\(\int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [418]

Optimal result

Integrand size = 24, antiderivative size = 204 \[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b^{3/2} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{7/2} d}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{7/2} d}+\frac {\left (a^2-3 a b+6 b^2\right ) \tan (c+d x)}{(a-b)^3 d}+\frac {2 (a-2 b) \tan ^3(c+d x)}{3 (a-b)^2 d}+\frac {\tan ^5(c+d x)}{5 (a-b) d} \]

[Out]

-1/2*b^(3/2)*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(3/4)/d/(a^(1/2)-b^(1/2))^(7/2)+1/2*b^(3/2)*
arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/a^(3/4)/d/(a^(1/2)+b^(1/2))^(7/2)+(a^2-3*a*b+6*b^2)*tan(d*x
+c)/(a-b)^3/d+2/3*(a-2*b)*tan(d*x+c)^3/(a-b)^2/d+1/5*tan(d*x+c)^5/(a-b)/d

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3303, 1184, 1180, 211} \[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {b^{3/2} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{7/2}}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{7/2}}+\frac {\left (a^2-3 a b+6 b^2\right ) \tan (c+d x)}{d (a-b)^3}+\frac {\tan ^5(c+d x)}{5 d (a-b)}+\frac {2 (a-2 b) \tan ^3(c+d x)}{3 d (a-b)^2} \]

[In]

Int[Sec[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/2*(b^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^(7/2)*d) +
(b^(3/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(7/2)*d) + ((a
^2 - 3*a*b + 6*b^2)*Tan[c + d*x])/((a - b)^3*d) + (2*(a - 2*b)*Tan[c + d*x]^3)/(3*(a - b)^2*d) + Tan[c + d*x]^
5/(5*(a - b)*d)

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 {\left (1+x^2\right )^4}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^2-3 a b+6 b^2}{(a-b)^3}+\frac {2 (a-2 b) x^2}{(a-b)^2}+\frac {x^4}{a-b}-\frac {b^2 (3 a+b)+4 b^2 (a+b) x^2}{(a-b)^3 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\left (a^2-3 a b+6 b^2\right ) \tan (c+d x)}{(a-b)^3 d}+\frac {2 (a-2 b) \tan ^3(c+d x)}{3 (a-b)^2 d}+\frac {\tan ^5(c+d x)}{5 (a-b) d}-\frac {\text {Subst}\left (\int \frac {b^2 (3 a+b)+4 b^2 (a+b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{(a-b)^3 d} \\ & = \frac {\left (a^2-3 a b+6 b^2\right ) \tan (c+d x)}{(a-b)^3 d}+\frac {2 (a-2 b) \tan ^3(c+d x)}{3 (a-b)^2 d}+\frac {\tan ^5(c+d x)}{5 (a-b) d}+\frac {\left (\left (\sqrt {a}-\sqrt {b}\right )^4 b^{3/2}\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} (a-b)^3 d}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right )^4 b^{3/2}\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} (a-b)^3 d} \\ & = -\frac {b^{3/2} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{7/2} d}+\frac {b^{3/2} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{7/2} d}+\frac {\left (a^2-3 a b+6 b^2\right ) \tan (c+d x)}{(a-b)^3 d}+\frac {2 (a-2 b) \tan ^3(c+d x)}{3 (a-b)^2 d}+\frac {\tan ^5(c+d x)}{5 (a-b) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.77 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.24 \[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {15 \left (\sqrt {a}-\sqrt {b}\right )^3 b^{3/2} \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {15 \left (\sqrt {a}+\sqrt {b}\right )^3 b^{3/2} \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {-a+\sqrt {a} \sqrt {b}}}+2 \left (8 a^2-21 a b+73 b^2\right ) \tan (c+d x)+4 (2 a-7 b) (a-b) \sec ^2(c+d x) \tan (c+d x)+6 (a-b)^2 \sec ^4(c+d x) \tan (c+d x)}{30 (a-b)^3 d} \]

[In]

Integrate[Sec[c + d*x]^6/(a - b*Sin[c + d*x]^4),x]

[Out]

