Optimal. Leaf size=496 \[ \frac{2 b \left (-189 a^2 d^2+54 a b c d+b^2 \left (-\left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{2 \left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (189 a^2 b d^2 \left (c^2+3 d^2\right )+420 a^3 c d^3-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (33 c^2 d^2+8 c^4+147 d^4\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f} \]
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Rubi [A] time = 1.02559, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 b \left (-189 a^2 d^2+54 a b c d+b^2 \left (-\left (8 c^2+49 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}-\frac{2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}-\frac{2 \left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (189 a^2 b d^2 \left (c^2+3 d^2\right )+420 a^3 c d^3-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (33 c^2 d^2+8 c^4+147 d^4\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{3/2} \, dx &=-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{2 \int (c+d \sin (e+f x))^{3/2} \left (\frac{1}{2} \left (2 b^3 c+9 a^3 d+5 a b^2 d\right )-\frac{1}{2} b \left (2 a b c-27 a^2 d-7 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-5 a d) \sin ^2(e+f x)\right ) \, dx}{9 d}\\ &=\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{4 \int (c+d \sin (e+f x))^{3/2} \left (-\frac{3}{4} d \left (2 b^3 c-21 a^3 d-45 a b^2 d\right )-\frac{1}{4} b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{63 d^2}\\ &=\frac{2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{8 \int \sqrt{c+d \sin (e+f x)} \left (\frac{3}{8} d \left (105 a^3 c d+171 a b^2 c d+189 a^2 b d^2-b^3 \left (2 c^2-49 d^2\right )\right )+\frac{3}{8} \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \sin (e+f x)\right ) \, dx}{315 d^2}\\ &=-\frac{2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}+\frac{2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{16 \int \frac{\frac{3}{16} d \left (756 a^2 b c d^2+105 a^3 d \left (3 c^2+d^2\right )+9 a b^2 d \left (51 c^2+25 d^2\right )+2 b^3 \left (c^3+93 c d^2\right )\right )+\frac{3}{16} \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{945 d^2}\\ &=-\frac{2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}+\frac{2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}-\frac{\left (\left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{315 d^3}+\frac{\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{315 d^3}\\ &=-\frac{2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}+\frac{2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{\left (\left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{315 d^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (\left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\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}{315 d^3 \sqrt{c+d \sin (e+f x)}}\\ &=-\frac{2 \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{315 d^2 f}+\frac{2 b \left (54 a b c d-189 a^2 d^2-b^2 \left (8 c^2+49 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d^2 f}+\frac{8 b^2 (b c-5 a d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{5/2}}{9 d f}+\frac{2 \left (420 a^3 c d^3+189 a^2 b d^2 \left (c^2+3 d^2\right )-a b^2 \left (54 c^3 d-738 c d^3\right )+b^3 \left (8 c^4+33 c^2 d^2+147 d^4\right )\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)}}{315 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (c^2-d^2\right ) \left (189 a^2 b c d^2+105 a^3 d^3-9 a b^2 d \left (6 c^2-25 d^2\right )+b^3 \left (8 c^3+39 c d^2\right )\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}}}{315 d^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 2.36794, size = 410, normalized size = 0.83 \[ \frac{d (c+d \sin (e+f x)) \left (b d \left (10 b d (27 a d+10 b c) \cos (3 (e+f x))-2 \sin (2 (e+f x)) \left (378 a^2 d^2+432 a b c d+b^2 \left (6 c^2+133 d^2\right )-35 b^2 d^2 \cos (2 (e+f x))\right )\right )-2 \left (1512 a^2 b c d^2+420 a^3 d^3+9 a b^2 d \left (12 c^2+115 d^2\right )+b^3 \left (402 c d^2-16 c^3\right )\right ) \cos (e+f x)\right )-8 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (756 a^2 b c d^2+105 a^3 d \left (3 c^2+d^2\right )+9 a b^2 d \left (51 c^2+25 d^2\right )+2 b^3 \left (c^3+93 c d^2\right )\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (189 a^2 b d^2 \left (c^2+3 d^2\right )+420 a^3 c d^3+a b^2 \left (738 c d^3-54 c^3 d\right )+b^3 \left (33 c^2 d^2+8 c^4+147 d^4\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )}{1260 d^3 f \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 5.964, size = 2112, normalized size = 4.3 \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{\left (b \sin \left (f x + e\right ) + a\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\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 ({\left (b^{3} d \cos \left (f x + e\right )^{4} -{\left (3 \, a b^{2} c +{\left (3 \, a^{2} b + 2 \, b^{3}\right )} d\right )} \cos \left (f x + e\right )^{2} +{\left (a^{3} + 3 \, a b^{2}\right )} c +{\left (3 \, a^{2} b + b^{3}\right )} d -{\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} \cos \left (f x + e\right )^{2} -{\left (3 \, a^{2} b + b^{3}\right )} c -{\left (a^{3} + 3 \, a b^{2}\right )} d\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}, 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(-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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