*(256*b^15 - 50464*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 212736*a^5*b^10 - 111296*a^6*b^9 + 57088*a^7
*b^8 - 20096*a^8*b^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*
b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3
*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b
^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 +
20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6
*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) - (sin(c + d*x)*(6802*a*b^12 + 1257*b^13 - 857*a^2*b^
11 + 892*a^3*b^10 + 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56
*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 2
0*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*
a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (((18064*a*b^13 + 256*b^14 + 119760*a^2*b^12 - 27588
8*a^3*b^11 + 116624*a^4*b^10 + 28848*a^5*b^9 - 13712*a^6*b^8 + 6768*a^7*b^7 - 720*a^8*b^6)/(64*(a^8 - 8*a^7*b
- 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) - (((4096*a*b^15 + 12288*a^
2*b^14 - 251904*a^3*b^13 + 1087488*a^4*b^12 - 2457600*a^5*b^11 + 3440640*a^6*b^10 - 3182592*a^7*b^9 + 2002944*
a^8*b^8 - 872448*a^9*b^7 + 266240*a^10*b^6 - 55296*a^11*b^5 + 6144*a^12*b^4)/(64*(a^8 - 8*a^7*b - 8*a*b^7 + b^
8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)) + (sin(c + d*x)*((a^3*(a^3*b^5)^(1/2) + b
^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))
/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*(8192*a^2*b^15 - 737
28*a^3*b^14 + 286720*a^4*b^13 - 614400*a^5*b^12 + 737280*a^6*b^11 - 344064*a^7*b^10 - 344064*a^8*b^9 + 737280*
a^9*b^8 - 614400*a^10*b^7 + 286720*a^11*b^6 - 73728*a^12*b^5 + 8192*a^13*b^4))/(16*(a^8 - 8*a^7*b - 8*a*b^7 +
b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^
(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6
*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (sin(c + d*x)*(256*b^15 - 50464
*a^2*b^13 + 190720*a^3*b^12 - 280960*a^4*b^11 + 212736*a^5*b^10 - 111296*a^6*b^9 + 57088*a^7*b^8 - 20096*a^8*b
^7 + 2304*a^9*b^6 - 288*a^10*b^5))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 -
56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 +
15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 -
20*a^6*b^3 + 15*a^7*b^2)))^(1/2))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4
*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*
b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) + (sin(c + d*x)*(6802*a*b^12 + 1257*b^13 - 857*a^2*b^11 + 892*a^3*b^10
+ 71*a^4*b^9 + 18*a^5*b^8 + 9*a^6*b^7))/(16*(a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*
b^4 - 56*a^5*b^3 + 28*a^6*b^2)))*((a^3*(a^3*b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*
b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b
^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2) - (1505*b^12 - 748*a*b^11 + 318*a^2*b^10 - 60*a^3*b^9 + 9*a^4*b^8)/(32*(
a^8 - 8*a^7*b - 8*a*b^7 + b^8 + 28*a^2*b^6 - 56*a^3*b^5 + 70*a^4*b^4 - 56*a^5*b^3 + 28*a^6*b^2))))*((a^3*(a^3*
b^5)^(1/2) + b^3*(a^3*b^5)^(1/2) + 6*a^2*b^5 + 20*a^3*b^4 + 6*a^4*b^3 + 15*a*b^2*(a^3*b^5)^(1/2) + 15*a^2*b*(a
^3*b^5)^(1/2))/(16*(a^9 - 6*a^8*b + a^3*b^6 - 6*a^4*b^5 + 15*a^5*b^4 - 20*a^6*b^3 + 15*a^7*b^2)))^(1/2)*2i)/d
+ ((sin(c + d*x)*(5*a - 13*b))/(8*(a^2 - 2*a*b + b^2)) - (sin(c + d*x)^3*(3*a - 11*b))/(8*(a^2 - 2*a*b + b^2))
)/(d*(cos(c + d*x)^2 - sin(c + d*x)^2 + sin(c + d*x)^4)) + (log(sin(c + d*x) + 1)*(3*a^2 - 6*a*b + 35*b^2))/(1
6*d*(a - b)^3)