Optimal. Leaf size=193 \[ \frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}-\frac {8 a b E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{5 d f g^4 \sqrt {\sin (2 e+2 f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}} \]
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Rubi [A] time = 0.47, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {2888, 2838, 2563, 2571, 2572, 2639} \[ \frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}-\frac {8 a b E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{5 d f g^4 \sqrt {\sin (2 e+2 f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2563
Rule 2571
Rule 2572
Rule 2639
Rule 2838
Rule 2888
Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt {d \sin (e+f x)}} \, dx &=\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}+\frac {(4 a) \int \frac {a+b \sin (e+f x)}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \, dx}{5 g^2}\\ &=\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}+\frac {\left (4 a^2\right ) \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \, dx}{5 g^2}+\frac {(4 a b) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}} \, dx}{5 d g^2}\\ &=\frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {(8 a b) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{5 d g^4}\\ &=\frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {\left (8 a b \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{5 d g^4 \sqrt {\sin (2 e+2 f x)}}\\ &=\frac {8 a^2 \sqrt {d \sin (e+f x)}}{5 d f g^3 \sqrt {g \cos (e+f x)}}+\frac {8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt {g \cos (e+f x)}}+\frac {2 \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac {8 a b \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 d f g^4 \sqrt {\sin (2 e+2 f x)}}\\ \end {align*}
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Mathematica [C] time = 0.66, size = 105, normalized size = 0.54 \[ \frac {2 \tan (e+f x) \left (3 \left (b^2-4 a^2\right ) \sin ^2(e+f x)+15 a^2+10 a b \sin (e+f x) \cos ^2(e+f x)^{5/4} \, _2F_1\left (\frac {3}{4},\frac {9}{4};\frac {7}{4};\sin ^2(e+f x)\right )\right )}{15 f g^2 \sqrt {d \sin (e+f x)} (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}}{d g^{4} \cos \left (f x + e\right )^{4} \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {7}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.93, size = 615, normalized size = 3.19 \[ \frac {\left (8 \EllipticE \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \left (\cos ^{3}\left (f x +e \right )\right ) a b -4 \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{3}\left (f x +e \right )\right ) a b +8 \EllipticE \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \left (\cos ^{2}\left (f x +e \right )\right ) a b -4 \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) a b -4 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}\, a b +4 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}\, a^{2}-\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}\, b^{2}+2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}\, a b +\sin \left (f x +e \right ) \sqrt {2}\, a^{2}+\sin \left (f x +e \right ) \sqrt {2}\, b^{2}+2 a b \sqrt {2}\right ) \cos \left (f x +e \right ) \sqrt {2}}{5 f \left (g \cos \left (f x +e \right )\right )^{\frac {7}{2}} \sqrt {d \sin \left (f x +e \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 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {7}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]
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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