Optimal. Leaf size=107 \[ \frac {4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}-\frac {4 \cos (e+f x) E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (e+f x)}}{5 f \sqrt {\sin (2 e+2 f x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2613, 2615, 2572, 2639} \[ \frac {4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}-\frac {4 \cos (e+f x) E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (e+f x)}}{5 f \sqrt {\sin (2 e+2 f x)}} \]
Antiderivative was successfully verified.
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Rule 2572
Rule 2613
Rule 2615
Rule 2639
Rubi steps
\begin {align*} \int \sec ^3(e+f x) \sqrt {d \tan (e+f x)} \, dx &=\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2}{5} \int \sec (e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}-\frac {4}{5} \int \cos (e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}-\frac {\left (4 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)} \, dx}{5 \sqrt {\sin (e+f x)}}\\ &=\frac {4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}-\frac {\left (4 \cos (e+f x) \sqrt {d \tan (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{5 \sqrt {\sin (2 e+2 f x)}}\\ &=-\frac {4 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{5 f \sqrt {\sin (2 e+2 f x)}}+\frac {4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}\\ \end {align*}
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Mathematica [C] time = 0.44, size = 102, normalized size = 0.95 \[ \frac {2 \sqrt {d \tan (e+f x)} \left (3 \sqrt {\sec ^2(e+f x)} (2 \sin (e+f x)+\tan (e+f x) \sec (e+f x))-4 \tan (e+f x) \sec (e+f x) \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(e+f x)\right )\right )}{15 f \sqrt {\sec ^2(e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d \tan \left (f x + e\right )} \sec \left (f x + e\right )^{3}, x\right ) \]
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 \[ \int \sqrt {d \tan \left (f x + e\right )} \sec \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.57, size = 559, normalized size = 5.22 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (2 \left (\cos ^{3}\left (f x +e \right )\right ) \EllipticF \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \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 )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}-4 \left (\cos ^{3}\left (f x +e \right )\right ) \EllipticE \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \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 )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}+2 \left (\cos ^{2}\left (f x +e \right )\right ) \EllipticF \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \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 )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}-4 \left (\cos ^{2}\left (f x +e \right )\right ) \EllipticE \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \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 )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}+2 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}-\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-\sqrt {2}\right ) \sqrt {\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \left (1+\cos \left (f x +e \right )\right )^{2} \sqrt {2}}{5 f \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{5}} \]
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 \[ \int \sqrt {d \tan \left (f x + e\right )} \sec \left (f x + e\right )^{3}\,{d x} \]
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 \[ \int \frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{{\cos \left (e+f\,x\right )}^3} \,d x \]
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 \[ \int \sqrt {d \tan {\left (e + f x \right )}} \sec ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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