Optimal. Leaf size=163 \[ -\frac {a (A-3 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}-\frac {a (A-3 B) \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2967, 2857, 2750, 2650, 2649, 206} \[ -\frac {a (A-3 B) \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {a (A-3 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}-\frac {a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}+\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2750
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))^{7/2}} \, dx &=(a c) \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{9/2}} \, dx\\ &=\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}+\frac {a \int \frac {-A c-7 B c-6 B c \sin (e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx}{6 c^2}\\ &=\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {(a (A-3 B)) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{16 c^2}\\ &=\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a (A-3 B) \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {(a (A-3 B)) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{64 c^3}\\ &=\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a (A-3 B) \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {(a (A-3 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 )}{32 c^3 f}\\ &=-\frac {a (A-3 B) \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{32 \sqrt {2} c^{7/2} f}+\frac {a (A+B) \cos (e+f x)}{3 f (c-c \sin (e+f x))^{7/2}}-\frac {a (A+13 B) \cos (e+f x)}{24 c f (c-c \sin (e+f x))^{5/2}}-\frac {a (A-3 B) \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 3.45, size = 217, normalized size = 1.33 \[ -\frac {a (\sin (e+f x)-1) (\sin (e+f x)+1) \left (\frac {\sqrt {c} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (4 (5 A+17 B) \sin (e+f x)+3 (A-3 B) \cos (2 (e+f x))+47 A-13 B)}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+3 \sqrt {2} (A-3 B) \sec (e+f x) \sqrt {-c (\sin (e+f x)+1)} \tan ^{-1}\left (\frac {\sqrt {-c (\sin (e+f x)+1)}}{\sqrt {2} \sqrt {c}}\right )\right )}{192 c^{7/2} f \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.45, size = 490, normalized size = 3.01 \[ -\frac {3 \, \sqrt {2} {\left ({\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{4} - 3 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{3} - 8 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{2} + 4 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right ) + 8 \, {\left (A - 3 \, B\right )} a + {\left ({\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{3} + 4 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{2} - 4 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right ) - 8 \, {\left (A - 3 \, B\right )} a\right )} \sin \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 (3 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{3} - {\left (7 \, A + 43 \, B\right )} a \cos \left (f x + e\right )^{2} + 2 \, {\left (11 \, A - B\right )} a \cos \left (f x + e\right ) + 32 \, {\left (A + B\right )} a + {\left (3 \, {\left (A - 3 \, B\right )} a \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, A + 17 \, B\right )} a \cos \left (f x + e\right ) + 32 \, {\left (A + B\right )} a\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{384 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} 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 = 1.73, size = 352, normalized size = 2.16 \[ \frac {a \left (-3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} \left (A -3 B \right ) \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+12 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} \left (A -3 B \right ) \sin \left (f x +e \right )+9 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4} \left (A -3 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+24 A \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}+32 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}-6 A \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}-72 B \sqrt {c +c \sin \left (f x +e \right )}\, c^{\frac {7}{2}}+32 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {3}{2}} c^{\frac {5}{2}}+18 B \left (c +c \sin \left (f x +e \right )\right )^{\frac {5}{2}} c^{\frac {3}{2}}-12 A \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}+36 B \sqrt {2}\, \arctanh \left (\frac {\sqrt {c +c \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{4}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{192 c^{\frac {15}{2}} \left (\sin \left (f x +e \right )-1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,\left (a+a\,\sin \left (e+f\,x\right )\right )}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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