Optimal. Leaf size=130 \[ -\frac {2}{b f (d \sec (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}-\frac {24 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{5 b^2 d^2 f \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}-\frac {12 (b \tan (e+f x))^{3/2}}{5 b^3 f (d \sec (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2689, 2692,
2696, 2721, 2719} \begin {gather*} -\frac {12 (b \tan (e+f x))^{3/2}}{5 b^3 f (d \sec (e+f x))^{5/2}}-\frac {24 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{5 b^2 d^2 f \sqrt {\sin (e+f x)} \sqrt {d \sec (e+f x)}}-\frac {2}{b f \sqrt {b \tan (e+f x)} (d \sec (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2689
Rule 2692
Rule 2696
Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int \frac {1}{(d \sec (e+f x))^{5/2} (b \tan (e+f x))^{3/2}} \, dx &=-\frac {2}{b f (d \sec (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}-\frac {6 \int \frac {\sqrt {b \tan (e+f x)}}{(d \sec (e+f x))^{5/2}} \, dx}{b^2}\\ &=-\frac {2}{b f (d \sec (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}-\frac {12 (b \tan (e+f x))^{3/2}}{5 b^3 f (d \sec (e+f x))^{5/2}}-\frac {12 \int \frac {\sqrt {b \tan (e+f x)}}{\sqrt {d \sec (e+f x)}} \, dx}{5 b^2 d^2}\\ &=-\frac {2}{b f (d \sec (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}-\frac {12 (b \tan (e+f x))^{3/2}}{5 b^3 f (d \sec (e+f x))^{5/2}}-\frac {\left (12 \sqrt {b \tan (e+f x)}\right ) \int \sqrt {b \sin (e+f x)} \, dx}{5 b^2 d^2 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}\\ &=-\frac {2}{b f (d \sec (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}-\frac {12 (b \tan (e+f x))^{3/2}}{5 b^3 f (d \sec (e+f x))^{5/2}}-\frac {\left (12 \sqrt {b \tan (e+f x)}\right ) \int \sqrt {\sin (e+f x)} \, dx}{5 b^2 d^2 \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}\\ &=-\frac {2}{b f (d \sec (e+f x))^{5/2} \sqrt {b \tan (e+f x)}}-\frac {24 E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right ) \sqrt {b \tan (e+f x)}}{5 b^2 d^2 f \sqrt {d \sec (e+f x)} \sqrt {\sin (e+f x)}}-\frac {12 (b \tan (e+f x))^{3/2}}{5 b^3 f (d \sec (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.78, size = 81, normalized size = 0.62 \begin {gather*} \frac {-11+\cos (2 (e+f x))+24 \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {3}{4};\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}}{5 b d^2 f \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.36, size = 570, normalized size = 4.38
method | result | size |
default | \(\frac {\left (24 \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-i-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-12 \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-i-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+\left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}+24 \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-i-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )-12 \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {-\frac {i \cos \left (f x +e \right )-i-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+6 \cos \left (f x +e \right ) \sqrt {2}-12 \sqrt {2}\right ) \sin \left (f x +e \right ) \sqrt {2}}{5 f \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}} \left (\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}} \cos \left (f x +e \right )^{4}}\) | \(570\) |
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 [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.12, size = 153, normalized size = 1.18 \begin {gather*} -\frac {2 \, {\left (6 i \, \sqrt {-2 i \, b d} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 6 i \, \sqrt {2 i \, b d} \sin \left (f x + e\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (\cos \left (f x + e\right )^{4} - 6 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}\right )}}{5 \, b^{2} d^{3} f \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {1}{{\left (b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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