Optimal. Leaf size=82 \[ \frac {\cos (c+d x) \left (a (b \csc (c+d x))^p\right )^n \, _2F_1\left (\frac {1}{2},\frac {1}{2} (1-n p);\frac {1}{2} (3-n p);\sin ^2(c+d x)\right ) \sin (c+d x)}{d (1-n p) \sqrt {\cos ^2(c+d x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 = {4208, 3857,
2722} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3857
Rule 4208
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*}
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Mathematica [A]
time = 0.09, size = 73, normalized size = 0.89 \begin {gather*} -\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 {3}{2};\cos ^2(c+d x)\right ) \sin (c+d x) \sin ^2(c+d x)^{\frac {1}{2} (-1+n p)}}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \left (a \left (b \csc \left (d x +c \right )\right )^{p}\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a \left (b \csc {\left (c + d x \right )}\right )^{p}\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a\,{\left (\frac {b}{\sin \left (c+d\,x\right )}\right )}^p\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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