Optimal. Leaf size=178 \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}} \]
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Rubi [A] time = 0.38, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2743, 2742} \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}} \]
Antiderivative was successfully verified.
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Rule 2742
Rule 2743
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {3 \int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{14 c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{28 c^2}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\int \frac {(a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{280 c^3}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{14 f (c-c \sin (e+f x))^{15/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{56 c f (c-c \sin (e+f x))^{13/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{280 c^2 f (c-c \sin (e+f x))^{11/2}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{2240 c^3 f (c-c \sin (e+f x))^{9/2}}\\ \end {align*}
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Mathematica [A] time = 6.69, size = 333, normalized size = 1.87 \[ -\frac {(a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}{4 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {6 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}{5 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {2 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}{f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {8 (a (\sin (e+f x)+1))^{7/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{7 f (c-c \sin (e+f x))^{15/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 192, normalized size = 1.08 \[ \frac {{\left (63 \, a^{3} \cos \left (f x + e\right )^{2} - 76 \, a^{3} + 7 \, {\left (5 \, a^{3} \cos \left (f x + e\right )^{2} - 12 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{140 \, {\left (7 \, c^{8} f \cos \left (f x + e\right )^{7} - 56 \, c^{8} f \cos \left (f x + e\right )^{5} + 112 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right ) - {\left (c^{8} f \cos \left (f x + e\right )^{7} - 24 \, c^{8} f \cos \left (f x + e\right )^{5} + 80 \, c^{8} f \cos \left (f x + e\right )^{3} - 64 \, c^{8} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
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 [A] time = 0.35, size = 302, normalized size = 1.70 \[ -\frac {\sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} \left (13 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+13 \left (\cos ^{7}\left (f x +e \right )\right )+91 \sin \left (f x +e \right ) \left (\cos ^{5}\left (f x +e \right )\right )-104 \left (\cos ^{6}\left (f x +e \right )\right )-403 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-312 \left (\cos ^{5}\left (f x +e \right )\right )-637 \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+1040 \left (\cos ^{4}\left (f x +e \right )\right )+1712 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+1075 \left (\cos ^{3}\left (f x +e \right )\right )+756 \sin \left (f x +e \right ) \cos \left (f x +e \right )-2468 \left (\cos ^{2}\left (f x +e \right )\right )-1672 \sin \left (f x +e \right )-916 \cos \left (f x +e \right )+1672\right )}{140 f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {15}{2}} \left (\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+\cos ^{4}\left (f x +e \right )-4 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \left (\cos ^{3}\left (f x +e \right )\right )-4 \sin \left (f x +e \right ) \cos \left (f x +e \right )-8 \left (\cos ^{2}\left (f x +e \right )\right )+8 \sin \left (f x +e \right )-4 \cos \left (f x +e \right )+8\right )} \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.32, size = 647, normalized size = 3.63 \[ \text {result too large to display} \]
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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