((15*(Sqrt[a] - Sqrt[b])^3*b^(3/2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt
[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (15*(Sqrt[a] + Sqrt[b])^3*b^(3/2)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/
Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 2*(8*a^2 - 21*a*b + 73*b^2)*Tan[c + d*x] +
4*(2*a - 7*b)*(a - b)*Sec[c + d*x]^2*Tan[c + d*x] + 6*(a - b)^2*Sec[c + d*x]^4*Tan[c + d*x])/(30*(a - b)^3*d)

Maple [A] (verified)

Time = 3.42 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.52

method	result	size
derivativedivides	\(\frac {\frac {\frac {a^{2} \left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {2 a b \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {b^{2} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {2 a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 a b \left (\tan ^{3}\left (d x +c \right )\right )+\frac {4 b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right ) a^{2}-3 a b \tan \left (d x +c \right )+6 \tan \left (d x +c \right ) b^{2}}{\left (a -b \right )^{3}}-\frac {b^{2} \left (\frac {\left (4 a \sqrt {a b}+4 \sqrt {a b}\, b +a^{2}+6 a b +b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}+4 \sqrt {a b}\, b -a^{2}-6 a b -b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{\left (a -b \right )^{2}}}{d}\)	\(311\)
default	\(\frac {\frac {\frac {a^{2} \left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {2 a b \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {b^{2} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {2 a^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 a b \left (\tan ^{3}\left (d x +c \right )\right )+\frac {4 b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right ) a^{2}-3 a b \tan \left (d x +c \right )+6 \tan \left (d x +c \right ) b^{2}}{\left (a -b \right )^{3}}-\frac {b^{2} \left (\frac {\left (4 a \sqrt {a b}+4 \sqrt {a b}\, b +a^{2}+6 a b +b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (4 a \sqrt {a b}+4 \sqrt {a b}\, b -a^{2}-6 a b -b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{\left (a -b \right )^{2}}}{d}\)	\(311\)
risch	\(\text {Expression too large to display}\)	\(1365\)

[In]

int(sec(d*x+c)^6/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a-b)^3*(1/5*a^2*tan(d*x+c)^5-2/5*a*b*tan(d*x+c)^5+1/5*b^2*tan(d*x+c)^5+2/3*a^2*tan(d*x+c)^3-2*a*b*tan(
d*x+c)^3+4/3*b^2*tan(d*x+c)^3+tan(d*x+c)*a^2-3*a*b*tan(d*x+c)+6*tan(d*x+c)*b^2)-b^2/(a-b)^2*(1/2*(4*a*(a*b)^(1
/2)+4*(a*b)^(1/2)*b+a^2+6*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((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))+1/2*(4*a*(a*b)^(1/2)+4*(a*b)^(1/2)*b-a^2-6*a*b-b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2
)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5587 vs. \(2 (160) = 320\).

Time = 1.59 (sec) , antiderivative size = 5587, normalized size of antiderivative = 27.39 \[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**6/(a-b*sin(d*x+c)**4),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sec \left (d x + c\right )^{6}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]

[In]

