Optimal. Leaf size=159 \[ \frac {\left (2 a^2-b^2\right ) \sqrt {\sin (2 e+2 f x)} F\left (\left .\frac {1}{4} (4 e-\pi )+f x\right |2\right )}{3 f g^2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}+\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}} \]
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Rubi [A] time = 0.43, antiderivative size = 161, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {2911, 2563, 3202, 457, 329, 237, 335, 275, 232} \[ -\frac {2 \left (2 a^2-b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2} F\left (\left .\frac {1}{2} \csc ^{-1}(\sin (e+f x))\right |2\right )}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 232
Rule 237
Rule 275
Rule 329
Rule 335
Rule 457
Rule 2563
Rule 2911
Rule 3202
Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx &=\frac {(2 a b) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{5/2}} \, dx}{d}+\int \frac {a^2+b^2 \sin ^2(e+f x)}{(g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)}} \, dx\\ &=\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\cos ^2(e+f x)^{3/4} \operatorname {Subst}\left (\int \frac {a^2+b^2 x^2}{\sqrt {d x} \left (1-x^2\right )^{7/4}} \, dx,x,\sin (e+f x)\right )}{f g (g \cos (e+f x))^{3/2}}\\ &=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {\left (\left (-2 a^2+b^2\right ) \cos ^2(e+f x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d x} \left (1-x^2\right )^{3/4}} \, dx,x,\sin (e+f x)\right )}{3 f g (g \cos (e+f x))^{3/2}}\\ &=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {\left (2 \left (-2 a^2+b^2\right ) \cos ^2(e+f x)^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^4}{d^2}\right )^{3/4}} \, dx,x,\sqrt {d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {\left (2 \left (-2 a^2+b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {d^2}{x^4}\right )^{3/4} x^3} \, dx,x,\sqrt {d \sin (e+f x)}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\left (2 \left (-2 a^2+b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-d^2 x^4\right )^{3/4}} \, dx,x,\frac {1}{\sqrt {d \sin (e+f x)}}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}+\frac {\left (\left (-2 a^2+b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} (d \sin (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-d^2 x^2\right )^{3/4}} \, dx,x,\frac {\csc (e+f x)}{d}\right )}{3 d f g (g \cos (e+f x))^{3/2}}\\ &=\frac {2 \left (a^2+b^2\right ) \sqrt {d \sin (e+f x)}}{3 d f g (g \cos (e+f x))^{3/2}}+\frac {4 a b (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}-\frac {2 \left (2 a^2-b^2\right ) \left (1-\csc ^2(e+f x)\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}(\csc (e+f x))\right |2\right ) (d \sin (e+f x))^{3/2}}{3 d^2 f g (g \cos (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.45, size = 127, normalized size = 0.80 \[ \frac {2 \tan (e+f x) \left (15 a^2 \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {1}{4},\frac {7}{4};\frac {5}{4};\sin ^2(e+f x)\right )+b \sin (e+f x) \left (10 a+3 b \sin (e+f x) \cos ^2(e+f x)^{3/4} \, _2F_1\left (\frac {5}{4},\frac {7}{4};\frac {9}{4};\sin ^2(e+f x)\right )\right )\right )}{15 f g^2 \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {d \sin \left (f x + e\right )}}{d g^{3} \cos \left (f x + e\right )^{3} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.93, size = 371, normalized size = 2.33 \[ \frac {\left (-2 \sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right ) a^{2}+\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right ) b^{2}+2 \sqrt {2}\, \cos \left (f x +e \right ) \sin \left (f x +e \right ) a b +\sqrt {2}\, \cos \left (f x +e \right ) a^{2}+\sqrt {2}\, \cos \left (f x +e \right ) b^{2}-2 \sqrt {2}\, \sin \left (f x +e \right ) a b -\sqrt {2}\, a^{2}-\sqrt {2}\, b^{2}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right ) \sqrt {2}}{3 f \left (-1+\cos \left (f x +e \right )\right ) \left (g \cos \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {d \sin \left (f x +e \right )}} \]
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 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \]
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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