Optimal. Leaf size=418 \[ -\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{4 \sqrt {2} b^{5/2} f}+\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{4 \sqrt {2} b^{5/2} f}+\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{8 \sqrt {2} b^{5/2} f}-\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{8 \sqrt {2} b^{5/2} f}+\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}} \]
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Rubi [A] time = 0.36, antiderivative size = 418, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2582, 2585, 2574, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{4 \sqrt {2} b^{5/2} f}+\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{4 \sqrt {2} b^{5/2} f}+\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{8 \sqrt {2} b^{5/2} f}-\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{8 \sqrt {2} b^{5/2} f}+\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2574
Rule 2582
Rule 2585
Rubi steps
\begin {align*} \int \frac {\sqrt {a \sin (e+f x)}}{(b \sec (e+f x))^{3/2}} \, dx &=\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}}+\frac {\int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \, dx}{4 b^2}\\ &=\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}} \, dx}{4 b^2}\\ &=\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (a \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{2 b f}\\ &=\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}}-\frac {\left (a \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {a-b x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{4 b^2 f}+\frac {\left (a \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {a+b x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{4 b^2 f}\\ &=\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (a \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}+x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{8 b^3 f}+\frac {\left (a \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}+x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{8 b^3 f}+\frac {\left (\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {b}}+2 x}{-\frac {a}{b}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}-x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{8 \sqrt {2} b^{5/2} f}+\frac {\left (\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {b}}-2 x}{-\frac {a}{b}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}-x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{8 \sqrt {2} b^{5/2} f}\\ &=\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{8 \sqrt {2} b^{5/2} f}-\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{8 \sqrt {2} b^{5/2} f}+\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{4 \sqrt {2} b^{5/2} f}-\frac {\left (\sqrt {a} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{4 \sqrt {2} b^{5/2} f}\\ &=-\frac {\sqrt {a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{4 \sqrt {2} b^{5/2} f}+\frac {\sqrt {a} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{4 \sqrt {2} b^{5/2} f}+\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{8 \sqrt {2} b^{5/2} f}-\frac {\sqrt {a} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{8 \sqrt {2} b^{5/2} f}+\frac {(a \sin (e+f x))^{3/2}}{2 a b f \sqrt {b \sec (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 76, normalized size = 0.18 \[ \frac {\sec ^2(e+f x) \sqrt {a \sin (e+f x)} \left (2 \tan (e+f x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(e+f x)\right )+3 \sin (2 (e+f x))\right )}{12 f (b \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {\sqrt {a \sin \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.21, size = 516, normalized size = 1.23 \[ \frac {\left (i \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 )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \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 )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \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 )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \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 )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-2 \cos \left (f x +e \right ) \sqrt {2}\right ) \sqrt {a \sin \left (f x +e \right )}\, \sin \left (f x +e \right ) \sqrt {2}}{8 f \left (-1+\cos \left (f x +e \right )\right ) \cos \left (f x +e \right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {a \sin \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {a\,\sin \left (e+f\,x\right )}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \sin {\left (e + f x \right )}}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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