Optimal. Leaf size=136 \[ \frac {(3 A-B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {(3 A-B) \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-B) \sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.33, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2967, 2855, 2650, 2649, 206} \[ \frac {(3 A-B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {(3 A-B) \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-B) \sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2855
Rule 2967
Rubi steps
\begin {align*} \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx &=\frac {\int \frac {\sec ^2(e+f x) (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx}{a c}\\ &=-\frac {(A-B) \sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {(3 A-B) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{2 a}\\ &=\frac {(3 A-B) \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-B) \sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {(3 A-B) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{8 a c}\\ &=\frac {(3 A-B) \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-B) \sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}-\frac {(3 A-B) \operatorname {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{4 a c f}\\ &=\frac {(3 A-B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {(3 A-B) \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {(A-B) \sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.54, size = 284, normalized size = 2.09 \[ \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 (B-A) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+(A+B) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 (A+B) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-(1+i) \sqrt [4]{-1} (3 A-B) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac {1}{4} (e+f x)\right )+1\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2\right )}{4 a f (\sin (e+f x)+1) (c-c \sin (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 231, normalized size = 1.70 \[ -\frac {\sqrt {2} {\left ({\left (3 \, A - B\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - {\left (3 \, A - B\right )} \cos \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left ({\left (3 \, A - B\right )} \sin \left (f x + e\right ) - A + 3 \, B\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{16 \, {\left (a c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c^{2} f \cos \left (f x + e\right )\right )}} \]
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 = 1.28, size = 225, normalized size = 1.65 \[ -\frac {\sin \left (f x +e \right ) \left (3 A \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -B \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -6 A \,c^{\frac {3}{2}}+2 B \,c^{\frac {3}{2}}\right )-3 A \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +B \sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 A \,c^{\frac {3}{2}}-6 B \,c^{\frac {3}{2}}}{8 c^{\frac {5}{2}} a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {B \sin \left (f x + e\right ) + A}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {A+B\,\sin \left (e+f\,x\right )}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {A}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin {\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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