\(\int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [412]

Optimal result

Integrand size = 24, antiderivative size = 186 \[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {11 x}{8 b}-\frac {(a+3 b) x}{b^2}+\frac {\left (\sqrt {a}-\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac {11 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d} \]

[Out]

-11/8*x/b-(a+3*b)*x/b^2-11/8*cos(d*x+c)*sin(d*x+c)/b/d-1/4*cos(d*x+c)^3*sin(d*x+c)/b/d+1/2*arctan((a^(1/2)-b^(
1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(a^(1/2)-b^(1/2))^(7/2)/a^(3/4)/b^2/d+1/2*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d
*x+c)/a^(1/4))*(a^(1/2)+b^(1/2))^(7/2)/a^(3/4)/b^2/d

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3303, 1184, 205, 209, 1180, 211} \[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\left (\sqrt {a}-\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac {x (a+3 b)}{b^2}-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 b d}-\frac {11 \sin (c+d x) \cos (c+d x)}{8 b d}-\frac {11 x}{8 b} \]

[In]

Int[Cos[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]

[Out]

(-11*x)/(8*b) - ((a + 3*b)*x)/b^2 + ((Sqrt[a] - Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a
^(1/4)])/(2*a^(3/4)*b^2*d) + ((Sqrt[a] + Sqrt[b])^(7/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)]
)/(2*a^(3/4)*b^2*d) - (11*Cos[c + d*x]*Sin[c + d*x])/(8*b*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(4*b*d)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 209

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

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 1180

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

Rule 1184

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

Rule 3303

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(m/2 + 2
*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{b \left (1+x^2\right )^3}-\frac {2}{b \left (1+x^2\right )^2}+\frac {-a-3 b}{b^2 \left (1+x^2\right )}+\frac {a^2+6 a b+b^2+(a-b) (a+3 b) x^2}{b^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a^2+6 a b+b^2+(a-b) (a+3 b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {(a+3 b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d} \\ & = -\frac {(a+3 b) x}{b^2}-\frac {\cos (c+d x) \sin (c+d x)}{b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\left (\left (\sqrt {a}-\sqrt {b}\right )^4 \left (\sqrt {a}+\sqrt {b}\right )\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt {a} b^2 d}+\frac {\left (\left (\sqrt {a}-\sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right )^4\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt {a} b^2 d}-\frac {3 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 b d}-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d} \\ & = -\frac {x}{b}-\frac {(a+3 b) x}{b^2}+\frac {\left (\sqrt {a}-\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac {11 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac {3 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 b d} \\ & = -\frac {11 x}{8 b}-\frac {(a+3 b) x}{b^2}+\frac {\left (\sqrt {a}-\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}+\frac {\left (\sqrt {a}+\sqrt {b}\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^2 d}-\frac {11 \cos (c+d x) \sin (c+d x)}{8 b d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {4 (8 a+35 b) (c+d x)-\frac {16 \left (\sqrt {a}+\sqrt {b}\right )^4 \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {16 \left (\sqrt {a}-\sqrt {b}\right )^4 \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {-a+\sqrt {a} \sqrt {b}}}+24 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{32 b^2 d} \]

[In]

Integrate[Cos[c + d*x]^8/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/32*(4*(8*a + 35*b)*(c + d*x) - (16*(Sqrt[a] + Sqrt[b])^4*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a +
Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (16*(Sqrt[a] - Sqrt[b])^4*ArcTanh[((Sqrt[a] - Sqrt[b
])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 24*b*Sin[2*(c + d*x)] + b
*Sin[4*(c + d*x)])/(b^2*d)

Maple [A] (verified)

Time = 2.68 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\frac {\frac {\left (a -b \right ) \left (\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b +3 a b +b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b -3 a b -b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b^{2}}-\frac {\frac {\frac {11 \left (\tan ^{3}\left (d x +c \right )\right ) b}{8}+\frac {13 \tan \left (d x +c \right ) b}{8}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (35 b +8 a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{b^{2}}}{d}\)	\(224\)
default	\(\frac {\frac {\left (a -b \right ) \left (\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b +3 a b +b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b -3 a b -b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b^{2}}-\frac {\frac {\frac {11 \left (\tan ^{3}\left (d x +c \right )\right ) b}{8}+\frac {13 \tan \left (d x +c \right ) b}{8}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (35 b +8 a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{b^{2}}}{d}\)	\(224\)
risch	\(\text {Expression too large to display}\)	\(1433\)

[In]

int(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/b^2*(a-b)*(1/2*(a*(a*b)^(1/2)+3*(a*b)^(1/2)*b+3*a*b+b^2)/(a*b)^(1/2)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arct
anh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+1/2*(a*(a*b)^(1/2)+3*(a*b)^(1/2)*b-3*a*b-b^2)/(a*b)^(1/2)
/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2)))-1/b^2*((11/8*tan(d*x+c)
^3*b+13/8*tan(d*x+c)*b)/(1+tan(d*x+c)^2)^2+1/8*(35*b+8*a)*arctan(tan(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2433 vs. \(2 (144) = 288\).

