\(\int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [208]

Optimal result

Integrand size = 24, antiderivative size = 139 \[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} \sqrt {\sqrt {a}+\sqrt {b}} d}-\frac {\cot (c+d x)}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 139, 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 = {3296, 1301, 1144, 211} \[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{5/4} d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\cot (c+d x)}{a d} \]

[In]

Int[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]

[Out]

(Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) - (Sqrt
[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(5/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d) - Cot[c + d*
x]/(a*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 1144

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2/2)*(b/q + 1), Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2/2)*(b/q - 1), 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 1301

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 3296

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]

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

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {\sqrt {b} \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}}}+2 \cot (c+d x)}{2 a d} \]

[In]

Integrate[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/2*((Sqrt[b]*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]]
+ (Sqrt[b]*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])/(a*d)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.12

method	result	size
risch	\(-\frac {2 i}{a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{6} d^{4}-65536 a^{5} b \,d^{4}\right ) \textit {\_Z}^{4}+512 a^{3} b \,d^{2} \textit {\_Z}^{2}+b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {8192 i d^{3} a^{5}}{b^{2}}-\frac {8192 i d^{3} a^{4}}{b}\right ) \textit {\_R}^{3}+\left (-\frac {512 d^{2} a^{4}}{b^{2}}+\frac {512 d^{2} a^{3}}{b}\right ) \textit {\_R}^{2}+\frac {64 i a^{2} d \textit {\_R}}{b}-\frac {2 a}{b}-1\right )\right )\)	\(155\)
derivativedivides	\(\frac {-\frac {1}{a \tan \left (d x +c \right )}+\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a}}{d}\)	\(164\)
default	\(\frac {-\frac {1}{a \tan \left (d x +c \right )}+\frac {b \left (a -b \right ) \left (\frac {\left (\sqrt {a b}+a \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \left (a -b \right ) \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}-a \right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a}}{d}\)	\(164\)

[In]

int(csc(d*x+c)^2/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

-2*I/a/d/(exp(2*I*(d*x+c))-1)-4*sum(_R*ln(exp(2*I*(d*x+c))+(8192*I/b^2*d^3*a^5-8192*I/b*d^3*a^4)*_R^3+(-512/b^
2*d^2*a^4+512/b*d^2*a^3)*_R^2+64*I/b*a^2*d*_R-2/b*a-1),_R=RootOf((65536*a^6*d^4-65536*a^5*b*d^4)*_Z^4+512*a^3*
b*d^2*_Z^2+b^2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1229 vs. \(2 (99) = 198\).

Time = 0.41 (sec) , antiderivative size = 1229, normalized size of antiderivative = 8.84 \[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/8*(a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(1/4
*b^2*cos(d*x + c)^2 - 1/4*b^2 - 1/4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2 - (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7 -
2*a^6*b + a^5*b^2)*d^4)) + 1/2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4))*cos(d*x + c)*sin(d
*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)
) + b)/((a^3 - a^2*b)*d^2)))*sin(d*x + c) - a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d
^4)) + b)/((a^3 - a^2*b)*d^2))*log(1/4*b^2*cos(d*x + c)^2 - 1/4*b^2 - 1/4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2
- (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - 1/2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6
*b + a^5*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) - a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3 - a^2*b)*d^2*sq
rt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2)))*sin(d*x + c) + a*d*sqrt(((a^3 - a^2*b)*d^2*
sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(-1/4*b^2*cos(d*x + c)^2 + 1/4*b^2 - 1/
4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2 - (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) + 1/2*((a
^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4))*cos(d*x + c)*sin(d*x + c) + a^2*b*d*cos(d*x + c)*sin
(d*x + c))*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2)))*sin(d*
x + c) - a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(-
1/4*b^2*cos(d*x + c)^2 + 1/4*b^2 - 1/4*(2*(a^4 - a^3*b)*d^2*cos(d*x + c)^2 - (a^4 - a^3*b)*d^2)*sqrt(b^3/((a^7
- 2*a^6*b + a^5*b^2)*d^4)) - 1/2*((a^5 - a^4*b)*d^3*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^4))*cos(d*x + c)*si
n(d*x + c) + a^2*b*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3 - a^2*b)*d^2*sqrt(b^3/((a^7 - 2*a^6*b + a^5*b^2)*d^
4)) - b)/((a^3 - a^2*b)*d^2)))*sin(d*x + c) + 8*cos(d*x + c))/(a*d*sin(d*x + c))

