Optimal. Leaf size=79 \[ \frac {2 \sec (e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {2 \sqrt {\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{3 f \sqrt {d \tan (e+f x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2613, 2614, 2573, 2641} \[ \frac {2 \sec (e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {2 \sqrt {\sin (2 e+2 f x)} \sec (e+f x) F\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{3 f \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2573
Rule 2613
Rule 2614
Rule 2641
Rubi steps
\begin {align*} \int \frac {\sec ^3(e+f x)}{\sqrt {d \tan (e+f x)}} \, dx &=\frac {2 \sec (e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {2}{3} \int \frac {\sec (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx\\ &=\frac {2 \sec (e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {\left (2 \sqrt {\sin (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)}} \, dx}{3 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}}\\ &=\frac {2 \sec (e+f x) \sqrt {d \tan (e+f x)}}{3 d f}+\frac {\left (2 \sec (e+f x) \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{3 \sqrt {d \tan (e+f x)}}\\ &=\frac {2 F\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sec (e+f x) \sqrt {\sin (2 e+2 f x)}}{3 f \sqrt {d \tan (e+f x)}}+\frac {2 \sec (e+f x) \sqrt {d \tan (e+f x)}}{3 d f}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 68, normalized size = 0.86 \[ \frac {2 \sin (e+f x) \left (2 \sqrt {\sec ^2(e+f x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\tan ^2(e+f x)\right )+\sec ^2(e+f x)\right )}{3 f \sqrt {d \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \tan \left (f x + e\right )} \sec \left (f x + e\right )^{3}}{d \tan \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\sec \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.63, size = 196, normalized size = 2.48 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (2 \cos \left (f x +e \right ) \sin \left (f x +e \right ) \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {-\frac {-\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )-\cos \left (f x +e \right ) \sqrt {2}+\sqrt {2}\right ) \left (1+\cos \left (f x +e \right )\right )^{2} \sqrt {2}}{3 f \cos \left (f x +e \right )^{2} \sqrt {\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sin \left (f x +e \right )^{3}} \]
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 \[ \int \frac {\sec \left (f x + e\right )^{3}}{\sqrt {d \tan \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (e+f\,x\right )}^3\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sqrt {d \tan {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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