integrate(sec(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

1/15*(300*(a*b - 5*b^2)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) - 10*(48*b^2*sin(6*d*x + 6*c) + 3*(a*b + 3*b^2)*sin(
8*d*x + 8*c) + 2*(8*a^2 - 21*a*b + 49*b^2)*sin(4*d*x + 4*c) + 8*(a^2 - 3*a*b + 8*b^2)*sin(2*d*x + 2*c))*cos(10
*d*x + 10*c) + 50*(6*(a*b - 5*b^2)*sin(6*d*x + 6*c) - 16*(a^2 - 3*a*b + 5*b^2)*sin(4*d*x + 4*c) - (8*a^2 - 27*
a*b + 55*b^2)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) - 200*((8*a^2 - 21*a*b + 25*b^2)*sin(4*d*x + 4*c) + 4*(a^2 -
3*a*b + 5*b^2)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 15*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(10*d*x + 10*c)^2
+ 25*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(8*d*x + 8*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(6*d*x +
6*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c)^2 + 25*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(
2*d*x + 2*c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(10*d*x + 10*c)^2 + 25*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d
*sin(8*d*x + 8*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(6*d*x + 6*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2
- b^3)*d*sin(4*d*x + 4*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*(a^
3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(2*d*x + 2*c)^2 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a
^3 - 3*a^2*b + 3*a*b^2 - b^3)*d + 2*(5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(8*d*x + 8*c) + 10*(a^3 - 3*a^2*b
+ 3*a*b^2 - b^3)*d*cos(6*d*x + 6*c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c) + 5*(a^3 - 3*a^2*b
+ 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*cos(10*d*x + 10*c) + 10*(10*(a^3 - 3
*a^2*b + 3*a*b^2 - b^3)*d*cos(6*d*x + 6*c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c) + 5*(a^3 -
3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*cos(8*d*x + 8*c) + 20*(10*(a^
3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c) + 5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3
- 3*a^2*b + 3*a*b^2 - b^3)*d)*cos(6*d*x + 6*c) + 20*(5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a
^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*cos(4*d*x + 4*c) + 10*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(8*d*x + 8*c) + 2
*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(6*d*x + 6*c) + 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c) + (
a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*
d*sin(6*d*x + 6*c) + 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*
sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c) + (a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(4*(4*(a*b^3 + 3*b^4)*cos(6*d*x + 6*c)^2 - 4
*(56*a^2*b^2 + 19*a*b^3 - 15*b^4)*cos(4*d*x + 4*c)^2 + 4*(a*b^3 + 3*b^4)*cos(2*d*x + 2*c)^2 + 4*(a*b^3 + 3*b^4
)*sin(6*d*x + 6*c)^2 - 4*(56*a^2*b^2 + 19*a*b^3 - 15*b^4)*sin(4*d*x + 4*c)^2 + 2*(8*a^2*b^2 - 7*a*b^3 - 29*b^4
)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(a*b^3 + 3*b^4)*sin(2*d*x + 2*c)^2 - ((a*b^3 + 3*b^4)*cos(6*d*x + 6*c)
- 2*(7*a*b^3 + 5*b^4)*cos(4*d*x + 4*c) + (a*b^3 + 3*b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - (a*b^3 + 3*b^4
- 2*(8*a^2*b^2 - 7*a*b^3 - 29*b^4)*cos(4*d*x + 4*c) - 8*(a*b^3 + 3*b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2
*(7*a*b^3 + 5*b^4 + (8*a^2*b^2 - 7*a*b^3 - 29*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (a*b^3 + 3*b^4)*cos(2*
d*x + 2*c) - ((a*b^3 + 3*b^4)*sin(6*d*x + 6*c) - 2*(7*a*b^3 + 5*b^4)*sin(4*d*x + 4*c) + (a*b^3 + 3*b^4)*sin(2*
d*x + 2*c))*sin(8*d*x + 8*c) + 2*((8*a^2*b^2 - 7*a*b^3 - 29*b^4)*sin(4*d*x + 4*c) + 4*(a*b^3 + 3*b^4)*sin(2*d*
x + 2*c))*sin(6*d*x + 6*c))/(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*cos(8
*d*x + 8*c)^2 + 16*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*cos(6*d*x + 6*c)^2 + 4*(64*a^5 - 240*a^4*b + 345*a^3*
b^2 - 235*a^2*b^3 + 75*a*b^4 - 9*b^5)*cos(4*d*x + 4*c)^2 + 16*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*cos(2*d*x
+ 2*c)^2 + (a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*sin(8*d*x + 8*c)^2 + 16*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)