Sympy [F]

\[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc ^{2}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]

[In]

integrate(csc(d*x+c)**2/(a-b*sin(d*x+c)**4),x)

[Out]

Integral(csc(c + d*x)**2/(a - b*sin(c + d*x)**4), x)

Maxima [F]

\[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]

[In]

integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

((a*d*cos(2*d*x + 2*c)^2 + a*d*sin(2*d*x + 2*c)^2 - 2*a*d*cos(2*d*x + 2*c) + a*d)*integrate(-4*(4*b^2*cos(6*d*
x + 6*c)^2 + 4*b^2*cos(2*d*x + 2*c)^2 + 4*b^2*sin(6*d*x + 6*c)^2 + 4*b^2*sin(2*d*x + 2*c)^2 - 4*(8*a*b - 3*b^2
)*cos(4*d*x + 4*c)^2 - b^2*cos(2*d*x + 2*c) - 4*(8*a*b - 3*b^2)*sin(4*d*x + 4*c)^2 + 2*(8*a*b - 7*b^2)*sin(4*d
*x + 4*c)*sin(2*d*x + 2*c) - (b^2*cos(6*d*x + 6*c) - 2*b^2*cos(4*d*x + 4*c) + b^2*cos(2*d*x + 2*c))*cos(8*d*x
+ 8*c) + (8*b^2*cos(2*d*x + 2*c) - b^2 + 2*(8*a*b - 7*b^2)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) + 2*(b^2 + (8*a*
b - 7*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (b^2*sin(6*d*x + 6*c) - 2*b^2*sin(4*d*x + 4*c) + b^2*sin(2*d*x
+ 2*c))*sin(8*d*x + 8*c) + 2*(4*b^2*sin(2*d*x + 2*c) + (8*a*b - 7*b^2)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c))/(a
*b^2*cos(8*d*x + 8*c)^2 + 16*a*b^2*cos(6*d*x + 6*c)^2 + 16*a*b^2*cos(2*d*x + 2*c)^2 + a*b^2*sin(8*d*x + 8*c)^2
+ 16*a*b^2*sin(6*d*x + 6*c)^2 + 16*a*b^2*sin(2*d*x + 2*c)^2 - 8*a*b^2*cos(2*d*x + 2*c) + a*b^2 + 4*(64*a^3 -
48*a^2*b + 9*a*b^2)*cos(4*d*x + 4*c)^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b - 3*
a*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - 2*(4*a*b^2*cos(6*d*x + 6*c) + 4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*
(8*a^2*b - 3*a*b^2)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*
a*b^2)*cos(4*d*x + 4*c))*cos(6*d*x + 6*c) - 4*(8*a^2*b - 3*a*b^2 - 4*(8*a^2*b - 3*a*b^2)*cos(2*d*x + 2*c))*cos
(4*d*x + 4*c) - 4*(2*a*b^2*sin(6*d*x + 6*c) + 2*a*b^2*sin(2*d*x + 2*c) + (8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c))
*sin(8*d*x + 8*c) + 16*(2*a*b^2*sin(2*d*x + 2*c) + (8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x)
- 2*sin(2*d*x + 2*c))/(a*d*cos(2*d*x + 2*c)^2 + a*d*sin(2*d*x + 2*c)^2 - 2*a*d*cos(2*d*x + 2*c) + a*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (99) = 198\).

Time = 0.75 (sec) , antiderivative size = 672, normalized size of antiderivative = 4.83 \[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {{\left ({\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} a^{2} {\left | a - b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2}\right )} {\left | a - b \right |}\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^{2} + \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} - \frac {{\left ({\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{2} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{3}\right )} a^{2} {\left | a - b \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{5} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{4} b - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{3} b^{2}\right )} {\left | a - b \right |}\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^{2} - \sqrt {a^{4} - {\left (a^{2} - a b\right )} a^{2}}}{a^{2} - a b}}}\right )\right )}}{{\left (3 \, a^{8} - 15 \, a^{7} b + 26 \, a^{6} b^{2} - 18 \, a^{5} b^{3} + 3 \, a^{4} b^{4} + a^{3} b^{5}\right )} {\left | a \right |}} + \frac {2}{a \tan \left (d x + c\right )}}{2 \, d} \]

[In]

integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

-1/2*(((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b
)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*sqrt(a*b)*a^5 - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*sqrt(a*b)*a^3*b^2)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 + sqrt(a^4 -
(a^2 - a*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a))
- ((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*
b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*a^2*abs(a - b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b
))*sqrt(a*b)*a^5 - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))
*sqrt(a*b)*a^3*b^2)*abs(a - b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - sqrt(a^4 - (a^
2 - a*b)*a^2))/(a^2 - a*b))))/((3*a^8 - 15*a^7*b + 26*a^6*b^2 - 18*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*abs(a)) + 2/
(a*tan(d*x + c)))/d

Mupad [B] (verification not implemented)

Time = 14.81 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.67 \[ \int \frac {\csc ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {2\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^4\,b^4-4\,a^6\,b^2\right )-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\sqrt {a^5\,b^3}+a^3\,b\right )\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )}{16\,\left (a^5\,b-a^6\right )}\right )\,\sqrt {\frac {\sqrt {a^5\,b^3}+a^3\,b}{16\,\left (a^5\,b-a^6\right )}}}{2\,a^3\,b^4-2\,a^4\,b^3}\right )\,\sqrt {\frac {\sqrt {a^5\,b^3}+a^3\,b}{16\,\left (a^5\,b-a^6\right )}}}{d}+\frac {2\,\mathrm {atanh}\left (\frac {2\,\left (\mathrm {tan}\left (c+d\,x\right )\,\left (4\,a^4\,b^4-4\,a^6\,b^2\right )+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\sqrt {a^5\,b^3}-a^3\,b\right )\,\left (64\,a^9\,b-128\,a^8\,b^2+64\,a^7\,b^3\right )}{16\,\left (a^5\,b-a^6\right )}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^3}-a^3\,b}{16\,\left (a^5\,b-a^6\right )}}}{2\,a^3\,b^4-2\,a^4\,b^3}\right )\,\sqrt {-\frac {\sqrt {a^5\,b^3}-a^3\,b}{16\,\left (a^5\,b-a^6\right )}}}{d}-\frac {\mathrm {cot}\left (c+d\,x\right )}{a\,d} \]

[In]

int(1/(sin(c + d*x)^2*(a - b*sin(c + d*x)^4)),x)

[Out]

(2*atanh((2*(tan(c + d*x)*(4*a^4*b^4 - 4*a^6*b^2) - (tan(c + d*x)*((a^5*b^3)^(1/2) + a^3*b)*(64*a^9*b + 64*a^7
*b^3 - 128*a^8*b^2))/(16*(a^5*b - a^6)))*(((a^5*b^3)^(1/2) + a^3*b)/(16*(a^5*b - a^6)))^(1/2))/(2*a^3*b^4 - 2*
a^4*b^3))*(((a^5*b^3)^(1/2) + a^3*b)/(16*(a^5*b - a^6)))^(1/2))/d + (2*atanh((2*(tan(c + d*x)*(4*a^4*b^4 - 4*a
^6*b^2) + (tan(c + d*x)*((a^5*b^3)^(1/2) - a^3*b)*(64*a^9*b + 64*a^7*b^3 - 128*a^8*b^2))/(16*(a^5*b - a^6)))*(
-((a^5*b^3)^(1/2) - a^3*b)/(16*(a^5*b - a^6)))^(1/2))/(2*a^3*b^4 - 2*a^4*b^3))*(-((a^5*b^3)^(1/2) - a^3*b)/(16
*(a^5*b - a^6)))^(1/2))/d - cot(c + d*x)/(a*d)