3.2.93 \(\int \frac {\csc ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [193]

Optimal. Leaf size=281 \[ -\frac {2 (-1)^{2/3} b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}+\frac {2 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} \sqrt {a^{2/3}-b^{2/3}} d}-\frac {2 \sqrt [3]{-1} b^{2/3} \tan ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{4/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {\cot (c+d x)}{a d} \]

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3299, 3852, 8, 2739, 632, 210} \begin {gather*} -\frac {2 (-1)^{2/3} b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}+\frac {2 b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {a^{2/3}-b^{2/3}}}-\frac {2 \sqrt [3]{-1} b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{4/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}-\frac {\cot (c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 210

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.20, size = 196, normalized size = 0.70 \begin {gather*} \frac {-3 \cot \left (\frac {1}{2} (c+d x)\right )+2 b \text {RootSum}\left [-b+3 b \text {$\#$1}^2-8 i a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \tan \left (\frac {1}{2} (c+d x)\right )}{6 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^2/(a + b*Sin[c + d*x]^3),x]

[Out]

(-3*Cot[(c + d*x)/2] + 2*b*RootSum[-b + 3*b*#1^2 - (8*I)*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-2*ArcTan[Sin[c + d*x
]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2
- I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*Tan[(c + d*x)/2])/(6
*a*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.76, size = 114, normalized size = 0.41

method	result	size
derivativedivides	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\)	\(114\)
default	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\)	\(114\)
risch	\(-\frac {2 i}{a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-4 \left (\munderset {\textit {\_R} =\RootOf \left (\left (2985984 a^{10} d^{6}-2985984 a^{8} b^{2} d^{6}\right ) \textit {\_Z}^{6}+62208 a^{6} b^{2} d^{4} \textit {\_Z}^{4}+b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {248832 d^{5} a^{10}}{a^{2} b^{3}+b^{5}}-\frac {248832 d^{5} a^{8} b^{2}}{a^{2} b^{3}+b^{5}}\right ) \textit {\_R}^{5}+\left (\frac {20736 i d^{4} a^{9}}{a^{2} b^{3}+b^{5}}-\frac {20736 i d^{4} a^{7} b^{2}}{a^{2} b^{3}+b^{5}}\right ) \textit {\_R}^{4}+\left (\frac {3456 d^{3} a^{6} b^{2}}{a^{2} b^{3}+b^{5}}+\frac {1728 d^{3} b^{4} a^{4}}{a^{2} b^{3}+b^{5}}\right ) \textit {\_R}^{3}+\left (\frac {288 i d^{2} a^{5} b^{2}}{a^{2} b^{3}+b^{5}}+\frac {144 i d^{2} b^{4} a^{3}}{a^{2} b^{3}+b^{5}}\right ) \textit {\_R}^{2}+\left (-\frac {12 d \,a^{4} b^{2}}{a^{2} b^{3}+b^{5}}-\frac {24 d \,b^{4} a^{2}}{a^{2} b^{3}+b^{5}}\right ) \textit {\_R} -\frac {i b^{4} a}{a^{2} b^{3}+b^{5}}\right )\right )\)	\(362\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^2/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/2/a*tan(1/2*d*x+1/2*c)-1/2/a/tan(1/2*d*x+1/2*c)-2/3/a*b*sum((_R^3+_R)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*l
n(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [C] Result contains complex when optimal does not.
time = 1.55, size = 21243, normalized size = 75.60 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm="fricas")

[Out]

