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

Optimal. Leaf size=204 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}-\sqrt{b}\right )^{7/2}}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}+\sqrt{b}\right )^{7/2}}+\frac{\left (a^2-3 a b+6 b^2\right ) \tan (c+d x)}{d (a-b)^3}+\frac{\tan ^5(c+d x)}{5 d (a-b)}+\frac{2 (a-2 b) \tan ^3(c+d x)}{3 d (a-b)^2} \]

[Out]

-(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) + (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)

________________________________________________________________________________________

Rubi [A]  time = 0.377966, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3224, 1170, 1166, 205} \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}-\sqrt{b}\right )^{7/2}}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt{a}+\sqrt{b}\right )^{7/2}}+\frac{\left (a^2-3 a b+6 b^2\right ) \tan (c+d x)}{d (a-b)^3}+\frac{\tan ^5(c+d x)}{5 d (a-b)}+\frac{2 (a-2 b) \tan ^3(c+d x)}{3 d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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) + (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 3224

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]

Rule 1170

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 1166

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 205

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

Rubi steps

\begin{align*} \int \frac{\sec ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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} \tan ^{-1}\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} \tan ^{-1}\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]  time = 1.34661, size = 253, normalized size = 1.24 \[ \frac{2 \left (8 a^2-21 a b+73 b^2\right ) \tan (c+d x)+\frac{15 b^{3/2} \left (\sqrt{a}-\sqrt{b}\right )^3 \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{15 b^{3/2} \left (\sqrt{a}+\sqrt{b}\right )^3 \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}+6 (a-b)^2 \tan (c+d x) \sec ^4(c+d x)+4 (2 a-7 b) (a-b) \tan (c+d x) \sec ^2(c+d x)}{30 d (a-b)^3} \]

Antiderivative was successfully verified.

[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 [B]  time = 0.148, size = 839, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Fricas [B]  time = 14.1733, size = 13045, normalized size = 63.95 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*(15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 - (a^8 - 7*a^7*b +
21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 15
19*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3
+ 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10
- 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a
^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2))*cos(d*x + c)^5*log(7/4*a^3*b^5 + 35/4*a^2*b^6 + 21/4*a*b^7 + 1/
4*b^8 - 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*a*b^7 + b^8)*cos(d*x + c)^2 + 1/2*(4*(a^11 - 6*a^10*b + 14*a^9*b^2 -
14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*a^4*b^7 - a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9
+ 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13
*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b
^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))*cos(d*x + c)*sin(d*x + c) + (7*a^6*b^3 + 77*a^5*b^4 + 238*a^
4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5
+ 7*b^6 - (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49
*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b +
91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 20
02*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*
a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2 + 21*a^7
*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^8*b^2 + 2
1*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a^5*b^8 +
1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^
3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^1
0 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))) - 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt(-(
a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 - (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 +
7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^
12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3
432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^
14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2))*cos(
d*x + c)^5*log(7/4*a^3*b^5 + 35/4*a^2*b^6 + 21/4*a*b^7 + 1/4*b^8 - 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*a*b^7 + b^
8)*cos(d*x + c)^2 - 1/2*(4*(a^11 - 6*a^10*b + 14*a^9*b^2 - 14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*a^4*b^7 -
a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/
((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^
7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))*c
os(d*x + c)*sin(d*x + c) + (7*a^6*b^3 + 77*a^5*b^4 + 238*a^4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)*d*cos(d*x
+ c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 - (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3
+ 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^
10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a
^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*
b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*
a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3
*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 +
7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*
b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 -
3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*
b^14)*d^4))) + 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 + (a^8 - 7*
a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*
b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a
^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a
^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^
3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2))*cos(d*x + c)^5*log(-7/4*a^3*b^5 - 35/4*a^2*b^6 - 21/4*a
*b^7 - 1/4*b^8 + 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*a*b^7 + b^8)*cos(d*x + c)^2 + 1/2*(4*(a^11 - 6*a^10*b + 14*a
^9*b^2 - 14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*a^4*b^7 - a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519
*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 +
1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 -
364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))*cos(d*x + c)*sin(d*x + c) - (7*a^6*b^3 + 77*a^5*b^4
+ 238*a^4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 +
35*a*b^5 + 7*b^6 + (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2
*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*
a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9
*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^
7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2
+ 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^
8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a
^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 36
4*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 100
1*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4))) - 15*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*
d*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 + (a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*
a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11
+ 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^1
1*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13
+ a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d
^2))*cos(d*x + c)^5*log(-7/4*a^3*b^5 - 35/4*a^2*b^6 - 21/4*a*b^7 - 1/4*b^8 + 1/4*(7*a^3*b^5 + 35*a^2*b^6 + 21*
a*b^7 + b^8)*cos(d*x + c)^2 - 1/2*(4*(a^11 - 6*a^10*b + 14*a^9*b^2 - 14*a^8*b^3 + 14*a^6*b^5 - 14*a^5*b^6 + 6*
a^4*b^7 - a^3*b^8)*d^3*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^1
2 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 34
32*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^1
4)*d^4))*cos(d*x + c)*sin(d*x + c) - (7*a^6*b^3 + 77*a^5*b^4 + 238*a^4*b^5 + 162*a^3*b^6 + 27*a^2*b^7 + a*b^8)
*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^3*b^3 + 21*a^2*b^4 + 35*a*b^5 + 7*b^6 + (a^8 - 7*a^7*b + 21*a^6*b^2 - 3
5*a^5*b^3 + 35*a^4*b^4 - 21*a^3*b^5 + 7*a^2*b^6 - a*b^7)*d^2*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1
484*a^3*b^10 + 511*a^2*b^11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^
4 - 2002*a^12*b^5 + 3003*a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11
+ 91*a^5*b^12 - 14*a^4*b^13 + a^3*b^14)*d^4)))/((a^8 - 7*a^7*b + 21*a^6*b^2 - 35*a^5*b^3 + 35*a^4*b^4 - 21*a^
3*b^5 + 7*a^2*b^6 - a*b^7)*d^2)) - 1/4*(2*(a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*a^4*b
^6 + 7*a^3*b^7 - a^2*b^8)*d^2*cos(d*x + c)^2 - (a^9*b - 7*a^8*b^2 + 21*a^7*b^3 - 35*a^6*b^4 + 35*a^5*b^5 - 21*
a^4*b^6 + 7*a^3*b^7 - a^2*b^8)*d^2)*sqrt((49*a^6*b^7 + 490*a^5*b^8 + 1519*a^4*b^9 + 1484*a^3*b^10 + 511*a^2*b^
11 + 42*a*b^12 + b^13)/((a^17 - 14*a^16*b + 91*a^15*b^2 - 364*a^14*b^3 + 1001*a^13*b^4 - 2002*a^12*b^5 + 3003*
a^11*b^6 - 3432*a^10*b^7 + 3003*a^9*b^8 - 2002*a^8*b^9 + 1001*a^7*b^10 - 364*a^6*b^11 + 91*a^5*b^12 - 14*a^4*b
^13 + a^3*b^14)*d^4))) + 8*((8*a^2 - 21*a*b + 73*b^2)*cos(d*x + c)^4 + 2*(2*a^2 - 9*a*b + 7*b^2)*cos(d*x + c)^
2 + 3*a^2 - 6*a*b + 3*b^2)*sin(d*x + c))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*cos(d*x + c)^5)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError