Optimal. Leaf size=84 \[ \frac{\sin (a+b x) E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}+\frac{\sin ^3(a+b x)}{3 b d (d \tan (a+b x))^{3/2}} \]
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Rubi [A] time = 0.102488, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2596, 2601, 2572, 2639} \[ \frac{\sin (a+b x) E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{2 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}+\frac{\sin ^3(a+b x)}{3 b d (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2596
Rule 2601
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin ^3(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx &=\frac{\sin ^3(a+b x)}{3 b d (d \tan (a+b x))^{3/2}}+\frac{\int \frac{\sin (a+b x)}{\sqrt{d \tan (a+b x)}} \, dx}{2 d^2}\\ &=\frac{\sin ^3(a+b x)}{3 b d (d \tan (a+b x))^{3/2}}+\frac{\sqrt{\sin (a+b x)} \int \sqrt{\cos (a+b x)} \sqrt{\sin (a+b x)} \, dx}{2 d^2 \sqrt{\cos (a+b x)} \sqrt{d \tan (a+b x)}}\\ &=\frac{\sin ^3(a+b x)}{3 b d (d \tan (a+b x))^{3/2}}+\frac{\sin (a+b x) \int \sqrt{\sin (2 a+2 b x)} \, dx}{2 d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}\\ &=\frac{\sin ^3(a+b x)}{3 b d (d \tan (a+b x))^{3/2}}+\frac{E\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sin (a+b x)}{2 b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.617591, size = 97, normalized size = 1.15 \[ \frac{\sqrt{d \tan (a+b x)} \left (4 \tan (a+b x) \sec (a+b x) \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right )+(\sin (a+b x)+\sin (3 (a+b x))) \sqrt{\sec ^2(a+b x)}\right )}{12 b d^3 \sqrt{\sec ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.167, size = 544, normalized size = 6.5 \begin{align*} \text{result too large to display} \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 )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac{5}{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{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sqrt{d \tan \left (b x + a\right )} \sin \left (b x + a\right )}{d^{3} \tan \left (b x + a\right )^{3}}, 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 )^{3}}{\left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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