Optimal. Leaf size=870 \[ -\frac{\left (-\left (3 c^2+14 d c+16 d^2\right ) b^2-6 a d (4 c+d) b+3 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) (a+b)^{3/2}}{24 b^2 d \sqrt{c+d} f}+\frac{(c-d) \sqrt{c+d} \left (\left (3 c^2+16 d^2\right ) b^2+38 a c d b+3 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) \sqrt{a+b}}{24 b d (b c-a d) f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{\left (\left (3 c^2+16 d^2\right ) b^2+38 a c d b+3 a^2 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{c+d} (b c+a d) \left (-\left (c^2-12 d^2\right ) b^2+10 a c d b-a^2 d^2\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^2 d^2 f \sqrt{a+b}} \]
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Rubi [A] time = 3.35895, antiderivative size = 870, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2821, 3049, 3061, 3053, 2811, 2998, 2818, 2996} \[ -\frac{\left (-\left (3 c^2+14 d c+16 d^2\right ) b^2-6 a d (4 c+d) b+3 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x)) (a+b)^{3/2}}{24 b^2 d \sqrt{c+d} f}+\frac{(c-d) \sqrt{c+d} \left (\left (3 c^2+16 d^2\right ) b^2+38 a c d b+3 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x)) \sqrt{a+b}}{24 b d (b c-a d) f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{\left (\left (3 c^2+16 d^2\right ) b^2+38 a c d b+3 a^2 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{c+d} (b c+a d) \left (-\left (c^2-12 d^2\right ) b^2+10 a c d b-a^2 d^2\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^2 d^2 f \sqrt{a+b}} \]
Antiderivative was successfully verified.
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Rule 2821
Rule 3049
Rule 3061
Rule 3053
Rule 2811
Rule 2998
Rule 2818
Rule 2996
Rubi steps
\begin{align*} \int (a+b \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx &=-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac{\int \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{1}{2} d \left (6 a^2 c+b^2 c+3 a b d\right )+d \left (5 a b c+3 a^2 d+2 b^2 d\right ) \sin (e+f x)+\frac{1}{2} b d (3 b c+7 a d) \sin ^2(e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx}{3 d}\\ &=-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac{\int \frac{\frac{1}{4} b d \left (7 b^2 c^2+22 a b c d+a^2 \left (24 c^2+7 d^2\right )\right )+\frac{1}{2} b d (b c+a d) (17 a c+13 b d) \sin (e+f x)+\frac{1}{4} b d \left (38 a b c d+3 a^2 d^2+b^2 \left (3 c^2+16 d^2\right )\right ) \sin ^2(e+f x)}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{6 b d}\\ &=-\frac{\left (38 a b c d+3 a^2 d^2+b^2 \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac{\int \frac{\frac{1}{4} b d \left (79 a^2 b c d^2+a^3 d \left (48 c^2+17 d^2\right )-b^3 \left (3 c^3+16 c d^2\right )-a b^2 \left (21 c^2 d-16 d^3\right )\right )+\frac{1}{2} b d \left (7 b^3 c^2 d+31 a^3 c d^2-a b^2 c \left (3 c^2-32 d^2\right )+a^2 b d \left (20 c^2+33 d^2\right )\right ) \sin (e+f x)-\frac{3}{4} b d (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2-12 b^2 d^2\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx}{12 b d^2}\\ &=-\frac{\left (38 a b c d+3 a^2 d^2+b^2 \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac{\int \frac{\frac{3}{4} a^2 b d (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2-12 b^2 d^2\right )+\frac{1}{4} b^3 d \left (79 a^2 b c d^2+a^3 d \left (48 c^2+17 d^2\right )-b^3 \left (3 c^3+16 c d^2\right )-a b^2 \left (21 c^2 d-16 d^3\right )\right )+b \left (\frac{3}{2} a b d (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2-12 b^2 d^2\right )+\frac{1}{2} b^2 d \left (7 b^3 c^2 d+31 a^3 c d^2-a b^2 c \left (3 c^2-32 d^2\right )+a^2 b d \left (20 c^2+33 d^2\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx}{12 b^3 d^2}+\frac{\left ((b c+a d) \left (10 a b c d-a^2 d^2-b^2 \left (c^2-12 d^2\right )\right )\right ) \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx}{16 b^2 d}\\ &=\frac{\sqrt{c+d} (b c+a d) \left (10 a b c d-a^2 d^2-b^2 \left (c^2-12 d^2\right )\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^2 \sqrt{a+b} d^2 f}-\frac{\left (38 a b c d+3 a^2 d^2+b^2 \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}+\frac{\left (\frac{3}{4} a^2 b d (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2-12 b^2 d^2\right )+\frac{1}{4} b^3 d \left (79 a^2 b c d^2+a^3 d \left (48 c^2+17 d^2\right )-b^3 \left (3 c^3+16 c d^2\right )-a b^2 \left (21 c^2 d-16 d^3\right )\right )-b \left (\frac{3}{2} a b d (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2-12 b^2 d^2\right )+\frac{1}{2} b^2 d \left (7 b^3 c^2 d+31 a^3 c d^2-a b^2 c \left (3 c^2-32 d^2\right )+a^2 b d \left (20 c^2+33 d^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{12 (a-b) b^3 d^2}-\frac{\left (-a b \left (\frac{3}{2} a b d (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2-12 b^2 d^2\right )+\frac{1}{2} b^2 d \left (7 b^3 c^2 d+31 a^3 c d^2-a b^2 c \left (3 c^2-32 d^2\right )+a^2 b d \left (20 c^2+33 d^2\right )\right )\right )+b \left (\frac{3}{4} a^2 b d (b c+a d) \left (b^2 c^2-10 a b c d+a^2 d^2-12 b^2 d^2\right )+\frac{1}{4} b^3 d \left (79 a^2 b c d^2+a^3 d \left (48 c^2+17 d^2\right )-b^3 \left (3 c^3+16 c d^2\right )-a b^2 \left (21 c^2 d-16 d^3\right )\right )\right )\right ) \int \frac{1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx}{12 (a-b) b^3 d^2}\\ &=\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (38 a b c d+3 a^2 d^2+b^2 \left (3 c^2+16 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b d (b c-a d) f}+\frac{\sqrt{c+d} (b c+a d) \left (10 a b c d-a^2 d^2-b^2 \left (c^2-12 d^2\right )\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^2 \sqrt{a+b} d^2 f}-\frac{\left (38 a b c d+3 a^2 d^2+b^2 \left (3 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d f \sqrt{a+b \sin (e+f x)}}-\frac{(3 b c+7 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 f}-\frac{(a+b)^{3/2} \left (3 a^2 d^2-6 a b d (4 c+d)-b^2 \left (3 c^2+14 c d+16 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^2 d \sqrt{c+d} f}-\frac{b \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 f}\\ \end{align*}
Mathematica [B] time = 6.2906, size = 1922, normalized size = 2.21 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 8.178, size = 409584, normalized size = 470.8 \begin{align*} \text{output 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 )}^{\frac{3}{2}}{\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(-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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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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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