*sin(6*d*x + 6*c)^2 + 4*(64*a^5 - 240*a^4*b + 345*a^3*b^2 - 235*a^2*b^3 + 75*a*b^4 - 9*b^5)*sin(4*d*x + 4*c)^2
+ 16*(8*a^4*b - 27*a^3*b^2 + 33*a^2*b^3 - 17*a*b^4 + 3*b^5)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^3*b^2 -
3*a^2*b^3 + 3*a*b^4 - b^5)*sin(2*d*x + 2*c)^2 + 2*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 - 4*(a^3*b^2 - 3*a^2*b
^3 + 3*a*b^4 - b^5)*cos(6*d*x + 6*c) - 2*(8*a^4*b - 27*a^3*b^2 + 33*a^2*b^3 - 17*a*b^4 + 3*b^5)*cos(4*d*x + 4*
c) - 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a^3*b^2 - 3*a^2*b^3 + 3*a
*b^4 - b^5 - 2*(8*a^4*b - 27*a^3*b^2 + 33*a^2*b^3 - 17*a*b^4 + 3*b^5)*cos(4*d*x + 4*c) - 4*(a^3*b^2 - 3*a^2*b^
3 + 3*a*b^4 - b^5)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^4*b - 27*a^3*b^2 + 33*a^2*b^3 - 17*a*b^4 + 3*b^
5 - 4*(8*a^4*b - 27*a^3*b^2 + 33*a^2*b^3 - 17*a*b^4 + 3*b^5)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^3*b^2 -
3*a^2*b^3 + 3*a*b^4 - b^5)*cos(2*d*x + 2*c) - 4*(2*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*sin(6*d*x + 6*c) + (
8*a^4*b - 27*a^3*b^2 + 33*a^2*b^3 - 17*a*b^4 + 3*b^5)*sin(4*d*x + 4*c) + 2*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^
5)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^4*b - 27*a^3*b^2 + 33*a^2*b^3 - 17*a*b^4 + 3*b^5)*sin(4*d*x +
4*c) + 2*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) + 2*(240*b^2*cos(6*d*x
+ 6*c) + 8*a^2 - 21*a*b + 73*b^2 + 15*(a*b + 3*b^2)*cos(8*d*x + 8*c) + 10*(8*a^2 - 21*a*b + 49*b^2)*cos(4*d*x
+ 4*c) + 40*(a^2 - 3*a*b + 8*b^2)*cos(2*d*x + 2*c))*sin(10*d*x + 10*c) + 10*(8*a^2 - 24*a*b + 64*b^2 - 30*(a*
b - 5*b^2)*cos(6*d*x + 6*c) + 80*(a^2 - 3*a*b + 5*b^2)*cos(4*d*x + 4*c) + 5*(8*a^2 - 27*a*b + 55*b^2)*cos(2*d*
x + 2*c))*sin(8*d*x + 8*c) + 20*(8*a^2 - 21*a*b + 49*b^2 + 10*(8*a^2 - 21*a*b + 25*b^2)*cos(4*d*x + 4*c) + 40*
(a^2 - 3*a*b + 5*b^2)*cos(2*d*x + 2*c))*sin(6*d*x + 6*c) + 60*(8*b^2 - 5*(a*b - 5*b^2)*cos(2*d*x + 2*c))*sin(4
*d*x + 4*c) + 30*(a*b + 3*b^2)*sin(2*d*x + 2*c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(10*d*x + 10*c)^2 + 25*
(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(8*d*x + 8*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(6*d*x + 6*c)^
2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c)^2 + 25*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x
+ 2*c)^2 + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(10*d*x + 10*c)^2 + 25*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(8
*d*x + 8*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(6*d*x + 6*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)
*d*sin(4*d*x + 4*c)^2 + 100*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*(a^3 - 3*
a^2*b + 3*a*b^2 - b^3)*d*sin(2*d*x + 2*c)^2 + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3 - 3
*a^2*b + 3*a*b^2 - b^3)*d + 2*(5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(8*d*x + 8*c) + 10*(a^3 - 3*a^2*b + 3*a*
b^2 - b^3)*d*cos(6*d*x + 6*c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c) + 5*(a^3 - 3*a^2*b + 3*a
*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*cos(10*d*x + 10*c) + 10*(10*(a^3 - 3*a^2*b
+ 3*a*b^2 - b^3)*d*cos(6*d*x + 6*c) + 10*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c) + 5*(a^3 - 3*a^2*
b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d)*cos(8*d*x + 8*c) + 20*(10*(a^3 - 3*
a^2*b + 3*a*b^2 - b^3)*d*cos(4*d*x + 4*c) + 5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3 - 3*a^
2*b + 3*a*b^2 - b^3)*d)*cos(6*d*x + 6*c) + 20*(5*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(2*d*x + 2*c) + (a^3 - 3
*a^2*b + 3*a*b^2 - b^3)*d)*cos(4*d*x + 4*c) + 10*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(8*d*x + 8*c) + 2*(a^3
- 3*a^2*b + 3*a*b^2 - b^3)*d*sin(6*d*x + 6*c) + 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c) + (a^3 -
3*a^2*b + 3*a*b^2 - b^3)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(
6*d*x + 6*c) + 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(2*
d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sin(4*d*x + 4*c) + (a^3 - 3*a^2*b + 3*
a*b^2 - b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3105 vs. \(2 (160) = 320\).

