Optimal. Leaf size=79 \[ -\frac{2 (b \csc (e+f x))^m \, _2F_1\left (-\frac{1}{4},\frac{1}{4} (-2 m-1);\frac{1}{4} (3-2 m);\sin ^2(e+f x)\right )}{d f (2 m+1) \sqrt [4]{\cos ^2(e+f x)} \sqrt{d \tan (e+f x)}} \]
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Rubi [A] time = 0.173675, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2618, 2602, 2577} \[ -\frac{2 (b \csc (e+f x))^m \, _2F_1\left (-\frac{1}{4},\frac{1}{4} (-2 m-1);\frac{1}{4} (3-2 m);\sin ^2(e+f x)\right )}{d f (2 m+1) \sqrt [4]{\cos ^2(e+f x)} \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2618
Rule 2602
Rule 2577
Rubi steps
\begin{align*} \int \frac{(b \csc (e+f x))^m}{(d \tan (e+f x))^{3/2}} \, dx &=\left ((b \csc (e+f x))^m \left (\frac{\sin (e+f x)}{b}\right )^m\right ) \int \frac{\left (\frac{\sin (e+f x)}{b}\right )^{-m}}{(d \tan (e+f x))^{3/2}} \, dx\\ &=\frac{\left ((b \csc (e+f x))^m \left (\frac{\sin (e+f x)}{b}\right )^{\frac{1}{2}+m}\right ) \int \cos ^{\frac{3}{2}}(e+f x) \left (\frac{\sin (e+f x)}{b}\right )^{-\frac{3}{2}-m} \, dx}{b d \sqrt{\cos (e+f x)} \sqrt{d \tan (e+f x)}}\\ &=-\frac{2 (b \csc (e+f x))^m \, _2F_1\left (-\frac{1}{4},\frac{1}{4} (-1-2 m);\frac{1}{4} (3-2 m);\sin ^2(e+f x)\right )}{d f (1+2 m) \sqrt [4]{\cos ^2(e+f x)} \sqrt{d \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.5421, size = 87, normalized size = 1.1 \[ -\frac{2 \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left (\frac{1}{4} (-2 m-1),1-\frac{m}{2};\frac{1}{4} (3-2 m);-\tan ^2(e+f x)\right )}{d f (2 m+1) \sqrt{d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.146, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b\csc \left ( fx+e \right ) \right ) ^{m} \left ( d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \csc \left (f x + e\right )\right )^{m}}{\left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d \tan \left (f x + e\right )} \left (b \csc \left (f x + e\right )\right )^{m}}{d^{2} \tan \left (f x + e\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b \csc \left (f x + e\right )\right )^{m}}{\left (d \tan \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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