Optimal. Leaf size=154 \[ -\frac {a^2 (3 A-7 B) \cos (e+f x)}{120 c^2 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {a (3 A-7 B) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{40 c f (c-c \sin (e+f x))^{9/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{10 f (c-c \sin (e+f x))^{11/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2972, 2739, 2738} \[ -\frac {a^2 (3 A-7 B) \cos (e+f x)}{120 c^2 f \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {a (3 A-7 B) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{40 c f (c-c \sin (e+f x))^{9/2}}+\frac {(A+B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{10 f (c-c \sin (e+f x))^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2738
Rule 2739
Rule 2972
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x))}{(c-c \sin (e+f x))^{11/2}} \, dx &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{10 f (c-c \sin (e+f x))^{11/2}}+\frac {(3 A-7 B) \int \frac {(a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{10 c}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{10 f (c-c \sin (e+f x))^{11/2}}+\frac {a (3 A-7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{40 c f (c-c \sin (e+f x))^{9/2}}-\frac {(a (3 A-7 B)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{7/2}} \, dx}{40 c^2}\\ &=\frac {(A+B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{10 f (c-c \sin (e+f x))^{11/2}}+\frac {a (3 A-7 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{40 c f (c-c \sin (e+f x))^{9/2}}-\frac {a^2 (3 A-7 B) \cos (e+f x)}{120 c^2 f \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}\\ \end {align*}
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Mathematica [A] time = 1.95, size = 126, normalized size = 0.82 \[ -\frac {a \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (5 (3 A+B) \sin (e+f x)+9 (A+B)-10 B \cos (2 (e+f x)))}{60 c^5 f (\sin (e+f x)-1)^5 \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 155, normalized size = 1.01 \[ -\frac {{\left (20 \, B a \cos \left (f x + e\right )^{2} - 5 \, {\left (3 \, A + B\right )} a \sin \left (f x + e\right ) - {\left (9 \, A + 19 \, B\right )} a\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{60 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{5} - 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} f \cos \left (f x + e\right ) - {\left (c^{6} f \cos \left (f x + e\right )^{5} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} + 16 \, c^{6} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.69, size = 339, normalized size = 2.20 \[ -\frac {\sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (9 A \left (\cos ^{5}\left (f x +e \right )\right )+9 A \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-B \left (\cos ^{5}\left (f x +e \right )\right )-B \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-54 A \left (\cos ^{4}\left (f x +e \right )\right )+45 A \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+6 B \left (\cos ^{4}\left (f x +e \right )\right )-5 B \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-108 A \left (\cos ^{3}\left (f x +e \right )\right )-153 A \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+12 B \left (\cos ^{3}\left (f x +e \right )\right )+17 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+288 A \left (\cos ^{2}\left (f x +e \right )\right )-135 A \sin \left (f x +e \right ) \cos \left (f x +e \right )-52 B \left (\cos ^{2}\left (f x +e \right )\right )+35 B \sin \left (f x +e \right ) \cos \left (f x +e \right )+159 A \cos \left (f x +e \right )+294 A \sin \left (f x +e \right )-11 B \cos \left (f x +e \right )-46 B \sin \left (f x +e \right )-294 A +46 B \right )}{60 f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {11}{2}} \left (\cos ^{2}\left (f x +e \right )+\sin \left (f x +e \right ) \cos \left (f x +e \right )+\cos \left (f x +e \right )-2 \sin \left (f x +e \right )-2\right )} \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 20.23, size = 279, normalized size = 1.81 \[ \frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}\,\left (\frac {a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (A+B\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,48{}\mathrm {i}}{5\,c^6\,f}-\frac {B\,a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,32{}\mathrm {i}}{3\,c^6\,f}+\frac {16\,a\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (e+f\,x\right )\,\left (A\,3{}\mathrm {i}+B\,1{}\mathrm {i}\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}}{3\,c^6\,f}\right )}{\cos \left (e+f\,x\right )\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,264{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (3\,e+3\,f\,x\right )\,220{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\cos \left (5\,e+5\,f\,x\right )\,20{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (2\,e+2\,f\,x\right )\,330{}\mathrm {i}+{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (4\,e+4\,f\,x\right )\,88{}\mathrm {i}-{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sin \left (6\,e+6\,f\,x\right )\,2{}\mathrm {i}} \]
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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