Time = 0.97 (sec) , antiderivative size = 2433, normalized size of antiderivative = 13.08 \[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/8*(b^2*d*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 +
b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2))*log(7/4*a^6 + 7/2*a^5*b - 63/4*a^4*b^2 +
9*a^3*b^3 + 25/4*a^2*b^4 - 9/2*a*b^5 - 1/4*b^6 - 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4
- 18*a*b^5 - b^6)*cos(d*x + c)^2 + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1
484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) - (21*a^5*b^2 + 112*a^4*b
^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b +
1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^
3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4
+ 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 +
b^6)/(a^3*b^7*d^4))) - b^2*d*sqrt(-(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^
2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2))*log(7/4*a^6 + 7/2*a^5
*b - 63/4*a^4*b^2 + 9*a^3*b^3 + 25/4*a^2*b^4 - 9/2*a*b^5 - 1/4*b^6 - 1/4*(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a
^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 - 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b
+ 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*sin(d*x + c) - (21*
a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a*b^4*d^2*sqrt((4
9*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) + a^3 + 21*a^2*
b + 35*a*b^2 + 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (
a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a
^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) + b^2*d*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 148
4*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2))*log
(-7/4*a^6 - 7/2*a^5*b + 63/4*a^4*b^2 - 9*a^3*b^3 - 25/4*a^2*b^4 + 9/2*a*b^5 + 1/4*b^6 + 1/4*(7*a^6 + 14*a^5*b
- 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 + 1/2*((a^4*b^5 + 3*a^3*b^6)*d^3*sqrt(
(49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))*cos(d*x + c)*
sin(d*x + c) + (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(
(a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4
)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2*
cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 14
84*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) - b^2*d*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b +
1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3
)/(a*b^4*d^2))*log(-7/4*a^6 - 7/2*a^5*b + 63/4*a^4*b^2 - 9*a^3*b^3 - 25/4*a^2*b^4 + 9/2*a*b^5 + 1/4*b^6 + 1/4*
(7*a^6 + 14*a^5*b - 63*a^4*b^2 + 36*a^3*b^3 + 25*a^2*b^4 - 18*a*b^5 - b^6)*cos(d*x + c)^2 - 1/2*((a^4*b^5 + 3*
a^3*b^6)*d^3*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d
^4))*cos(d*x + c)*sin(d*x + c) + (21*a^5*b^2 + 112*a^4*b^3 + 98*a^3*b^4 + 24*a^2*b^5 + a*b^6)*d*cos(d*x + c)*s
in(d*x + c))*sqrt((a*b^4*d^2*sqrt((49*a^6 + 490*a^5*b + 1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 +
b^6)/(a^3*b^7*d^4)) - a^3 - 21*a^2*b - 35*a*b^2 - 7*b^3)/(a*b^4*d^2)) - 1/4*(2*(a^5*b^3 - 3*a^4*b^4 + 3*a^3*b
^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^5*b^3 - 3*a^4*b^4 + 3*a^3*b^5 - a^2*b^6)*d^2)*sqrt((49*a^6 + 490*a^5*b +
1519*a^4*b^2 + 1484*a^3*b^3 + 511*a^2*b^4 + 42*a*b^5 + b^6)/(a^3*b^7*d^4))) - (8*a + 35*b)*d*x - (2*b*cos(d*x
+ c)^3 + 11*b*cos(d*x + c))*sin(d*x + c))/(b^2*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**8/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\cos \left (d x + c\right )^{8}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]

