Optimal. Leaf size=171 \[ \frac {a \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {a x \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a x \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}} \]
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Rubi [A] time = 0.97, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {4604, 6741, 12, 6742, 4185, 4181, 2279, 2391, 3757, 3767, 8, 4413} \[ \frac {a \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}+\frac {a x \tan (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}}-\frac {a x \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c \sin (e+f x)+c}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2279
Rule 2391
Rule 3757
Rule 3767
Rule 4181
Rule 4185
Rule 4413
Rule 4604
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int \frac {x \sqrt {a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx &=\frac {\cos (e+f x) \int x \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 \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 \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 \sec ^3(e+f x)-2 x \sec ^2(e+f x) \tan (e+f x)+x \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 \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 \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 \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}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x \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 \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 \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 \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 ^2(e+f x) \, dx}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {a}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a x \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 x \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x \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 \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)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(a \cos (e+f x)) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int \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 \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}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x \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)) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(i a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(i a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,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 \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,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 \cos (e+f x)) \int \log \left (1-i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {(a \cos (e+f x)) \int \log \left (1+i e^{i (e+f x)}\right ) \, dx}{2 c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {a}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {i a \cos (e+f x) \text {Li}_2\left (-i e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {i a \cos (e+f x) \text {Li}_2\left (i e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x \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)) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {(i a \cos (e+f x)) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{2 c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=-\frac {a}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}-\frac {a x \sec (e+f x)}{c f \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a \sin (e+f x)}{c f^2 \sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}+\frac {a x \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 [A] time = 0.60, size = 150, normalized size = 0.88 \[ -\frac {\sqrt {a-a \sin (e+f x)} \sqrt {c (\sin (e+f x)+1)} \left (f x \sin \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}+f x\right )+\cos \left (\frac {e}{2}\right ) (f x-1)+\cos \left (\frac {e}{2}+f x\right )+\sin \left (\frac {e}{2}\right )\right )}{c^2 f^2 \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 72, normalized size = 0.42 \[ -\frac {{\left (f x + \cos \left (f x + e\right )\right )} \sqrt {-a \sin \left (f x + e\right ) + a} \sqrt {c \sin \left (f x + e\right ) + c}}{c^{2} f^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + c^{2} f^{2} \cos \left (f x + e\right )} \]
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 \[ \int \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x \sqrt {a -a \sin \left (f x +e \right )}}{\left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\sqrt {-a \sin \left (f x + e\right ) + a} x}{{\left (c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.32, size = 88, normalized size = 0.51 \[ -\frac {\sqrt {-a\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (\cos \left (2\,e+2\,f\,x\right )+2\,f\,x\,\cos \left (e+f\,x\right )+1-\cos \left (e+f\,x\right )\,2{}\mathrm {i}-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{c\,f^2\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,\sqrt {c\,\left (\sin \left (e+f\,x\right )+1\right )}} \]
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 \[ \int \frac {x \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right )}}{\left (c \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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