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

Optimal. Leaf size=178 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{2 \cot ^3(c+d x)}{3 a d} \]

[Out]

(b^(3/2)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (b^(3
/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*C
ot[c + d*x])/(a^2*d) - (2*Cot[c + d*x]^3)/(3*a*d) - Cot[c + d*x]^5/(5*a*d)

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Rubi [A]  time = 0.202967, antiderivative size = 178, 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 = {3217, 1287, 1130, 205} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} d \sqrt{\sqrt{a}+\sqrt{b}}}-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}-\frac{2 \cot ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[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^(9/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (b^(3
/2)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(9/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - ((a + b)*C
ot[c + d*x])/(a^2*d) - (2*Cot[c + d*x]^3)/(3*a*d) - Cot[c + d*x]^5/(5*a*d)

Rule 3217

Int[sin[(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^(m + 1)/f, Subst[Int[(x^m*(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 1287

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

Rule 1130

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

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{\csc ^6(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{x^6 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^6}+\frac{2}{a x^4}+\frac{a+b}{a^2 x^2}+\frac{b^2 x^2}{a^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{2 \cot ^3(c+d x)}{3 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{2 \cot ^3(c+d x)}{3 a d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{\left (\left (\sqrt{a}+\sqrt{b}\right ) 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 a^2 d}+\frac{\left (\left (1-\frac{\sqrt{a}}{\sqrt{b}}\right ) b^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 a^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^{9/4} \sqrt{\sqrt{a}-\sqrt{b}} d}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{9/4} \sqrt{\sqrt{a}+\sqrt{b}} d}-\frac{(a+b) \cot (c+d x)}{a^2 d}-\frac{2 \cot ^3(c+d x)}{3 a d}-\frac{\cot ^5(c+d x)}{5 a d}\\ \end{align*}

Mathematica [A]  time = 4.56645, size = 174, normalized size = 0.98 \[ -\frac{\frac{15 b^{3/2} \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{15 b^{3/2} \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}-a}}+2 \cot (c+d x) \left (3 a \csc ^4(c+d x)+4 a \csc ^2(c+d x)+8 a+15 b\right )}{30 a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((15*b^(3/2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]]
+ (15*b^(3/2)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b
]] + 2*Cot[c + d*x]*(8*a + 15*b + 4*a*Csc[c + d*x]^2 + 3*a*Csc[c + d*x]^4))/(30*a^2*d)

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Maple [B]  time = 0.137, size = 585, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

1/15*(300*b*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + 10*(3*b*sin(8*d*x + 8*c) - 12*b*sin(6*d*x + 6*c) + 2*(8*a + 9*
b)*sin(4*d*x + 4*c) - 4*(2*a + 3*b)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 50*(6*b*sin(6*d*x + 6*c) - 4*(4*a +
3*b)*sin(4*d*x + 4*c) + (8*a + 9*b)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 200*((8*a + 3*b)*sin(4*d*x + 4*c) -
(4*a + 3*b)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 15*(a^2*d*cos(10*d*x + 10*c)^2 + 25*a^2*d*cos(8*d*x + 8*c)^2
+ 100*a^2*d*cos(6*d*x + 6*c)^2 + 100*a^2*d*cos(4*d*x + 4*c)^2 + 25*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(10*d*x
+ 10*c)^2 + 25*a^2*d*sin(8*d*x + 8*c)^2 + 100*a^2*d*sin(6*d*x + 6*c)^2 + 100*a^2*d*sin(4*d*x + 4*c)^2 - 100*a
^2*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*d*sin(2*d*x + 2*c)^2 - 10*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(
5*a^2*d*cos(8*d*x + 8*c) - 10*a^2*d*cos(6*d*x + 6*c) + 10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) +
a^2*d)*cos(10*d*x + 10*c) - 10*(10*a^2*d*cos(6*d*x + 6*c) - 10*a^2*d*cos(4*d*x + 4*c) + 5*a^2*d*cos(2*d*x + 2*
c) - a^2*d)*cos(8*d*x + 8*c) - 20*(10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(6*d*x + 6
*c) - 20*(5*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 10*(a^2*d*sin(8*d*x + 8*c) - 2*a^2*d*sin(6*d*x
+ 6*c) + 2*a^2*d*sin(4*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) - 50*(2*a^2*d*sin(6*d*x + 6*c)
- 2*a^2*d*sin(4*d*x + 4*c) + a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 100*(2*a^2*d*sin(4*d*x + 4*c) - a^2*d*
sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-4*(4*b^3*cos(6*d*x + 6*c)^2 + 4*b^3*cos(2*d*x + 2*c)^2 + 4*b^3*
sin(6*d*x + 6*c)^2 + 4*b^3*sin(2*d*x + 2*c)^2 - b^3*cos(2*d*x + 2*c) - 4*(8*a*b^2 - 3*b^3)*cos(4*d*x + 4*c)^2
- 4*(8*a*b^2 - 3*b^3)*sin(4*d*x + 4*c)^2 + 2*(8*a*b^2 - 7*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^3*cos(6*
d*x + 6*c) - 2*b^3*cos(4*d*x + 4*c) + b^3*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) + (8*b^3*cos(2*d*x + 2*c) - b^3 +
2*(8*a*b^2 - 7*b^3)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) + 2*(b^3 + (8*a*b^2 - 7*b^3)*cos(2*d*x + 2*c))*cos(4*d
*x + 4*c) - (b^3*sin(6*d*x + 6*c) - 2*b^3*sin(4*d*x + 4*c) + b^3*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*(4*b^3
*sin(2*d*x + 2*c) + (8*a*b^2 - 7*b^3)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c))/(a^2*b^2*cos(8*d*x + 8*c)^2 + 16*a^2
*b^2*cos(6*d*x + 6*c)^2 + 16*a^2*b^2*cos(2*d*x + 2*c)^2 + a^2*b^2*sin(8*d*x + 8*c)^2 + 16*a^2*b^2*sin(6*d*x +
6*c)^2 + 16*a^2*b^2*sin(2*d*x + 2*c)^2 - 8*a^2*b^2*cos(2*d*x + 2*c) + a^2*b^2 + 4*(64*a^4 - 48*a^3*b + 9*a^2*b
^2)*cos(4*d*x + 4*c)^2 + 4*(64*a^4 - 48*a^3*b + 9*a^2*b^2)*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 3*a^2*b^2)*sin(4
*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*a^2*b^2*cos(6*d*x + 6*c) + 4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3
*b - 3*a^2*b^2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(4*a^2*b^2*cos(2*d*x + 2*c) - a^2*b^2 + 2*(8*a^3*b - 3*
a^2*b^2)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a^3*b - 3*a^2*b^2 - 4*(8*a^3*b - 3*a^2*b^2)*cos(2*d*x + 2*c
))*cos(4*d*x + 4*c) - 4*(2*a^2*b^2*sin(6*d*x + 6*c) + 2*a^2*b^2*sin(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4
*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2*a^2*b^2*sin(2*d*x + 2*c) + (8*a^3*b - 3*a^2*b^2)*sin(4*d*x + 4*c))*sin(6
*d*x + 6*c)), x) - 2*(15*b*cos(8*d*x + 8*c) - 60*b*cos(6*d*x + 6*c) + 10*(8*a + 9*b)*cos(4*d*x + 4*c) - 20*(2*
a + 3*b)*cos(2*d*x + 2*c) + 8*a + 15*b)*sin(10*d*x + 10*c) - 10*(30*b*cos(6*d*x + 6*c) - 20*(4*a + 3*b)*cos(4*
d*x + 4*c) + 5*(8*a + 9*b)*cos(2*d*x + 2*c) - 8*a - 12*b)*sin(8*d*x + 8*c) - 20*(10*(8*a + 3*b)*cos(4*d*x + 4*
c) - 10*(4*a + 3*b)*cos(2*d*x + 2*c) + 8*a + 9*b)*sin(6*d*x + 6*c) - 60*(5*b*cos(2*d*x + 2*c) - 2*b)*sin(4*d*x
+ 4*c) - 30*b*sin(2*d*x + 2*c))/(a^2*d*cos(10*d*x + 10*c)^2 + 25*a^2*d*cos(8*d*x + 8*c)^2 + 100*a^2*d*cos(6*d
*x + 6*c)^2 + 100*a^2*d*cos(4*d*x + 4*c)^2 + 25*a^2*d*cos(2*d*x + 2*c)^2 + a^2*d*sin(10*d*x + 10*c)^2 + 25*a^2
*d*sin(8*d*x + 8*c)^2 + 100*a^2*d*sin(6*d*x + 6*c)^2 + 100*a^2*d*sin(4*d*x + 4*c)^2 - 100*a^2*d*sin(4*d*x + 4*
c)*sin(2*d*x + 2*c) + 25*a^2*d*sin(2*d*x + 2*c)^2 - 10*a^2*d*cos(2*d*x + 2*c) + a^2*d - 2*(5*a^2*d*cos(8*d*x +
8*c) - 10*a^2*d*cos(6*d*x + 6*c) + 10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(10*d*x +
10*c) - 10*(10*a^2*d*cos(6*d*x + 6*c) - 10*a^2*d*cos(4*d*x + 4*c) + 5*a^2*d*cos(2*d*x + 2*c) - a^2*d)*cos(8*d
*x + 8*c) - 20*(10*a^2*d*cos(4*d*x + 4*c) - 5*a^2*d*cos(2*d*x + 2*c) + a^2*d)*cos(6*d*x + 6*c) - 20*(5*a^2*d*c
os(2*d*x + 2*c) - a^2*d)*cos(4*d*x + 4*c) - 10*(a^2*d*sin(8*d*x + 8*c) - 2*a^2*d*sin(6*d*x + 6*c) + 2*a^2*d*si
n(4*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) - 50*(2*a^2*d*sin(6*d*x + 6*c) - 2*a^2*d*sin(4*d*x
+ 4*c) + a^2*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 100*(2*a^2*d*sin(4*d*x + 4*c) - a^2*d*sin(2*d*x + 2*c))*s
in(6*d*x + 6*c))