-1/12*(sqrt(2/3)*sqrt(1/6)*a*d*sqrt(((a^4 - a^2*b^2)*(b^4*(-I*sqrt(3) + 1)/((a^4*d^2 - a^2*b^2*d^2)^2*(-1/1458
*b^4/(a^10*d^6 - a^8*b^2*d^6) - 1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1458*(a^2 + b^2)*b^4/((a^2 - b^2)^2*a^
8*d^6))^(1/3)) + 81*(-1/1458*b^4/(a^10*d^6 - a^8*b^2*d^6) - 1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1458*(a^2
+ b^2)*b^4/((a^2 - b^2)^2*a^8*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))*d^2 + 3*sqrt(1/3)*
(a^4 - a^2*b^2)*d^2*sqrt(-((a^8 - 2*a^6*b^2 + a^4*b^4)*(b^4*(-I*sqrt(3) + 1)/((a^4*d^2 - a^2*b^2*d^2)^2*(-1/14
58*b^4/(a^10*d^6 - a^8*b^2*d^6) - 1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1458*(a^2 + b^2)*b^4/((a^2 - b^2)^2*
a^8*d^6))^(1/3)) + 81*(-1/1458*b^4/(a^10*d^6 - a^8*b^2*d^6) - 1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 + 1/1458*(a^
2 + b^2)*b^4/((a^2 - b^2)^2*a^8*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))^2*d^4 - 972*b^4
- 36*(a^4*b^2 - a^2*b^4)*(b^4*(-I*sqrt(3) + 1)/((a^4*d^2 - a^2*b^2*d^2)^2*(-1/1458*b^4/(a^10*d^6 - a^8*b^2*d^6
) - 1/729* ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{2}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**2/(a+b*sin(d*x+c)**3),x)

[Out]

Integral(csc(c + d*x)**2/(a + b*sin(c + d*x)**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a+b*sin(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(csc(d*x + c)^2/(b*sin(d*x + c)^3 + a), x)

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Mupad [B]
time = 14.42, size = 697, normalized size = 2.48 \begin {gather*} \frac {\left (\sum _{k=1}^6\ln \left (8192\,a^7\,b^6-{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^2\,a^9\,b^6\,294912+{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^3\,a^{11}\,b^5\,1548288-{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^4\,a^{13}\,b^4\,1990656-{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^5\,a^{13}\,b^5\,7962624+{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^5\,a^{15}\,b^3\,5971968-65536\,a^6\,b^7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )\,a^8\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,196608-{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^2\,a^{10}\,b^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,294912-{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^3\,a^{10}\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1769472+{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^3\,a^{12}\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,221184-{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^4\,a^{12}\,b^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,2654208-{\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )}^5\,a^{14}\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1990656\right )\,\mathrm {root}\left (729\,a^8\,b^2\,d^6-729\,a^{10}\,d^6-243\,a^6\,b^2\,d^4-b^4,d,k\right )\right )-\frac {1}{2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(c + d*x)^2*(a + b*sin(c + d*x)^3)),x)

[Out]

(symsum(log(8192*a^7*b^6 - 294912*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 - b^4, d, k)^2*a^9*b^6
+ 1548288*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 - b^4, d, k)^3*a^11*b^5 - 1990656*root(729*a^
8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 - b^4, d, k)^4*a^13*b^4 - 7962624*root(729*a^8*b^2*d^6 - 729*a^10*d
^6 - 243*a^6*b^2*d^4 - b^4, d, k)^5*a^13*b^5 + 5971968*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 -
b^4, d, k)^5*a^15*b^3 - 65536*a^6*b^7*tan(c/2 + (d*x)/2) + 196608*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a
^6*b^2*d^4 - b^4, d, k)*a^8*b^6*tan(c/2 + (d*x)/2) - 294912*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*
d^4 - b^4, d, k)^2*a^10*b^5*tan(c/2 + (d*x)/2) - 1769472*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4
- b^4, d, k)^3*a^10*b^6*tan(c/2 + (d*x)/2) + 221184*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 - b
^4, d, k)^3*a^12*b^4*tan(c/2 + (d*x)/2) - 2654208*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 - b^4,
d, k)^4*a^12*b^5*tan(c/2 + (d*x)/2) - 1990656*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 - b^4, d,
k)^5*a^14*b^4*tan(c/2 + (d*x)/2))*root(729*a^8*b^2*d^6 - 729*a^10*d^6 - 243*a^6*b^2*d^4 - b^4, d, k), k, 1, 6
) - 1/(2*a*tan(c/2 + (d*x)/2)) + tan(c/2 + (d*x)/2)/(2*a))/d

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