Optimal. Leaf size=109 \[ \frac {64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.26, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2736, 2674, 2673} \[ \frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}+\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rule 2736
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{9} \left (8 a^2 c^3\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{63} \left (32 a^2 c^4\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 5.69, size = 96, normalized size = 0.88 \[ -\frac {a^2 c^2 \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5 (220 \sin (e+f x)+35 \cos (2 (e+f x))-249)}{315 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 201, normalized size = 1.84 \[ \frac {2 \, {\left (35 \, a^{2} c^{2} \cos \left (f x + e\right )^{5} - 5 \, a^{2} c^{2} \cos \left (f x + e\right )^{4} + 8 \, a^{2} c^{2} \cos \left (f x + e\right )^{3} - 16 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} + 64 \, a^{2} c^{2} \cos \left (f x + e\right ) + 128 \, a^{2} c^{2} + {\left (35 \, a^{2} c^{2} \cos \left (f x + e\right )^{4} + 40 \, a^{2} c^{2} \cos \left (f x + e\right )^{3} + 48 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} + 64 \, a^{2} c^{2} \cos \left (f x + e\right ) + 128 \, a^{2} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{315 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \]
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 = 0.84, size = 71, normalized size = 0.65 \[ -\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (35 \left (\sin ^{2}\left (f x +e \right )\right )-110 \sin \left (f x +e \right )+107\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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 \[ a^{2} \left (\int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- 2 c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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