Optimal. Leaf size=89 \[ \frac{2 \cos (a+b x) \sin ^{1-m}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2-3 m);\frac{3 (2-m)}{4};\sin ^2(a+b x)\right )}{b c (2-3 m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}} \]
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Rubi [A] time = 0.0405227, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3208, 2643} \[ \frac{2 \cos (a+b x) \sin ^{1-m}(a+b x) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2-3 m);\frac{3 (2-m)}{4};\sin ^2(a+b x)\right )}{b c (2-3 m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3208
Rule 2643
Rubi steps
\begin{align*} \int \frac{1}{\left (c \sin ^m(a+b x)\right )^{3/2}} \, dx &=\frac{\sin ^{\frac{m}{2}}(a+b x) \int \sin ^{-\frac{3 m}{2}}(a+b x) \, dx}{c \sqrt{c \sin ^m(a+b x)}}\\ &=\frac{2 \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2-3 m);\frac{3 (2-m)}{4};\sin ^2(a+b x)\right ) \sin ^{1-m}(a+b x)}{b c (2-3 m) \sqrt{\cos ^2(a+b x)} \sqrt{c \sin ^m(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.108938, size = 71, normalized size = 0.8 \[ \frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left (\frac{1}{2},\frac{1}{4} (2-3 m);-\frac{3}{4} (m-2);\sin ^2(a+b x)\right )}{\left (b-\frac{3 b m}{2}\right ) \left (c \sin ^m(a+b x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \sin \left ( bx+a \right ) \right ) ^{m} \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{1}{\left (c \sin \left (b x + a\right )^{m}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{1}{\left (c \sin ^{m}{\left (a + b x \right )}\right )^{\frac{3}{2}}}\, 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{1}{\left (c \sin \left (b x + a\right )^{m}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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