Optimal. Leaf size=200 \[ -\frac {2 a^3 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c (A+B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 (A+B) \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}}+\frac {8 \sqrt {2} a^3 (A+B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.52, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2967, 2860, 2679, 2649, 206} \[ -\frac {2 a^3 c^2 (A+B) \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 c (A+B) \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 (A+B) \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}}+\frac {8 \sqrt {2} a^3 (A+B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2679
Rule 2860
Rule 2967
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3 (A+B \sin (e+f x))}{\sqrt {c-c \sin (e+f x)}} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{7/2}} \, dx\\ &=-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}+\left (a^3 (A+B) c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx\\ &=-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}+\left (2 a^3 (A+B) c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\left (4 a^3 (A+B) c\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 (A+B) \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}}+\left (8 a^3 (A+B)\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 (A+B) \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}}-\frac {\left (16 a^3 (A+B)\right ) \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 )}{f}\\ &=\frac {8 \sqrt {2} a^3 (A+B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{\sqrt {c} f}-\frac {2 a^3 B c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^{7/2}}-\frac {2 a^3 (A+B) c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}-\frac {4 a^3 (A+B) c \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\frac {8 a^3 (A+B) \cos (e+f x)}{f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 1.35, size = 193, normalized size = 0.96 \[ -\frac {a^3 (\sin (e+f x)+1)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (-2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (-(448 A+673 B) \sin (e+f x)+6 (7 A+22 B) \cos (2 (e+f x))-2086 A+15 B \sin (3 (e+f x))-2236 B)+(6720+6720 i) \sqrt [4]{-1} (A+B) \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac {1}{4} (e+f x)\right )+1\right )\right )\right )}{420 f \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 353, normalized size = 1.76 \[ \frac {2 \, {\left (\frac {210 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} c \cos \left (f x + e\right ) - {\left (A + B\right )} a^{3} c \sin \left (f x + e\right ) + {\left (A + B\right )} a^{3} c\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} - {\left (15 \, B a^{3} \cos \left (f x + e\right )^{4} - 3 \, {\left (7 \, A + 22 \, B\right )} a^{3} \cos \left (f x + e\right )^{3} - {\left (133 \, A + 253 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} + 4 \, {\left (133 \, A + 148 \, B\right )} a^{3} \cos \left (f x + e\right ) + 4 \, {\left (161 \, A + 191 \, B\right )} a^{3} - {\left (15 \, B a^{3} \cos \left (f x + e\right )^{3} + 3 \, {\left (7 \, A + 27 \, B\right )} a^{3} \cos \left (f x + e\right )^{2} - 4 \, {\left (28 \, A + 43 \, B\right )} a^{3} \cos \left (f x + e\right ) - 4 \, {\left (161 \, A + 191 \, B\right )} a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{105 \, {\left (c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f\right )}} \]
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 = 1.56, size = 233, normalized size = 1.16 \[ -\frac {2 \left (\sin \left (f x +e \right )-1\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, a^{3} \left (420 c^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) A +420 c^{\frac {7}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) B -15 B \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}}-21 A \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c -21 B \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c -70 A \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{2}-70 B \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{2}-420 A \,c^{3} \sqrt {c \left (1+\sin \left (f x +e \right )\right )}-420 B \,c^{3} \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\right )}{105 c^{4} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt {-c \sin \left (f x + e\right ) + c}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]
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 \[ a^{3} \left (\int \frac {A}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 A \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 A \sin ^{2}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {A \sin ^{3}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin {\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 B \sin ^{2}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 B \sin ^{3}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {B \sin ^{4}{\left (e + f x \right )}}{\sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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