Optimal. Leaf size=107 \[ -\frac {4 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{5 f \sqrt {\sin (2 e+2 f x)}}+\frac {4 \cos (e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {2 \sec (e+f x) (d \tan (e+f x))^{3/2}}{5 d f} \]
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Rubi [A]
time = 0.09, 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 = {2693, 2695,
2652, 2719} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2652
Rule 2693
Rule 2695
Rule 2719
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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.48, size = 102, normalized size = 0.95 \begin {gather*} \frac {2 \sqrt {d \tan (e+f x)} \left (-4 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(e+f x)\right ) \sec (e+f x) \tan (e+f x)+3 \sqrt {\sec ^2(e+f x)} (2 \sin (e+f x)+\sec (e+f x) \tan (e+f x))\right )}{15 f \sqrt {\sec ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(558\) vs.
\(2(118)=236\).
time = 0.37, size = 559, normalized size = 5.22
method | result | size |
default | \(-\frac {\left (\cos \left (f x +e \right )-1\right )^{2} \left (2 \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 ) \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 )}{\sin \left (f x +e \right )}}\, \left (\cos ^{3}\left (f x +e \right )\right )-4 \EllipticE \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 ) \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 )}{\sin \left (f x +e \right )}}\, \left (\cos ^{3}\left (f x +e \right )\right )+2 \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 ) \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 )}{\sin \left (f x +e \right )}}\, \left (\cos ^{2}\left (f x +e \right )\right )-4 \EllipticE \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 ) \sqrt {\frac {\cos \left (f x +e \right )-1}{\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 )}{\sin \left (f x +e \right )}}\, \left (\cos ^{2}\left (f x +e \right )\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 ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {d \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \sqrt {2}}{5 f \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{5}}\) | \(559\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \sqrt {d \tan {\left (e + f x \right )}} \sec ^{3}{\left (e + f x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{{\cos \left (e+f\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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