[In]

integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/32*(32*b^2*d*integrate(-16*(4*(a*b^2 + b^3)*cos(6*d*x + 6*c)^2 + 2*(8*a^3 + 29*a^2*b - 20*a*b^2 + 3*b^3)*co
s(4*d*x + 4*c)^2 + 4*(a*b^2 + b^3)*cos(2*d*x + 2*c)^2 + 4*(a*b^2 + b^3)*sin(6*d*x + 6*c)^2 + 2*(8*a^3 + 29*a^2
*b - 20*a*b^2 + 3*b^3)*sin(4*d*x + 4*c)^2 + 2*(10*a^2*b + 13*a*b^2 - 5*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c)
+ 4*(a*b^2 + b^3)*sin(2*d*x + 2*c)^2 - ((a*b^2 + b^3)*cos(6*d*x + 6*c) + (a^2*b + 4*a*b^2 - b^3)*cos(4*d*x + 4
*c) + (a*b^2 + b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - (a*b^2 + b^3 - 2*(10*a^2*b + 13*a*b^2 - 5*b^3)*cos(4*
d*x + 4*c) - 8*(a*b^2 + b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - (a^2*b + 4*a*b^2 - b^3 - 2*(10*a^2*b + 13*a*
b^2 - 5*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*b^2 + b^3)*cos(2*d*x + 2*c) - ((a*b^2 + b^3)*sin(6*d*x +
6*c) + (a^2*b + 4*a*b^2 - b^3)*sin(4*d*x + 4*c) + (a*b^2 + b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((10*a^
2*b + 13*a*b^2 - 5*b^3)*sin(4*d*x + 4*c) + 4*(a*b^2 + b^3)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))/(b^4*cos(8*d*x
+ 8*c)^2 + 16*b^4*cos(6*d*x + 6*c)^2 + 16*b^4*cos(2*d*x + 2*c)^2 + b^4*sin(8*d*x + 8*c)^2 + 16*b^4*sin(6*d*x +
6*c)^2 + 16*b^4*sin(2*d*x + 2*c)^2 - 8*b^4*cos(2*d*x + 2*c) + b^4 + 4*(64*a^2*b^2 - 48*a*b^3 + 9*b^4)*cos(4*d
*x + 4*c)^2 + 4*(64*a^2*b^2 - 48*a*b^3 + 9*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c)*sin
(2*d*x + 2*c) - 2*(4*b^4*cos(6*d*x + 6*c) + 4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*b^3 - 3*b^4)*cos(4*d*x + 4*c
))*cos(8*d*x + 8*c) + 8*(4*b^4*cos(2*d*x + 2*c) - b^4 + 2*(8*a*b^3 - 3*b^4)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c)
- 4*(8*a*b^3 - 3*b^4 - 4*(8*a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*b^4*sin(6*d*x + 6*c) + 2
*b^4*sin(2*d*x + 2*c) + (8*a*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*b^4*sin(2*d*x + 2*c) + (8
*a*b^3 - 3*b^4)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + 4*(8*a + 35*b)*d*x + b*sin(4*d*x + 4*c) + 24*b*sin(2
*d*x + 2*c))/(b^2*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 836 vs. \(2 (144) = 288\).

Time = 0.95 (sec) , antiderivative size = 836, normalized size of antiderivative = 4.49 \[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {4 \, {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{4} + 12 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b - 34 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 12 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4} - 12 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} + 12 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b + 28 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} + 4 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} + \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}} + \frac {4 \, {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{4} + 12 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b - 34 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} - 12 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4} + 12 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} - 12 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 28 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - 4 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b^{2} - \sqrt {a^{2} b^{4} - {\left (a b^{2} - b^{3}\right )} a b^{2}}}{a b^{2} - b^{3}}}}\right )\right )} {\left | -a + b \right |}}{3 \, a^{5} b^{2} - 12 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}} - \frac {{\left (d x + c\right )} {\left (8 \, a + 35 \, b\right )}}{b^{2}} - \frac {11 \, \tan \left (d x + c\right )^{3} + 13 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} b}}{8 \, d} \]

[In]

integrate(cos(d*x+c)^8/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

1/8*(4*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4 + 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b - 34*sqrt(a^2
- a*b + sqrt(a*b)*(a - b))*a^2*b^2 - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b + sqrt(a*b
)*(a - b))*b^4 - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 + 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))
*sqrt(a*b)*a^2*b + 28*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 4*sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*sqrt(a*b)*b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 + sqrt(a^2*b^4 - (a*b^2 - b
^3)*a*b^2))/(a*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6) + 4*(3*sqrt
(a^2 - a*b - sqrt(a*b)*(a - b))*a^4 + 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b - 34*sqrt(a^2 - a*b - sqrt(
a*b)*(a - b))*a^2*b^2 - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^3 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^4
+ 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2
*b - 28*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*
b^3)*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a*b^2 - sqrt(a^2*b^4 - (a*b^2 - b^3)*a*b^2))/(a
*b^2 - b^3))))*abs(-a + b)/(3*a^5*b^2 - 12*a^4*b^3 + 14*a^3*b^4 - 4*a^2*b^5 - a*b^6) - (d*x + c)*(8*a + 35*b)/
b^2 - (11*tan(d*x + c)^3 + 13*tan(d*x + c))/((tan(d*x + c)^2 + 1)^2*b))/d