Time = 1.00 (sec) , antiderivative size = 3105, normalized size of antiderivative = 15.22 \[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^6/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

1/30*(2*(3*a^4*tan(d*x + c)^5 - 12*a^3*b*tan(d*x + c)^5 + 18*a^2*b^2*tan(d*x + c)^5 - 12*a*b^3*tan(d*x + c)^5
+ 3*b^4*tan(d*x + c)^5 + 10*a^4*tan(d*x + c)^3 - 50*a^3*b*tan(d*x + c)^3 + 90*a^2*b^2*tan(d*x + c)^3 - 70*a*b^
3*tan(d*x + c)^3 + 20*b^4*tan(d*x + c)^3 + 15*a^4*tan(d*x + c) - 75*a^3*b*tan(d*x + c) + 195*a^2*b^2*tan(d*x +
c) - 225*a*b^3*tan(d*x + c) + 90*b^4*tan(d*x + c))/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)
- 15*(4*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(
a*b)*a^2*b^3 - 7*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqr
t(a*b)*b^5)*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)^2*abs(-a + b) + (9*sqrt(a^2 - a*b + sqrt
(a*b)*(a - b))*a^9*b^2 - 69*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^8*b^3 + 216*sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*a^7*b^4 - 352*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^5 + 306*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^
6 - 114*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^7 - 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^8 + 24*sqrt
(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^9 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^10 - sqrt(a^2 - a*b + sqrt
(a*b)*(a - b))*b^11)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*abs(-a + b) - (3*sqrt(a^2 -
a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b - 18*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^13*b^2 - 19*sqr
t(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 508*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b
^4 - 2221*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 5314*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sq
rt(a*b)*a^9*b^6 - 8139*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 + 8328*sqrt(a^2 - a*b + sqrt(a*b)
*(a - b))*sqrt(a*b)*a^7*b^8 - 5631*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^9 + 2322*sqrt(a^2 - a*b
+ sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^10 - 417*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^11 - 68*sqr
t(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^12 + 41*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^1
3 - 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^14 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^1
5)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*
b^3 + 5*a^2*b^4 - a*b^5 + sqrt((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)^2 - (a^6 - 5*a^5*
b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b
^5 + b^6)))/(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6))))/((3*a^14 - 39*a^13*b + 2
30*a^12*b^2 - 814*a^11*b^3 + 1925*a^10*b^4 - 3201*a^9*b^5 + 3828*a^8*b^6 - 3300*a^7*b^7 + 2013*a^6*b^8 - 825*a
^5*b^9 + 198*a^4*b^10 - 14*a^3*b^11 - 5*a^2*b^12 + a*b^13)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b
^4 - b^5)) - 15*(4*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a
- b))*sqrt(a*b)*a^2*b^3 - 7*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(
a - b))*sqrt(a*b)*b^5)*(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)^2*abs(-a + b) + (9*sqrt(a^2 -
a*b - sqrt(a*b)*(a - b))*a^9*b^2 - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^8*b^3 + 216*sqrt(a^2 - a*b - sqrt
(a*b)*(a - b))*a^7*b^4 - 352*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^5 + 306*sqrt(a^2 - a*b - sqrt(a*b)*(a -
b))*a^5*b^6 - 114*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^7 - 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^
8 + 24*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^9 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^10 - sqrt(a^2 -
a*b - sqrt(a*b)*(a - b))*b^11)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*abs(-a + b) - (3*
sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b - 18*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^13*b
^2 - 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 508*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(
a*b)*a^11*b^4 - 2221*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 5314*sqrt(a^2 - a*b - sqrt(a*b)*
(a - b))*sqrt(a*b)*a^9*b^6 - 8139*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 + 8328*sqrt(a^2 - a*b
- sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^8 - 5631*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^9 + 2322*sqr
t(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^10 - 417*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^
11 - 68*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^12 + 41*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a
*b)*a^2*b^13 - 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^14 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sq
rt(a*b)*b^15)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^6 - 5*a^5*b + 10*a^4*b^
2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5 - sqrt((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)^2 - (a
^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*
b^4 - 6*a*b^5 + b^6)))/(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6))))/((3*a^14 - 39
*a^13*b + 230*a^12*b^2 - 814*a^11*b^3 + 1925*a^10*b^4 - 3201*a^9*b^5 + 3828*a^8*b^6 - 3300*a^7*b^7 + 2013*a^6*
b^8 - 825*a^5*b^9 + 198*a^4*b^10 - 14*a^3*b^11 - 5*a^2*b^12 + a*b^13)*abs(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*
b^3 + 5*a*b^4 - b^5)))/d

