Optimal. Leaf size=471 \[ -\frac {22 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5525 c^7 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac {44 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}} \]
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Rubi [A] time = 2.49, antiderivative size = 471, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2850, 2852, 2842, 2640, 2639} \[ \frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac {44 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {22 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5525 c^7 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2842
Rule 2850
Rule 2852
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{15/2}} \, dx &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {(3 a) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx}{5 c}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {\left (11 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{35 c^2}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {\left (11 a^3\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx}{85 c^3}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {\left (33 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{1105 c^4}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {\left (11 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{1105 c^5}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {\left (11 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{5525 c^6}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{5525 c^7}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{5525 c^7 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5525 c^7 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{25 f g (c-c \sin (e+f x))^{15/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{35 c f g (c-c \sin (e+f x))^{13/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{595 c^2 f g (c-c \sin (e+f x))^{11/2}}-\frac {44 a^4 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{5525 c^6 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {22 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5525 c^7 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.87, size = 668, normalized size = 1.42 \[ \frac {\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{15} \left (\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {4288 \sin \left (\frac {1}{2} (e+f x)\right )}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {9312 \sin \left (\frac {1}{2} (e+f x)\right )}{2975 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {832 \sin \left (\frac {1}{2} (e+f x)\right )}{175 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}}+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{25 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{13}}+\frac {22}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {22}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {2144}{5525 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {4656}{2975 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}-\frac {416}{175 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}}+\frac {32}{25 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{12}}+\frac {22}{5525}\right )}{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 {22 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{15}}{5525 f \cos ^{\frac {3}{2}}(e+f x) (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 [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right ) + {\left (a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{c^{8} \cos \left (f x + e\right )^{8} - 32 \, c^{8} \cos \left (f x + e\right )^{6} + 160 \, c^{8} \cos \left (f x + e\right )^{4} - 256 \, c^{8} \cos \left (f x + e\right )^{2} + 128 \, c^{8} + 8 \, {\left (c^{8} \cos \left (f x + e\right )^{6} - 10 \, c^{8} \cos \left (f x + e\right )^{4} + 24 \, c^{8} \cos \left (f x + e\right )^{2} - 16 \, c^{8}\right )} \sin \left (f x + e\right )}, x\right ) \]
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 [C] time = 0.74, size = 1644, normalized size = 3.49 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{15/2}} \,d x \]
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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