Optimal. Leaf size=115 \[ -\frac {a (A+5 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} c^{3/2} f}+\frac {a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac {2 a B \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.32, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2967, 2857, 2751, 2649, 206} \[ -\frac {a (A+5 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} c^{3/2} f}+\frac {a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac {2 a B \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2751
Rule 2857
Rule 2967
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{3/2}} \, dx &=(a c) \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac {a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac {a \int \frac {-A c-3 B c-2 B c \sin (e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{2 c^2}\\ &=\frac {a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac {2 a B \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}-\frac {(a (A+5 B)) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{2 c}\\ &=\frac {a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac {2 a B \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}+\frac {(a (A+5 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 )}{c f}\\ &=-\frac {a (A+5 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {2} c^{3/2} f}+\frac {a (A+B) \cos (e+f x)}{f (c-c \sin (e+f x))^{3/2}}+\frac {2 a B \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.61, size = 157, normalized size = 1.37 \[ \frac {a \sec (e+f x) \left (2 \sqrt {c} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 (A-2 B \sin (e+f x)+3 B)+\sqrt {2} (A+5 B) \sqrt {-c (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \tan ^{-1}\left (\frac {\sqrt {-c (\sin (e+f x)+1)}}{\sqrt {2} \sqrt {c}}\right )\right )}{2 c^{3/2} f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 318, normalized size = 2.77 \[ \frac {\frac {\sqrt {2} {\left ({\left (A + 5 \, B\right )} a c \cos \left (f x + e\right )^{2} - {\left (A + 5 \, B\right )} a c \cos \left (f x + e\right ) - 2 \, {\left (A + 5 \, B\right )} a c + {\left ({\left (A + 5 \, B\right )} a c \cos \left (f x + e\right ) + 2 \, {\left (A + 5 \, B\right )} a c\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} - 4 \, {\left (2 \, B a \cos \left (f x + e\right )^{2} + {\left (A + 3 \, B\right )} a \cos \left (f x + e\right ) + {\left (A + B\right )} a - {\left (2 \, B a \cos \left (f x + e\right ) - {\left (A + B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{4 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f + {\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
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 [B] time = 1.24, size = 227, normalized size = 1.97 \[ \frac {a \left (A \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c +5 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c -A \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -4 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c}\, B \sin \left (f x +e \right )-5 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +2 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c}\, A +6 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c}\, B \right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{2 c^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ a \left (\int \frac {A}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {A \sin {\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin {\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin ^{2}{\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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