Optimal. Leaf size=68 \[ \frac{2 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\cos (a+b x)}} \]
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Rubi [A] time = 0.0637746, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2566, 2640, 2639} \[ \frac{2 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{4 E\left (\left .\frac{1}{2} (a+b x)\right |2\right ) \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\cos (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2566
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx &=\frac{2 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{2 \int \sqrt{d \cos (a+b x)} \, dx}{d^2}\\ &=\frac{2 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{\left (2 \sqrt{d \cos (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \, dx}{d^2 \sqrt{\cos (a+b x)}}\\ &=-\frac{4 \sqrt{d \cos (a+b x)} E\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{b d^2 \sqrt{\cos (a+b x)}}+\frac{2 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.0817393, size = 60, normalized size = 0.88 \[ \frac{\sin ^3(a+b x) \sqrt [4]{\cos ^2(a+b x)} \, _2F_1\left (\frac{5}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right )}{3 b d \sqrt{d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 168, normalized size = 2.5 \begin{align*} -4\,{\frac{\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}d+ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}d} \left ( \sqrt{2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) - \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}\cos \left ( 1/2\,bx+a/2 \right ) \right ) }{d\sqrt{-d \left ( 2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}- \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2} \right ) }\sin \left ( 1/2\,bx+a/2 \right ) \sqrt{d \left ( 2\, \left ( \cos \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) }b}} \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{\sin \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}\,{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 \cos \left (b x + a\right )}{\left (\cos \left (b x + a\right )^{2} - 1\right )}}{d^{2} \cos \left (b x + a\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\sin \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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