Optimal. Leaf size=157 \[ -\frac {8 a^2 \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x)}{105 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 f}-\frac {4 d (7 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f} \]
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Rubi [A] time = 0.23, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2761, 2751, 2647, 2646} \[ -\frac {8 a^2 \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x)}{105 f \sqrt {a \sin (e+f x)+a}}-\frac {2 a \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 f}-\frac {4 d (7 c-d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 a f} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rule 2761
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^2 \, dx &=-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac {2 \int (a+a \sin (e+f x))^{3/2} \left (\frac {1}{2} a \left (7 c^2+5 d^2\right )+a (7 c-d) d \sin (e+f x)\right ) \, dx}{7 a}\\ &=-\frac {4 (7 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac {1}{35} \left (35 c^2+42 c d+19 d^2\right ) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac {2 a \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {4 (7 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}+\frac {1}{105} \left (4 a \left (35 c^2+42 c d+19 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {8 a^2 \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {2 a \left (35 c^2+42 c d+19 d^2\right ) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {4 (7 c-d) d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 a f}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 136, normalized size = 0.87 \[ -\frac {a \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\left (140 c^2+504 c d+253 d^2\right ) \sin (e+f x)+700 c^2-6 d (14 c+13 d) \cos (2 (e+f x))+1092 c d-15 d^2 \sin (3 (e+f x))+494 d^2\right )}{210 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 229, normalized size = 1.46 \[ \frac {2 \, {\left (15 \, a d^{2} \cos \left (f x + e\right )^{4} + 3 \, {\left (14 \, a c d + 13 \, a d^{2}\right )} \cos \left (f x + e\right )^{3} - 140 \, a c^{2} - 168 \, a c d - 76 \, a d^{2} - {\left (35 \, a c^{2} + 84 \, a c d + 43 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (175 \, a c^{2} + 294 \, a c d + 143 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (15 \, a d^{2} \cos \left (f x + e\right )^{3} + 140 \, a c^{2} + 168 \, a c d + 76 \, a d^{2} - 6 \, {\left (7 \, a c d + 4 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, a c^{2} + 126 \, a c d + 67 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{105 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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 = 0.89, size = 130, normalized size = 0.83 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{2} \left (\sin \left (f x +e \right )-1\right ) \left (15 d^{2} \left (\sin ^{3}\left (f x +e \right )\right )+42 c d \left (\sin ^{2}\left (f x +e \right )\right )+39 d^{2} \left (\sin ^{2}\left (f x +e \right )\right )+35 c^{2} \sin \left (f x +e \right )+126 c d \sin \left (f x +e \right )+52 d^{2} \sin \left (f x +e \right )+175 c^{2}+252 c d +104 d^{2}\right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2 \,d x \]
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 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sin {\left (e + f x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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