Mupad [B] (verification not implemented)

Time = 16.88 (sec) , antiderivative size = 8773, normalized size of antiderivative = 47.17 \[ \int \frac {\cos ^8(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^8/(a - b*sin(c + d*x)^4),x)

[Out]

- (atan(((((373728*a^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296*a^4*b^7 - 240464*a^5*b^6 + 29040
*a^6*b^5 + 27648*a^7*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2*b^11 - 129024*a^3*b^10 + 69632*a
^4*b^9 + 14336*a^5*b^8 - 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2
) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a
^3*b^8))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*(-(7*a^3*(a^3*b^9)
^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a
^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) + (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b
^8 + 53616*a^4*b^7 - 12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(
a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)
^(1/2))/(16*a^3*b^8))^(1/2))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*
b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) + (tan(c + d*x)*(336*
a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 9223*a
^6*b^3 + 1721*a^7*b^2))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21
*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*1i - (((373728*a
^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296*a^4*b^7 - 240464*a^5*b^6 + 29040*a^6*b^5 + 27648*a^7
*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2*b^11 - 129024*a^3*b^10 + 69632*a^4*b^9 + 14336*a^5*b
^8 - 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a
^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*(1228
8*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^
9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2)
)/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7 -
12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a
^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8)
)^(1/2))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a
*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(336*a^8*b - 1497*a*b^8 +
96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 9223*a^6*b^3 + 1721*a^7*b^
2))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 +
21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*1i)/((11696*a*b^8 + 1247*a^8*b - 344
*a^9 - 1505*b^9 - 39388*a^2*b^7 + 74648*a^3*b^6 - 86086*a^4*b^5 + 60200*a^5*b^4 - 22876*a^6*b^3 + 2408*a^7*b^2
)/(32*b^5) + (((373728*a^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296*a^4*b^7 - 240464*a^5*b^6 + 2
9040*a^6*b^5 + 27648*a^7*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2*b^11 - 129024*a^3*b^10 + 696
32*a^4*b^9 + 14336*a^5*b^8 - 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^
(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(
16*a^3*b^8))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*(-(7*a^3*(a^3*
b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) +
35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) + (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a
^3*b^8 + 53616*a^4*b^7 - 12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b
^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*
b^9)^(1/2))/(16*a^3*b^8))^(1/2))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*
a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) + (tan(c + d*x)*(
336*a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 92
23*a^6*b^3 + 1721*a^7*b^2))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6
+ 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) + (((373728*
a^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296*a^4*b^7 - 240464*a^5*b^6 + 29040*a^6*b^5 + 27648*a^
7*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2*b^11 - 129024*a^3*b^10 + 69632*a^4*b^9 + 14336*a^5*
b^8 - 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*
a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*(122
88*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b
^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2
))/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7
- 12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*
a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8
))^(1/2))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*