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Fricas [B]  time = 3.64392, size = 3224, normalized size = 18.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(8*(8*a + 15*b)*cos(d*x + c)^5 - 80*(2*a + 3*b)*cos(d*x + c)^3 - 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos
(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b
)*d^2))*log(1/4*b^5*cos(d*x + c)^2 - 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5*b^2)
*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) + 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^8 - a^7*b)*s
qrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11
- 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 15*(a^2*d*cos(d*x + c)^4 - 2*a^2*
d*cos(d*x + c)^2 + a^2*d)*sqrt(-((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 -
a^4*b)*d^2))*log(1/4*b^5*cos(d*x + c)^2 - 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b - a^5
*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) - 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) - (a^8 - a^7
*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^5 - a^4*b)*sqrt(b^7/(
(a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 + b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 15*(a^2*d*cos(d*x + c)^4 - 2
*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^
5 - a^4*b)*d^2))*log(-1/4*b^5*cos(d*x + c)^2 + 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^6*b
- a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) + 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) + (a^8
- a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5 - a^4*b)*sqrt(b
^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) - 15*(a^2*d*cos(d*x + c)^4
- 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sqrt(((a^5 - a^4*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/
((a^5 - a^4*b)*d^2))*log(-1/4*b^5*cos(d*x + c)^2 + 1/4*b^5 - 1/4*(2*(a^6*b - a^5*b^2)*d^2*cos(d*x + c)^2 - (a^
6*b - a^5*b^2)*d^2)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4)) - 1/2*(a^3*b^3*d*cos(d*x + c)*sin(d*x + c) + (
a^8 - a^7*b)*sqrt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^3*cos(d*x + c)*sin(d*x + c))*sqrt(((a^5 - a^4*b)*sq
rt(b^7/((a^11 - 2*a^10*b + a^9*b^2)*d^4))*d^2 - b^3)/((a^5 - a^4*b)*d^2)))*sin(d*x + c) + 120*(a + b)*cos(d*x
+ c))/((a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d)*sin(d*x + c))

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

[Out]

Timed out

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

[Out]

Exception raised: NotImplementedError