Optimal. Leaf size=228 \[ \frac {a^{3/2} (c-11 d) (c+d)^2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{3/2} f}+\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.29, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2842, 21, 2849,
2854, 211} \begin {gather*} \frac {a^{3/2} (c-11 d) (c+d)^2 \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{3/2} f}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a \sin (e+f x)+a}}+\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 211
Rule 2842
Rule 2849
Rule 2854
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{3/2} \, dx &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\left (-\frac {1}{2} a^2 (c-11 d)-\frac {1}{2} a^2 (c-11 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 d}\\ &=-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-11 d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx}{6 d}\\ &=\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-11 d) (c+d)) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{8 d}\\ &=\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}-\frac {\left (a (c-11 d) (c+d)^2\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{16 d}\\ &=\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c-11 d) (c+d)^2\right ) \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d f}\\ &=\frac {a^{3/2} (c-11 d) (c+d)^2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{3/2} f}+\frac {a^2 (c-11 d) (c+d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-11 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{3 d f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 281, normalized size = 1.23 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{3/2} \left (\frac {(c-11 d) (c+d)^2 \left (-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )+\log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right )\right )}{d^{3/2}}-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (3 c^2+52 c d+37 d^2-4 d^2 \cos (2 (e+f x))+2 d (7 c+11 d) \sin (e+f x)\right )}{3 d}\right )}{16 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 459 vs.
\(2 (208) = 416\).
time = 0.76, size = 1391, normalized size = 6.10 \begin {gather*} \left [-\frac {3 \, {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3} + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \cos \left (f x + e\right ) + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{d}} \log \left (\frac {128 \, a d^{4} \cos \left (f x + e\right )^{5} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} + 128 \, {\left (2 \, a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{4} - 32 \, {\left (5 \, a c^{2} d^{2} - 14 \, a c d^{3} + 13 \, a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (a c^{3} d - 2 \, a c^{2} d^{2} + 9 \, a c d^{3} - 4 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, d^{4} \cos \left (f x + e\right )^{4} - c^{3} d + 17 \, c^{2} d^{2} - 59 \, c d^{3} + 51 \, d^{4} + 24 \, {\left (c d^{3} - d^{4}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{2} d^{2} - 26 \, c d^{3} + 33 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - {\left (c^{3} d - 7 \, c^{2} d^{2} + 31 \, c d^{3} - 25 \, d^{4}\right )} \cos \left (f x + e\right ) + {\left (16 \, d^{4} \cos \left (f x + e\right )^{3} + c^{3} d - 17 \, c^{2} d^{2} + 59 \, c d^{3} - 51 \, d^{4} - 8 \, {\left (3 \, c d^{3} - 5 \, d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, c^{2} d^{2} - 14 \, c d^{3} + 13 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {-\frac {a}{d}} + {\left (a c^{4} - 28 \, a c^{3} d + 230 \, a c^{2} d^{2} - 476 \, a c d^{3} + 289 \, a d^{4}\right )} \cos \left (f x + e\right ) + {\left (128 \, a d^{4} \cos \left (f x + e\right )^{4} + a c^{4} + 4 \, a c^{3} d + 6 \, a c^{2} d^{2} + 4 \, a c d^{3} + a d^{4} - 256 \, {\left (a c d^{3} - a d^{4}\right )} \cos \left (f x + e\right )^{3} - 32 \, {\left (5 \, a c^{2} d^{2} - 6 \, a c d^{3} + 5 \, a d^{4}\right )} \cos \left (f x + e\right )^{2} + 32 \, {\left (a c^{3} d - 7 \, a c^{2} d^{2} + 15 \, a c d^{3} - 9 \, a d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right ) - 8 \, {\left (8 \, a d^{2} \cos \left (f x + e\right )^{3} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} - 14 \, {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (3 \, a c^{2} + 52 \, a c d + 41 \, a d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, a d^{2} \cos \left (f x + e\right )^{2} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} + 2 \, {\left (7 \, a c d + 11 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{192 \, {\left (d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f\right )}}, -\frac {3 \, {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3} + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \cos \left (f x + e\right ) + {\left (a c^{3} - 9 \, a c^{2} d - 21 \, a c d^{2} - 11 \, a d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{d}} \arctan \left (\frac {{\left (8 \, d^{2} \cos \left (f x + e\right )^{2} - c^{2} + 6 \, c d - 9 \, d^{2} - 8 \, {\left (c d - d^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c} \sqrt {\frac {a}{d}}}{4 \, {\left (2 \, a d^{2} \cos \left (f x + e\right )^{3} - {\left (3 \, a c d - a d^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (a c^{2} - a c d + 2 \, a d^{2}\right )} \cos \left (f x + e\right )\right )}}\right ) - 4 \, {\left (8 \, a d^{2} \cos \left (f x + e\right )^{3} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} - 14 \, {\left (a c d + a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (3 \, a c^{2} + 52 \, a c d + 41 \, a d^{2}\right )} \cos \left (f x + e\right ) - {\left (8 \, a d^{2} \cos \left (f x + e\right )^{2} - 3 \, a c^{2} - 38 \, a c d - 19 \, a d^{2} + 2 \, {\left (7 \, a c d + 11 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{96 \, {\left (d f \cos \left (f x + e\right ) + d f \sin \left (f x + e\right ) + d f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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