Optimal. Leaf size=74 \[ \frac{2 \sin (a+b x) \cos (a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{m+2}{4};\frac{m+6}{4};\sin ^2(a+b x)\right )}{b (m+2) \sqrt{\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.0360372, antiderivative size = 74, 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} \[ \frac{2 \sin (a+b x) \cos (a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{m+2}{4};\frac{m+6}{4};\sin ^2(a+b x)\right )}{b (m+2) \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3208
Rule 2643
Rubi steps
\begin{align*} \int \sqrt{c \sin ^m(a+b x)} \, dx &=\left (\sin ^{-\frac{m}{2}}(a+b x) \sqrt{c \sin ^m(a+b x)}\right ) \int \sin ^{\frac{m}{2}}(a+b x) \, dx\\ &=\frac{2 \cos (a+b x) \, _2F_1\left (\frac{1}{2},\frac{2+m}{4};\frac{6+m}{4};\sin ^2(a+b x)\right ) \sin (a+b x) \sqrt{c \sin ^m(a+b x)}}{b (2+m) \sqrt{\cos ^2(a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0716404, size = 68, normalized size = 0.92 \[ \frac{2 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{m+2}{4};\frac{m+6}{4};\sin ^2(a+b x)\right )}{b (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.213, size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \left ( \sin \left ( bx+a \right ) \right ) ^{m}}\, 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 \sqrt{c \sin \left (b x + a\right )^{m}}\,{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 \sqrt{c \sin ^{m}{\left (a + b 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 \sqrt{c \sin \left (b x + a\right )^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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