Optimal. Leaf size=78 \[ -\frac{3 \sin (a+b x) E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}-\frac{2 \sin (a+b x)}{b d (d \tan (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0865126, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2597, 2601, 2572, 2639} \[ -\frac{3 \sin (a+b x) E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}-\frac{2 \sin (a+b x)}{b d (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2597
Rule 2601
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sin (a+b x)}{(d \tan (a+b x))^{5/2}} \, dx &=-\frac{2 \sin (a+b x)}{b d (d \tan (a+b x))^{3/2}}-\frac{3 \int \frac{\sin (a+b x)}{\sqrt{d \tan (a+b x)}} \, dx}{d^2}\\ &=-\frac{2 \sin (a+b x)}{b d (d \tan (a+b x))^{3/2}}-\frac{\left (3 \sqrt{\sin (a+b x)}\right ) \int \sqrt{\cos (a+b x)} \sqrt{\sin (a+b x)} \, dx}{d^2 \sqrt{\cos (a+b x)} \sqrt{d \tan (a+b x)}}\\ &=-\frac{2 \sin (a+b x)}{b d (d \tan (a+b x))^{3/2}}-\frac{(3 \sin (a+b x)) \int \sqrt{\sin (2 a+2 b x)} \, dx}{d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}\\ &=-\frac{2 \sin (a+b x)}{b d (d \tan (a+b x))^{3/2}}-\frac{3 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sin (a+b x)}{b d^2 \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.376484, size = 69, normalized size = 0.88 \[ -\frac{2 \cos (a+b x) \left (\tan ^2(a+b x) \sqrt{\sec ^2(a+b x)} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\tan ^2(a+b x)\right )+1\right )}{b d^2 \sqrt{d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.138, size = 511, normalized size = 6.6 \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 )}{\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{\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 )}{\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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