a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(336*a^8*b - 1497*a*b^8
+ 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 9223*a^6*b^3 + 1721*a^7*b
^2))/(16*b^4))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4
+ 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)))*(-(7*a^3*(a^3*b^9)^(1/2) + b^3*(a
^3*b^9)^(1/2) + 7*a^2*b^7 + 35*a^3*b^6 + 21*a^4*b^5 + a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^
(1/2))/(16*a^3*b^8))^(1/2)*2i)/d - (atan(((((373728*a^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296
*a^4*b^7 - 240464*a^5*b^6 + 29040*a^6*b^5 + 27648*a^7*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2
*b^11 - 129024*a^3*b^10 + 69632*a^4*b^9 + 14336*a^5*b^8 - 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*((7*a^3*(a^3
*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) +
35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^
8))/(16*b^4))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 +
21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) + (tan(c + d*x)*(256*a*b^10 + 256*b^1
1 - 70832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7 - 12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(
(7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b
^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a
^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8)
)^(1/2) + (tan(c + d*x)*(336*a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^
4*b^5 + 3077*a^5*b^4 - 9223*a^6*b^3 + 1721*a^7*b^2))/(16*b^4))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) -
7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*
b^8))^(1/2)*1i - (((373728*a^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296*a^4*b^7 - 240464*a^5*b^6
+ 29040*a^6*b^5 + 27648*a^7*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2*b^11 - 129024*a^3*b^10 +
69632*a^4*b^9 + 14336*a^5*b^8 - 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^
9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2)
)/(16*a^3*b^8))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*((7*a^3*(a^
3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2)
+ 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136
*a^3*b^8 + 53616*a^4*b^7 - 12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*((7*a^3*(a^3*b^9)^(1/2) +
b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3
*b^9)^(1/2))/(16*a^3*b^8))^(1/2))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*
a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(
336*a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 92
23*a^6*b^3 + 1721*a^7*b^2))/(16*b^4))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 -
21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*1i)/((11696*a
*b^8 + 1247*a^8*b - 344*a^9 - 1505*b^9 - 39388*a^2*b^7 + 74648*a^3*b^6 - 86086*a^4*b^5 + 60200*a^5*b^4 - 22876
*a^6*b^3 + 2408*a^7*b^2)/(32*b^5) + (((373728*a^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296*a^4*b
^7 - 240464*a^5*b^6 + 29040*a^6*b^5 + 27648*a^7*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2*b^11
- 129024*a^3*b^10 + 69632*a^4*b^9 + 14336*a^5*b^8 - 12288*a^6*b^7)/(64*b^5) - (tan(c + d*x)*((7*a^3*(a^3*b^9)^
(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^
2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(1
6*b^4))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b
^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) + (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70
832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7 - 12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*((7*a^3
*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1
/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7
- 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2
) + (tan(c + d*x)*(336*a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5
+ 3077*a^5*b^4 - 9223*a^6*b^3 + 1721*a^7*b^2))/(16*b^4))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2
*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^
(1/2) + (((373728*a^3*b^8 - 256*b^11 - 208208*a^2*b^9 - 17552*a*b^10 + 35296*a^4*b^7 - 240464*a^5*b^6 + 29040*
a^6*b^5 + 27648*a^7*b^4 + 768*a^8*b^3)/(64*b^5) - (((4096*a*b^12 + 53248*a^2*b^11 - 129024*a^3*b^10 + 69632*a^
