Optimal. Leaf size=79 \[ \frac{2 \sqrt [4]{\cos ^2(e+f x)} \sqrt{d \tan (e+f x)} (b \csc (e+f x))^m \, _2F_1\left (\frac{1}{4},\frac{1}{4} (1-2 m);\frac{1}{4} (5-2 m);\sin ^2(e+f x)\right )}{d f (1-2 m)} \]
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Rubi [A] time = 0.150533, 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} \[ \frac{2 \sqrt [4]{\cos ^2(e+f x)} \sqrt{d \tan (e+f x)} (b \csc (e+f x))^m \, _2F_1\left (\frac{1}{4},\frac{1}{4} (1-2 m);\frac{1}{4} (5-2 m);\sin ^2(e+f x)\right )}{d f (1-2 m)} \]
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}{\sqrt{d \tan (e+f x)}} \, 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}}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{\left (\sqrt{\cos (e+f x)} (b \csc (e+f x))^{1+m} \left (\frac{\sin (e+f x)}{b}\right )^{\frac{1}{2}+m} \sqrt{d \tan (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \left (\frac{\sin (e+f x)}{b}\right )^{-\frac{1}{2}-m} \, dx}{b d}\\ &=\frac{2 \sqrt [4]{\cos ^2(e+f x)} (b \csc (e+f x))^{1+m} \, _2F_1\left (\frac{1}{4},\frac{1}{4} (1-2 m);\frac{1}{4} (5-2 m);\sin ^2(e+f x)\right ) \sin (e+f x) \sqrt{d \tan (e+f x)}}{b d f (1-2 m)}\\ \end{align*}
Mathematica [A] time = 1.22483, size = 87, normalized size = 1.1 \[ -\frac{2 \sqrt{d \tan (e+f x)} \sec ^2(e+f x)^{-m/2} (b \csc (e+f x))^m \, _2F_1\left (\frac{1}{4} (1-2 m),1-\frac{m}{2};\frac{1}{4} (5-2 m);-\tan ^2(e+f x)\right )}{d f (2 m-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.168, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b\csc \left ( fx+e \right ) \right ) ^{m}{\frac{1}{\sqrt{d\tan \left ( fx+e \right ) }}}}\, 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}}{\sqrt{d \tan \left (f x + e\right )}}\,{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 \tan \left (f x + e\right )}, x\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{\left (b \csc{\left (e + f x \right )}\right )^{m}}{\sqrt{d \tan{\left (e + f x \right )}}}\, dx \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}}{\sqrt{d \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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