Optimal. Leaf size=110 \[ \frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{5/2} f}-\frac {2 a+3 b}{2 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^2(e+f x)}{2 a f \sqrt {a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3194, 78, 51, 63, 208} \[ -\frac {2 a+3 b}{2 a^2 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{5/2} f}-\frac {\csc ^2(e+f x)}{2 a f \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 208
Rule 3194
Rubi steps
\begin {align*} \int \frac {\cot ^3(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x}{x^2 (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=-\frac {\csc ^2(e+f x)}{2 a f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(2 a+3 b) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{4 a f}\\ &=-\frac {2 a+3 b}{2 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^2(e+f x)}{2 a f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(2 a+3 b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 a^2 f}\\ &=-\frac {2 a+3 b}{2 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^2(e+f x)}{2 a f \sqrt {a+b \sin ^2(e+f x)}}-\frac {(2 a+3 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{2 a^2 b f}\\ &=\frac {(2 a+3 b) \tanh ^{-1}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{5/2} f}-\frac {2 a+3 b}{2 a^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {\csc ^2(e+f x)}{2 a f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 70, normalized size = 0.64 \[ \frac {-(2 a+3 b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b \sin ^2(e+f x)}{a}+1\right )-a \csc ^2(e+f x)}{2 a^2 f \sqrt {a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 406, normalized size = 3.69 \[ \left [\frac {{\left ({\left (2 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (2 \, a^{2} + 7 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right )} \sqrt {a} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) + 2 \, {\left ({\left (2 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 3 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{4 \, {\left (a^{3} b f \cos \left (f x + e\right )^{4} - {\left (a^{4} + 2 \, a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} + a^{3} b\right )} f\right )}}, -\frac {{\left ({\left (2 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - {\left (2 \, a^{2} + 7 \, a b + 6 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left ({\left (2 \, a^{2} + 3 \, a b\right )} \cos \left (f x + e\right )^{2} - 3 \, a^{2} - 3 \, a b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{2 \, {\left (a^{3} b f \cos \left (f x + e\right )^{4} - {\left (a^{4} + 2 \, a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} + a^{3} b\right )} f\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.01, size = 525, normalized size = 4.77 \[ -\frac {\frac {{\left (\frac {{\left (a^{5} b + a^{4} b^{2}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{6} b + a^{5} b^{2}} + \frac {2 \, {\left (5 \, a^{5} b + 11 \, a^{4} b^{2} + 6 \, a^{3} b^{3}\right )}}{a^{6} b + a^{5} b^{2}}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \frac {9 \, a^{5} b + 17 \, a^{4} b^{2} + 8 \, a^{3} b^{3}}{a^{6} b + a^{5} b^{2}}}{\sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}} + \frac {2 \, {\left (2 \, a^{\frac {3}{2}} + 3 \, \sqrt {a} b\right )} \log \left ({\left | -{\left (\sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )} a - a^{\frac {3}{2}} - 2 \, \sqrt {a} b \right |}\right )}{a^{3}} - \frac {2 \, {\left ({\left (\sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )} a^{\frac {3}{2}} + 2 \, {\left (\sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )} \sqrt {a} b + a^{2}\right )}}{{\left ({\left (\sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a}\right )}^{2} \sqrt {a} - a^{\frac {3}{2}}\right )} a^{2}}}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.78, size = 159, normalized size = 1.45 \[ -\frac {1}{a f \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}+\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f \,a^{\frac {3}{2}}}-\frac {1}{2 f a \sin \left (f x +e \right )^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}-\frac {3 b}{2 f \,a^{2} \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}+\frac {3 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{2 f \,a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 117, normalized size = 1.06 \[ \frac {\frac {2 \, \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {3}{2}}} + \frac {3 \, b \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right )}{a^{\frac {5}{2}}} - \frac {2}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a} - \frac {3 \, b}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2}} - \frac {1}{\sqrt {b \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right )^{2}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^3}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{3}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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