Optimal. Leaf size=369 \[ -\frac{2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{15 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{15 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}+\frac{2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d f \left (c^2-d^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d f \left (c^2-d^2\right )^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
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Rubi [A] time = 0.526389, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \cos (e+f x)}{15 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \cos (e+f x)}{15 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{3/2}}-\frac{2 (b c-a d) \cos (e+f x)}{5 f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{5/2}}+\frac{2 \left (-8 a c d+3 b c^2+5 b d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d f \left (c^2-d^2\right )^2 \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d f \left (c^2-d^2\right )^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}} \]
Antiderivative was successfully verified.
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Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{a+b \sin (e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx &=-\frac{2 (b c-a d) \cos (e+f x)}{5 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac{2 \int \frac{-\frac{5}{2} (a c-b d)-\frac{3}{2} (b c-a d) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{5 \left (c^2-d^2\right )}\\ &=-\frac{2 (b c-a d) \cos (e+f x)}{5 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac{2 \left (3 b c^2-8 a c d+5 b d^2\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}+\frac{4 \int \frac{\frac{3}{4} \left (5 a c^2-8 b c d+3 a d^2\right )+\frac{1}{4} \left (3 b c^2-8 a c d+5 b d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{15 \left (c^2-d^2\right )^2}\\ &=-\frac{2 (b c-a d) \cos (e+f x)}{5 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac{2 \left (3 b c^2-8 a c d+5 b d^2\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 \left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}-\frac{8 \int \frac{\frac{1}{8} \left (-15 a c^3+27 b c^2 d-17 a c d^2+5 b d^3\right )+\frac{1}{8} \left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 \left (c^2-d^2\right )^3}\\ &=-\frac{2 (b c-a d) \cos (e+f x)}{5 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac{2 \left (3 b c^2-8 a c d+5 b d^2\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 \left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (3 b c^2-8 a c d+5 b d^2\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 d \left (c^2-d^2\right )^2}-\frac{\left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{15 d \left (c^2-d^2\right )^3}\\ &=-\frac{2 (b c-a d) \cos (e+f x)}{5 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac{2 \left (3 b c^2-8 a c d+5 b d^2\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 \left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}-\frac{\left (\left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{15 d \left (c^2-d^2\right )^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (\left (3 b c^2-8 a c d+5 b d^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{15 d \left (c^2-d^2\right )^2 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{2 (b c-a d) \cos (e+f x)}{5 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{5/2}}-\frac{2 \left (3 b c^2-8 a c d+5 b d^2\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{3/2}}-\frac{2 \left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) \cos (e+f x)}{15 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 \left (3 b c^3-23 a c^2 d+29 b c d^2-9 a d^3\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{15 d \left (c^2-d^2\right )^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 \left (3 b c^2-8 a c d+5 b d^2\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{15 d \left (c^2-d^2\right )^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.9782, size = 297, normalized size = 0.8 \[ \frac{2 \left (\frac{\cos (e+f x) \left (d^2 \left (23 a c^2 d+9 a d^3-3 b c^3-29 b c d^2\right ) \sin ^2(e+f x)+d \left (54 a c^3 d+10 a c d^3-60 b c^2 d^2-9 b c^4+5 b d^4\right ) \sin (e+f x)+a d \left (-5 c^2 d^2+34 c^4+3 d^4\right )+b \left (-25 c^3 d^2-9 c^5+2 c d^4\right )\right )}{\left (c^2-d^2\right )^3}+\frac{\left (\frac{c+d \sin (e+f x)}{c+d}\right )^{5/2} \left (\left (-23 a c^2 d-9 a d^3+3 b c^3+29 b c d^2\right ) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-(c-d) \left (-8 a c d+3 b c^2+5 b d^2\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )}{d (c-d)^3}\right )}{15 f (c+d \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 6.319, size = 1049, normalized size = 2.8 \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{b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{7}{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 (b \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\right ) + c}}{d^{4} \cos \left (f x + e\right )^{4} + c^{4} + 6 \, c^{2} d^{2} + d^{4} - 2 \,{\left (3 \, c^{2} d^{2} + d^{4}\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left (c d^{3} \cos \left (f x + e\right )^{2} - c^{3} d - c d^{3}\right )} \sin \left (f x + e\right )}, 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{b \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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