Optimal. Leaf size=167 \[ \frac {(A+C) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {(A+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.65, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {3036, 2969, 2737, 2667, 31} \[ \frac {(A+C) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {(A+C) \cos (e+f x) \log (\sin (e+f x)+1)}{4 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2667
Rule 2737
Rule 2969
Rule 3036
Rubi steps
\begin {align*} \int \frac {A+C \sin ^2(e+f x)}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx &=\frac {(A+C) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\int \frac {-2 a^2 (A-C)+4 a^2 C \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{4 a^2 c}\\ &=\frac {(A+C) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac {(A-3 C) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{4 a c}+\frac {(A+C) \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx}{4 c^2}\\ &=\frac {(A+C) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}+\frac {((A-3 C) \cos (e+f x)) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{4 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {(a (A+C) \cos (e+f x)) \int \frac {\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{4 c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {(A+C) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {((A-3 C) \cos (e+f x)) \operatorname {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{4 c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {((A+C) \cos (e+f x)) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (e+f x)\right )}{4 c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {(A+C) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-3 C) \cos (e+f x) \log (1-\sin (e+f x))}{4 c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {(A+C) \cos (e+f x) \log (1+\sin (e+f x))}{4 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 = 0.62, size = 190, normalized size = 1.14 \[ \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (-\left ((A-3 C) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )\right )+(A+C) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+A+C\right )}{2 f \sqrt {a (\sin (e+f x)+1)} (c-c \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (f x + e\right )^{2} - A - C\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{a c^{2} \cos \left (f x + e\right )^{2} \sin \left (f x + e\right ) - a c^{2} \cos \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {C \sin \left (f x + e\right )^{2} + A}{\sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.83, size = 343, normalized size = 2.05 \[ -\frac {\left (A \sin \left (f x +e \right ) \ln \left (\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-A \sin \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-2 C \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )+C \ln \left (\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \sin \left (f x +e \right )+3 C \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right ) \sin \left (f x +e \right )-A \sin \left (f x +e \right )-A \ln \left (\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+A \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-C \sin \left (f x +e \right )+2 C \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-C \ln \left (\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-3 C \ln \left (-\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )\right ) \cos \left (f x +e \right )}{2 f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {C \sin \left (f x + e\right )^{2} + A}{\sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {C\,{\sin \left (e+f\,x\right )}^2+A}{\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {A + C \sin ^{2}{\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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