Optimal. Leaf size=82 \[ \frac{\sin (c+d x) \cos (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (1-n p),\frac{1}{2} (3-n p),\sin ^2(c+d x)\right ) \left (a (b \csc (c+d x))^p\right )^n}{d (1-n p) \sqrt{\cos ^2(c+d x)}} \]
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Rubi [A] time = 0.0491908, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {4123, 3772, 2643} \[ \frac{\sin (c+d x) \cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\sin ^2(c+d x)\right ) \left (a (b \csc (c+d x))^p\right )^n}{d (1-n p) \sqrt{\cos ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3772
Rule 2643
Rubi steps
\begin{align*} \int \left (a (b \csc (c+d x))^p\right )^n \, dx &=\left ((b \csc (c+d x))^{-n p} \left (a (b \csc (c+d x))^p\right )^n\right ) \int (b \csc (c+d x))^{n p} \, dx\\ &=\left (\left (a (b \csc (c+d x))^p\right )^n \left (\frac{\sin (c+d x)}{b}\right )^{n p}\right ) \int \left (\frac{\sin (c+d x)}{b}\right )^{-n p} \, dx\\ &=\frac{\cos (c+d x) \left (a (b \csc (c+d x))^p\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\sin ^2(c+d x)\right ) \sin (c+d x)}{d (1-n p) \sqrt{\cos ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.088213, size = 73, normalized size = 0.89 \[ -\frac{\sin (c+d x) \cos (c+d x) \sin ^2(c+d x)^{\frac{1}{2} (n p-1)} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (n p+1),\frac{3}{2},\cos ^2(c+d x)\right ) \left (a (b \csc (c+d x))^p\right )^n}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.343, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( b\csc \left ( dx+c \right ) \right ) ^{p} \right ) ^{n}\, 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 \left (\left (b \csc \left (d x + c\right )\right )^{p} a\right )^{n}\,{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 (\left (\left (b \csc \left (d x + c\right )\right )^{p} a\right )^{n}, 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 \left (a \left (b \csc{\left (c + d x \right )}\right )^{p}\right )^{n}\, 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 \left (\left (b \csc \left (d x + c\right )\right )^{p} a\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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