Mupad [B] (verification not implemented)

Time = 17.54 (sec) , antiderivative size = 6534, normalized size of antiderivative = 32.03 \[ \int \frac {\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^6*(a - b*sin(c + d*x)^4)),x)

[Out]

(atan(((((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^4*b^5 - 44*a^5*b^4 + 12*a^6*b^3))/(5*a*b^4 - 5*a^4*b + a^
5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) - (4*tan(c + d*x)*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6
+ 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^1
0 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*(16*a^9*b - 16*a^2*b^8 +
112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^4 + 336*a^7*b^3 - 112*a^8*b^2))/(5*a*b^4 - 5*a^4*b + a^5 -
b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^3*b^5 + 21*a
^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*
b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) - (4*tan(c + d*x)*(28*a*b^7 + b^8 + 70*a^2*b^
6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(a^3*b^7)^(1/2)
+ b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a
^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2))
)^(1/2)*1i - (((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^4*b^5 - 44*a^5*b^4 + 12*a^6*b^3))/(5*a*b^4 - 5*a^4*
b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) + (4*tan(c + d*x)*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a
^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b
- a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*(16*a^9*b - 16*a^2*
b^8 + 112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^4 + 336*a^7*b^3 - 112*a^8*b^2))/(5*a*b^4 - 5*a^4*b +
a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^3*b^5
+ 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 -
7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) + (4*tan(c + d*x)*(28*a*b^7 + b^8 + 70*
a^2*b^6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(a^3*b^7)^
(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^
2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8
*b^2)))^(1/2)*1i)/((((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^4*b^5 - 44*a^5*b^4 + 12*a^6*b^3))/(5*a*b^4 -
5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) - (4*tan(c + d*x)*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2)
+ 7*a^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7
*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*(16*a^9*b - 1
6*a^2*b^8 + 112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^4 + 336*a^7*b^3 - 112*a^8*b^2))/(5*a*b^4 - 5*a
^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^
3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*
b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) - (4*tan(c + d*x)*(28*a*b^7 + b^8
+ 70*a^2*b^6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(a^3
*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) +
35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 -
21*a^8*b^2)))^(1/2) + (((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^4*b^5 - 44*a^5*b^4 + 12*a^6*b^3))/(5*a*b^4
- 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) + (4*tan(c + d*x)*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1
/2) + 7*a^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16
*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*(16*a^9*b
- 16*a^2*b^8 + 112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^4 + 336*a^7*b^3 - 112*a^8*b^2))/(5*a*b^4 -
5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35
*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a
^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) + (4*tan(c + d*x)*(28*a*b^7 +
b^8 + 70*a^2*b^6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*((7*a^3*(
a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2
) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3
- 21*a^8*b^2)))^(1/2) - (8*(a*b^6 + b^7))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2)))*((7*a^3
*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) + 7*a^2*b^6 + 35*a^3*b^5 + 21*a^4*b^4 + a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1
/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b
