Optimal. Leaf size=231 \[ \frac {4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a \sin (e+f x)+a}}+\frac {4 d (c-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{315 f} \]
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Rubi [A] time = 0.38, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2763, 21, 2770, 2761, 2751, 2646} \[ \frac {4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt {a \sin (e+f x)+a}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a \sin (e+f x)+a}}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a \sin (e+f x)+a}}+\frac {4 d (c-17 d) (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{105 f}+\frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{315 f} \]
Antiderivative was successfully verified.
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Rule 21
Rule 2646
Rule 2751
Rule 2761
Rule 2763
Rule 2770
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^3 \, dx &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}+\frac {2 \int \frac {\left (-\frac {1}{2} a^2 (c-17 d)-\frac {1}{2} a^2 (c-17 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{\sqrt {a+a \sin (e+f x)}} \, dx}{9 d}\\ &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-17 d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx}{9 d}\\ &=\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {(2 a (c-17 d) (c+d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx}{21 d}\\ &=\frac {4 (c-17 d) d (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {(4 (c-17 d) (c+d)) \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{105 d}\\ &=\frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}+\frac {4 (c-17 d) d (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}-\frac {\left (2 a (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx}{315 d}\\ &=\frac {4 a^2 (c-17 d) (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{315 d f \sqrt {a+a \sin (e+f x)}}+\frac {8 a (c-17 d) (5 c-d) (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{315 f}+\frac {4 (c-17 d) d (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{105 f}+\frac {2 a^2 (c-17 d) \cos (e+f x) (c+d \sin (e+f x))^3}{63 d f \sqrt {a+a \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^4}{9 d f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.69, size = 203, normalized size = 0.88 \[ -\frac {a \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (840 c^3 \sin (e+f x)+4200 c^3-4 d \left (189 c^2+351 c d+137 d^2\right ) \cos (2 (e+f x))+4536 c^2 d \sin (e+f x)+9828 c^2 d+4554 c d^2 \sin (e+f x)-270 c d^2 \sin (3 (e+f x))+8892 c d^2+1598 d^3 \sin (e+f x)-170 d^3 \sin (3 (e+f x))+35 d^3 \cos (4 (e+f x))+2689 d^3\right )}{1260 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 339, normalized size = 1.47 \[ -\frac {2 \, {\left (35 \, a d^{3} \cos \left (f x + e\right )^{5} - 5 \, {\left (27 \, a c d^{2} + 10 \, a d^{3}\right )} \cos \left (f x + e\right )^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} - {\left (189 \, a c^{2} d + 351 \, a c d^{2} + 172 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (105 \, a c^{3} + 378 \, a c^{2} d + 387 \, a c d^{2} + 134 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (525 \, a c^{3} + 1323 \, a c^{2} d + 1287 \, a c d^{2} + 409 \, a d^{3}\right )} \cos \left (f x + e\right ) - {\left (35 \, a d^{3} \cos \left (f x + e\right )^{4} + 420 \, a c^{3} + 756 \, a c^{2} d + 684 \, a c d^{2} + 188 \, a d^{3} + 5 \, {\left (27 \, a c d^{2} + 17 \, a d^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (63 \, a c^{2} d + 72 \, a c d^{2} + 29 \, a d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (105 \, a c^{3} + 567 \, a c^{2} d + 603 \, a c d^{2} + 221 \, a d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{315 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.92, size = 195, normalized size = 0.84 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (35 d^{3} \left (\sin ^{4}\left (f x +e \right )\right )+135 c \,d^{2} \left (\sin ^{3}\left (f x +e \right )\right )+85 d^{3} \left (\sin ^{3}\left (f x +e \right )\right )+189 c^{2} d \left (\sin ^{2}\left (f x +e \right )\right )+351 c \,d^{2} \left (\sin ^{2}\left (f x +e \right )\right )+102 d^{3} \left (\sin ^{2}\left (f x +e \right )\right )+105 c^{3} \sin \left (f x +e \right )+567 c^{2} d \sin \left (f x +e \right )+468 c \,d^{2} \sin \left (f x +e \right )+136 d^{3} \sin \left (f x +e \right )+525 c^{3}+1134 c^{2} d +936 c \,d^{2}+272 d^{3}\right )}{315 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{3}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \]
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 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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