Optimal. Leaf size=513 \[ -\frac {2 \sqrt {2} b^3 \sqrt {g} \sqrt {\sin (e+f x)} \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^3 d^3 f \sqrt {b-a} \sqrt {a+b} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {g} \sqrt {\sin (e+f x)} \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^3 d^3 f \sqrt {b-a} \sqrt {a+b} \sqrt {d \sin (e+f x)}}-\frac {2 b^2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a^3 d^4 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt {d \sin (e+f x)}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{5 a d^4 f \sqrt {\sin (2 e+2 f x)}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}} \]
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Rubi [A] time = 1.45, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {2910, 2570, 2572, 2639, 2563, 2906, 2905, 490, 1218} \[ -\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 b^2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a^3 d^4 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {g} \sqrt {\sin (e+f x)} \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^3 d^3 f \sqrt {b-a} \sqrt {a+b} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {g} \sqrt {\sin (e+f x)} \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a^3 d^3 f \sqrt {b-a} \sqrt {a+b} \sqrt {d \sin (e+f x)}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {4 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{5 a d^4 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 490
Rule 1218
Rule 2563
Rule 2570
Rule 2572
Rule 2639
Rule 2905
Rule 2906
Rule 2910
Rubi steps
\begin {align*} \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx &=\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}} \, dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx}{a d}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}+\frac {2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{5 a d^2}+\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{a^2 d^2}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}} \, dx}{a^2 d}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {4 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{5 a d^4}-\frac {b^3 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a^3 d^3}+\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{a^3 d^2}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {\left (2 b^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{a^3 d^4}-\frac {\left (b^3 \sqrt {\sin (e+f x)}\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a^3 d^3 \sqrt {d \sin (e+f x)}}-\frac {\left (4 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{5 a d^4 \sqrt {\sin (2 e+2 f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {4 \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 a d^4 f \sqrt {\sin (2 e+2 f x)}}+\frac {\left (4 \sqrt {2} b^3 g \sqrt {\sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a^3 d^3 f \sqrt {d \sin (e+f x)}}-\frac {\left (2 b^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{a^3 d^4 \sqrt {\sin (2 e+2 f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {4 \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 a d^4 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^3 d^4 f \sqrt {\sin (2 e+2 f x)}}+\frac {\left (2 \sqrt {2} b^3 g \sqrt {\sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g-\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a^3 \sqrt {-a+b} d^3 f \sqrt {d \sin (e+f x)}}-\frac {\left (2 \sqrt {2} b^3 g \sqrt {\sin (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g+\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a^3 \sqrt {-a+b} d^3 f \sqrt {d \sin (e+f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{5 a d f g (d \sin (e+f x))^{5/2}}+\frac {2 b (g \cos (e+f x))^{3/2}}{3 a^2 d^2 f g (d \sin (e+f x))^{3/2}}-\frac {4 (g \cos (e+f x))^{3/2}}{5 a d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a^3 d^3 f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {g} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{a^3 \sqrt {-a+b} \sqrt {a+b} d^3 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {g} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{a^3 \sqrt {-a+b} \sqrt {a+b} d^3 f \sqrt {d \sin (e+f x)}}-\frac {4 \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 a d^4 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^3 d^4 f \sqrt {\sin (2 e+2 f x)}}\\ \end {align*}
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Mathematica [C] time = 23.61, size = 1726, normalized size = 3.36 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {g \cos \left (f x + e\right )}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.75, size = 6208, normalized size = 12.10 \[ \text {output too large to display} \]
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 \[ \int \frac {\sqrt {g \cos \left (f x + e\right )}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {g\,\cos \left (e+f\,x\right )}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{7/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]
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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