^3 - 21*a^8*b^2)))^(1/2)*2i)/d + (atan(((((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^4*b^5 - 44*a^5*b^4 + 12*
a^6*b^3))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) - (4*tan(c + d*x)*(-(7*a^3*(a^3*b^7)^(1/2)
+ b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(
a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)
))^(1/2)*(16*a^9*b - 16*a^2*b^8 + 112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^4 + 336*a^7*b^3 - 112*a^
8*b^2))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1
/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16
*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) - (4*tan(c
+ d*x)*(28*a*b^7 + b^8 + 70*a^2*b^6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10
*a^3*b^2))*(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21
*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35
*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*1i - (((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^4*b^5 - 44*a^5*
b^4 + 12*a^6*b^3))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) + (4*tan(c + d*x)*(-(7*a^3*(a^3*b
^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 3
5*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21
*a^8*b^2)))^(1/2)*(16*a^9*b - 16*a^2*b^8 + 112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^4 + 336*a^7*b^3
- 112*a^8*b^2))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^
3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(
1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) +
(4*tan(c + d*x)*(28*a*b^7 + b^8 + 70*a^2*b^6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2
*b^3 + 10*a^3*b^2))*(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5
*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5
*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*1i)/((((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^4*b^5
- 44*a^5*b^4 + 12*a^6*b^3))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) - (4*tan(c + d*x)*(-(7*a
^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^
(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7
*b^3 - 21*a^8*b^2)))^(1/2)*(16*a^9*b - 16*a^2*b^8 + 112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^4 + 33
6*a^7*b^3 - 112*a^8*b^2))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*(-(7*a^3*(a^3*b^7)^(1/2)
+ b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a
^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2))
)^(1/2) - (4*tan(c + d*x)*(28*a*b^7 + b^8 + 70*a^2*b^6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5 - b^5
- 10*a^2*b^3 + 10*a^3*b^2))*(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*
b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6
+ 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) + (((4*(4*a*b^8 - 4*a^2*b^7 - 24*a^3*b^6 + 56*a^
4*b^5 - 44*a^5*b^4 + 12*a^6*b^3))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2) + (4*tan(c + d*x)*
(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3
*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 +
35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*(16*a^9*b - 16*a^2*b^8 + 112*a^3*b^7 - 336*a^4*b^6 + 560*a^5*b^5 - 560*a^6*b^
4 + 336*a^7*b^3 - 112*a^8*b^2))/(5*a*b^4 - 5*a^4*b + a^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2))*(-(7*a^3*(a^3*b^7)^
(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^
2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8
*b^2)))^(1/2) + (4*tan(c + d*x)*(28*a*b^7 + b^8 + 70*a^2*b^6 + 28*a^3*b^5 + a^4*b^4))/(5*a*b^4 - 5*a^4*b + a^5
- b^5 - 10*a^2*b^3 + 10*a^3*b^2))*(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5 - 2
1*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 - 7*a
^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2) - (8*(a*b^6 + b^7))/(5*a*b^4 - 5*a^4*b + a
^5 - b^5 - 10*a^2*b^3 + 10*a^3*b^2)))*(-(7*a^3*(a^3*b^7)^(1/2) + b^3*(a^3*b^7)^(1/2) - 7*a^2*b^6 - 35*a^3*b^5
- 21*a^4*b^4 - a^5*b^3 + 21*a*b^2*(a^3*b^7)^(1/2) + 35*a^2*b*(a^3*b^7)^(1/2))/(16*(7*a^9*b - a^10 + a^3*b^7 -
7*a^4*b^6 + 21*a^5*b^5 - 35*a^6*b^4 + 35*a^7*b^3 - 21*a^8*b^2)))^(1/2)*2i)/d - (tan(c + d*x)^3*((2*a)/(3*(a -
b)^2) - 4/(3*(a - b))))/d + (tan(c + d*x)*(6/(a - b) - a/(a - b)^2 + (2*a*((2*a)/(a - b)^2 - 4/(a - b)))/(a -
b)))/d + tan(c + d*x)^5/(5*d*(a - b))