Optimal. Leaf size=57 \[ \frac{1}{a f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{a^{3/2} f} \]
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Rubi [A] time = 0.0756698, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3194, 51, 63, 208} \[ \frac{1}{a f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{a^{3/2} f} \]
Antiderivative was successfully verified.
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Rule 3194
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cot (e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,\sin ^2(e+f x)\right )}{2 f}\\ &=\frac{1}{a f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 a f}\\ &=\frac{1}{a f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sin ^2(e+f x)}\right )}{a b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sin ^2(e+f x)}}{\sqrt{a}}\right )}{a^{3/2} f}+\frac{1}{a f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.0567884, size = 46, normalized size = 0.81 \[ \frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b \sin ^2(e+f x)}{a}+1\right )}{a f \sqrt{a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.235, size = 64, normalized size = 1.1 \begin{align*}{\frac{1}{af}{\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}}-{\frac{1}{f}\ln \left ({\frac{1}{\sin \left ( fx+e \right ) } \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}} \right ) } \right ){a}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.54335, size = 536, normalized size = 9.4 \begin{align*} \left [\frac{{\left (b \cos \left (f x + e\right )^{2} - a - b\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 \, \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} a}{2 \,{\left (a^{2} b f \cos \left (f x + e\right )^{2} -{\left (a^{3} + a^{2} b\right )} f\right )}}, \frac{{\left (b \cos \left (f x + e\right )^{2} - a - b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a}}{a}\right ) - \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} a}{a^{2} b f \cos \left (f x + e\right )^{2} -{\left (a^{3} + a^{2} b\right )} f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1096, size = 77, normalized size = 1.35 \begin{align*} \frac{\arctan \left (\frac{\sqrt{b \sin \left (f x + e\right )^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a f} + \frac{1}{\sqrt{b \sin \left (f x + e\right )^{2} + a} a f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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