Optimal. Leaf size=111 \[ \frac {7 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f} \]
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Rubi [A]
time = 0.10, antiderivative size = 111, 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 = {2692, 2695,
2652, 2719} \begin {gather*} \frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {7 \cos (e+f x) E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2652
Rule 2692
Rule 2695
Rule 2719
Rubi steps
\begin {align*} \int \cos ^5(e+f x) \sqrt {d \tan (e+f x)} \, dx &=\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7}{10} \int \cos ^3(e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {7}{20} \int \cos (e+f x) \sqrt {d \tan (e+f x)} \, dx\\ &=\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {\left (7 \sqrt {\cos (e+f x)} \sqrt {d \tan (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \sqrt {\sin (e+f x)} \, dx}{20 \sqrt {\sin (e+f x)}}\\ &=\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(e+f x) (d \tan (e+f x))^{3/2}}{5 d f}+\frac {\left (7 \cos (e+f x) \sqrt {d \tan (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{20 \sqrt {\sin (2 e+2 f x)}}\\ &=\frac {7 \cos (e+f x) E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \tan (e+f x)}}{20 f \sqrt {\sin (2 e+2 f x)}}+\frac {7 \cos ^3(e+f x) (d \tan (e+f x))^{3/2}}{30 d f}+\frac {\cos ^5(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 = 0.83, size = 86, normalized size = 0.77 \begin {gather*} \frac {\cos (e+f x) \sqrt {d \tan (e+f x)} \left (20 \sin (2 (e+f x))+3 \sin (4 (e+f x))+28 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\tan ^2(e+f x)\right ) \sqrt {\sec ^2(e+f x)} \tan (e+f x)\right )}{120 f} \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. \(549\) vs.
\(2(122)=244\).
time = 0.35, size = 550, normalized size = 4.95
method | result | size |
default | \(-\frac {\left (\cos \left (f x +e \right )-1\right )^{2} \left (12 \sqrt {2}\, \left (\cos ^{6}\left (f x +e \right )\right )+2 \sqrt {2}\, \left (\cos ^{4}\left (f x +e \right )\right )-21 \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 )}}\, \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 ) \cos \left (f x +e \right )+42 \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 )}}\, \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 ) \cos \left (f x +e \right )-21 \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 )}}\, \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 )+42 \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 )}}\, \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 )+7 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-21 \cos \left (f x +e \right ) \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}}{120 f \sin \left (f x +e \right )^{5}}\) | \(550\) |
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]
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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\cos \left (e+f\,x\right )}^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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