4*b^9 + 14336*a^5*b^8 - 12288*a^6*b^7)/(64*b^5) + (tan(c + d*x)*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2)
- 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3
*b^8))^(1/2)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(16*b^4))*((7*a^3*(a^3*b^9)^(1
/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*
b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b^8
+ 53616*a^4*b^7 - 12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*
b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/
2))/(16*a^3*b^8))^(1/2))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 -
a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2) - (tan(c + d*x)*(336*a^8*b
- 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 9223*a^6*b^
3 + 1721*a^7*b^2))/(16*b^4))*((7*a^3*(a^3*b^9)^(1/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b
^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)))*((7*a^3*(a^3*b^9)^(1
/2) + b^3*(a^3*b^9)^(1/2) - 7*a^2*b^7 - 35*a^3*b^6 - 21*a^4*b^5 - a^5*b^4 + 21*a*b^2*(a^3*b^9)^(1/2) + 35*a^2*
b*(a^3*b^9)^(1/2))/(16*a^3*b^8))^(1/2)*2i)/d - ((13*tan(c + d*x))/(8*b) + (11*tan(c + d*x)^3)/(8*b))/(d*(2*tan
(c + d*x)^2 + tan(c + d*x)^4 + 1)) - (atan((((a*8i + b*35i)*(((((11679*a^3*b^8)/2 - 4*b^11 - (13013*a^2*b^9)/4
- (1097*a*b^10)/4 + (1103*a^4*b^7)/2 - (15029*a^5*b^6)/4 + (1815*a^6*b^5)/4 + 432*a^7*b^4 + 12*a^8*b^3)/b^5 -
(((((64*a*b^12 + 832*a^2*b^11 - 2016*a^3*b^10 + 1088*a^4*b^9 + 224*a^5*b^8 - 192*a^6*b^7)/b^5 - (tan(c + d*x)
*(a*8i + b*35i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(256*b^6))*(a*8i + b*35i))/
(16*b^2) + (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7 - 12432*a^5*b^
6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(a*8i + b*35i))/(16*b^2))*(a*8i + b*35i))/(16*b^2) + (tan(c + d*x
)*(336*a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 -
9223*a^6*b^3 + 1721*a^7*b^2))/(16*b^4))*1i)/(16*b^2) - ((a*8i + b*35i)*(((((11679*a^3*b^8)/2 - 4*b^11 - (1301
3*a^2*b^9)/4 - (1097*a*b^10)/4 + (1103*a^4*b^7)/2 - (15029*a^5*b^6)/4 + (1815*a^6*b^5)/4 + 432*a^7*b^4 + 12*a^
8*b^3)/b^5 - (((((64*a*b^12 + 832*a^2*b^11 - 2016*a^3*b^10 + 1088*a^4*b^9 + 224*a^5*b^8 - 192*a^6*b^7)/b^5 + (
tan(c + d*x)*(a*8i + b*35i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(256*b^6))*(a*8
i + b*35i))/(16*b^2) - (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7 -
12432*a^5*b^6 - 29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(a*8i + b*35i))/(16*b^2))*(a*8i + b*35i))/(16*b^2) -
(tan(c + d*x)*(336*a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 30
77*a^5*b^4 - 9223*a^6*b^3 + 1721*a^7*b^2))/(16*b^4))*1i)/(16*b^2))/(((731*a*b^8)/2 + (1247*a^8*b)/32 - (43*a^9
)/4 - (1505*b^9)/32 - (9847*a^2*b^7)/8 + (9331*a^3*b^6)/4 - (43043*a^4*b^5)/16 + (7525*a^5*b^4)/4 - (5719*a^6*
b^3)/8 + (301*a^7*b^2)/4)/b^5 + ((a*8i + b*35i)*(((((11679*a^3*b^8)/2 - 4*b^11 - (13013*a^2*b^9)/4 - (1097*a*b
^10)/4 + (1103*a^4*b^7)/2 - (15029*a^5*b^6)/4 + (1815*a^6*b^5)/4 + 432*a^7*b^4 + 12*a^8*b^3)/b^5 - (((((64*a*b
^12 + 832*a^2*b^11 - 2016*a^3*b^10 + 1088*a^4*b^9 + 224*a^5*b^8 - 192*a^6*b^7)/b^5 - (tan(c + d*x)*(a*8i + b*3
5i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(256*b^6))*(a*8i + b*35i))/(16*b^2) + (
tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7 - 12432*a^5*b^6 - 29696*a^
6*b^5 - 2304*a^7*b^4))/(16*b^4))*(a*8i + b*35i))/(16*b^2))*(a*8i + b*35i))/(16*b^2) + (tan(c + d*x)*(336*a^8*b
- 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 9223*a^6*b^
3 + 1721*a^7*b^2))/(16*b^4)))/(16*b^2) + ((a*8i + b*35i)*(((((11679*a^3*b^8)/2 - 4*b^11 - (13013*a^2*b^9)/4 -
(1097*a*b^10)/4 + (1103*a^4*b^7)/2 - (15029*a^5*b^6)/4 + (1815*a^6*b^5)/4 + 432*a^7*b^4 + 12*a^8*b^3)/b^5 - ((
(((64*a*b^12 + 832*a^2*b^11 - 2016*a^3*b^10 + 1088*a^4*b^9 + 224*a^5*b^8 - 192*a^6*b^7)/b^5 + (tan(c + d*x)*(a
*8i + b*35i)*(12288*a^2*b^11 - 12288*a^3*b^10 - 12288*a^4*b^9 + 12288*a^5*b^8))/(256*b^6))*(a*8i + b*35i))/(16
*b^2) - (tan(c + d*x)*(256*a*b^10 + 256*b^11 - 70832*a^2*b^9 + 61136*a^3*b^8 + 53616*a^4*b^7 - 12432*a^5*b^6 -
29696*a^6*b^5 - 2304*a^7*b^4))/(16*b^4))*(a*8i + b*35i))/(16*b^2))*(a*8i + b*35i))/(16*b^2) - (tan(c + d*x)*(
336*a^8*b - 1497*a*b^8 + 96*a^9 - 1257*b^9 + 21499*a^2*b^7 - 41861*a^3*b^6 + 27109*a^4*b^5 + 3077*a^5*b^4 - 92
23*a^6*b^3 + 1721*a^7*b^2))/(16*b^4)))/(16*b^2)))*(a*8i + b*35i)*1i)/(8*b^2*d)