Optimal. Leaf size=280 \[ \frac {2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {2 a \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {2 a x \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {2 a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {a x^2 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}} \]
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Rubi [A] time = 2.16, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.424, Rules used = {4604, 6741, 12, 6742, 4186, 3770, 4181, 2531, 2282, 6589, 3757, 4184, 3475, 4413} \[ \frac {2 a x \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {2 a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {2 a \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {a x^2 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2282
Rule 2531
Rule 3475
Rule 3757
Rule 3770
Rule 4181
Rule 4184
Rule 4186
Rule 4413
Rule 4604
Rule 6589
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx &=\frac {\cos (e+f x) \int x^2 \sec ^3(e+f x) (a-a \sin (e+f x))^2 \, dx}{a c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {\cos (e+f x) \int a^2 x^2 \sec ^3(e+f x) (1-\sin (e+f x))^2 \, dx}{a c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {(a \cos (e+f x)) \int x^2 \sec ^3(e+f x) (1-\sin (e+f x))^2 \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {(a \cos (e+f x)) \int \left (x^2 \sec ^3(e+f x)-2 x^2 \sec ^2(e+f x) \tan (e+f x)+x^2 \sec (e+f x) \tan ^2(e+f x)\right ) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {(a \cos (e+f x)) \int x^2 \sec ^3(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^2 \sec (e+f x) \tan ^2(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(2 a \cos (e+f x)) \int x^2 \sec ^2(e+f x) \tan (e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^2 \tan (e+f x)}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^2 \sec (e+f x) \, dx}{2 c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \int x^2 \sec (e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^2 \sec ^3(e+f x) \, dx}{c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int \sec (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(2 a \cos (e+f x)) \int x \sec ^2(e+f x) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {2 a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a x^2 \tan ^{-1}\left (e^{i (e+f x)}\right ) \cos (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a x \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^2 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x^2 \sec (e+f x) \, dx}{2 c \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int \sec (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(2 a \cos (e+f x)) \int \tan (e+f x) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \int x \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(2 a \cos (e+f x)) \int x \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(2 a \cos (e+f x)) \int x \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {2 a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {i a x \cos (e+f x) \text {Li}_2\left (-i e^{i (e+f x)}\right )}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a x \cos (e+f x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a x \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^2 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(i a \cos (e+f x)) \int \text {Li}_2\left (-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(i a \cos (e+f x)) \int \text {Li}_2\left (i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(2 i a \cos (e+f x)) \int \text {Li}_2\left (-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(2 i a \cos (e+f x)) \int \text {Li}_2\left (i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \int x \log \left (1-i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int x \log \left (1+i e^{i (e+f x)}\right ) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {2 a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a x \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^2 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(2 a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(2 a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(i a \cos (e+f x)) \int \text {Li}_2\left (-i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(i a \cos (e+f x)) \int \text {Li}_2\left (i e^{i (e+f x)}\right ) \, dx}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {2 a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a \cos (e+f x) \text {Li}_3\left (-i e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a \cos (e+f x) \text {Li}_3\left (i e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a x \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^2 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {2 a x}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \tanh ^{-1}(\sin (e+f x)) \cos (e+f x)}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x^2 \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {2 a x \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x^2 \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 1.56, size = 154, normalized size = 0.55 \[ -\frac {\sqrt {a-a \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (-4 \log \left (e^{i (e+f x)}+i\right )+2 f x \cos (e+f x)+\left (2 i f x-4 \log \left (e^{i (e+f x)}+i\right )\right ) \sin (e+f x)+f^2 x^2+2 i f x\right )}{f^3 (c (\sin (e+f x)+1))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x^{2}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sqrt {a -a \sin \left (f x +e \right )}}{\left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x^{2}}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\sqrt {a-a\,\sin \left (e+f\,x\right )}}{{\left (c+c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}{\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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