Optimal. Leaf size=178 \[ \frac {3 d^3 \text {ArcTan}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}{4 \sqrt {b} f \sqrt {b \tan (e+f x)}}+\frac {3 d^3 \tanh ^{-1}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}{4 \sqrt {b} f \sqrt {b \tan (e+f x)}}+\frac {d^2 (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}{2 b f} \]
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Rubi [A]
time = 0.12, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {2693, 2696,
2644, 335, 218, 212, 209} \begin {gather*} \frac {3 d^3 \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)} \text {ArcTan}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right )}{4 \sqrt {b} f \sqrt {b \tan (e+f x)}}+\frac {3 d^3 \sqrt {b \sin (e+f x)} \sqrt {d \sec (e+f x)} \tanh ^{-1}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right )}{4 \sqrt {b} f \sqrt {b \tan (e+f x)}}+\frac {d^2 \sqrt {b \tan (e+f x)} (d \sec (e+f x))^{3/2}}{2 b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 2644
Rule 2693
Rule 2696
Rubi steps
\begin {align*} \int \frac {(d \sec (e+f x))^{7/2}}{\sqrt {b \tan (e+f x)}} \, dx &=\frac {d^2 (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}{2 b f}+\frac {1}{4} \left (3 d^2\right ) \int \frac {(d \sec (e+f x))^{3/2}}{\sqrt {b \tan (e+f x)}} \, dx\\ &=\frac {d^2 (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}{2 b f}+\frac {\left (3 d^3 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}\right ) \int \frac {\sec (e+f x)}{\sqrt {b \sin (e+f x)}} \, dx}{4 \sqrt {b \tan (e+f x)}}\\ &=\frac {d^2 (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}{2 b f}+\frac {\left (3 d^3 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{b^2}\right )} \, dx,x,b \sin (e+f x)\right )}{4 b f \sqrt {b \tan (e+f x)}}\\ &=\frac {d^2 (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}{2 b f}+\frac {\left (3 d^3 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{b^2}} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{2 b f \sqrt {b \tan (e+f x)}}\\ &=\frac {d^2 (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}{2 b f}+\frac {\left (3 d^3 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{4 f \sqrt {b \tan (e+f x)}}+\frac {\left (3 d^3 \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \sin (e+f x)}\right )}{4 f \sqrt {b \tan (e+f x)}}\\ &=\frac {3 d^3 \tan ^{-1}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}{4 \sqrt {b} f \sqrt {b \tan (e+f x)}}+\frac {3 d^3 \tanh ^{-1}\left (\frac {\sqrt {b \sin (e+f x)}}{\sqrt {b}}\right ) \sqrt {d \sec (e+f x)} \sqrt {b \sin (e+f x)}}{4 \sqrt {b} f \sqrt {b \tan (e+f x)}}+\frac {d^2 (d \sec (e+f x))^{3/2} \sqrt {b \tan (e+f x)}}{2 b f}\\ \end {align*}
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Mathematica [A]
time = 6.82, size = 136, normalized size = 0.76 \begin {gather*} \frac {d^3 \sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)} \left (-3 \text {ArcTan}\left (\frac {\sqrt {\sec (e+f x)}}{\sqrt [4]{\tan ^2(e+f x)}}\right )+3 \tanh ^{-1}\left (\frac {\sqrt {\sec (e+f x)}}{\sqrt [4]{\tan ^2(e+f x)}}\right )+2 \sec ^{\frac {3}{2}}(e+f x) \sqrt [4]{\tan ^2(e+f x)}\right )}{4 b f \sqrt {\sec (e+f x)} \sqrt [4]{\tan ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.36, size = 758, normalized size = 4.26
method | result | size |
default | \(-\frac {\left (3 i \left (\cos ^{2}\left (f x +e \right )\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 )}}\, \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+3 i \left (\cos ^{2}\left (f x +e \right )\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 )}}\, \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-6 i \left (\cos ^{2}\left (f x +e \right )\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 )}}\, \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\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 )-3 \left (\cos ^{2}\left (f x +e \right )\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 )}}\, \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+3 \left (\cos ^{2}\left (f x +e \right )\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 )}}\, \sqrt {-\frac {i \left (\cos \left (f x +e \right )-1\right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {i \cos \left (f x +e \right )-i+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (f x +e \right ) \sqrt {2}+2 \sqrt {2}\right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {2}}{8 f \left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}}\) | \(758\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 421 vs.
\(2 (154) = 308\).
time = 0.59, size = 850, normalized size = 4.78 \begin {gather*} \left [-\frac {6 \, b d^{3} \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (\cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) + 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) + 4\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {-\frac {d}{b}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{4 \, {\left (d \cos \left (f x + e\right )^{2} - {\left (d \cos \left (f x + e\right ) + d\right )} \sin \left (f x + e\right ) - d\right )}}\right ) \cos \left (f x + e\right ) - 3 \, b d^{3} \sqrt {-\frac {d}{b}} \cos \left (f x + e\right ) \log \left (\frac {d \cos \left (f x + e\right )^{4} - 72 \, d \cos \left (f x + e\right )^{2} + 8 \, {\left (7 \, \cos \left (f x + e\right )^{3} - {\left (\cos \left (f x + e\right )^{3} - 8 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 8 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {-\frac {d}{b}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} + 28 \, {\left (d \cos \left (f x + e\right )^{2} - 2 \, d\right )} \sin \left (f x + e\right ) + 72 \, d}{\cos \left (f x + e\right )^{4} - 8 \, \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} - 2\right )} \sin \left (f x + e\right ) + 8}\right ) - 16 \, d^{3} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{32 \, b f \cos \left (f x + e\right )}, \frac {6 \, b d^{3} \sqrt {\frac {d}{b}} \arctan \left (\frac {{\left (\cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} + 6 \, \cos \left (f x + e\right ) + 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) + 4\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{b}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{4 \, {\left (d \cos \left (f x + e\right )^{2} + {\left (d \cos \left (f x + e\right ) + d\right )} \sin \left (f x + e\right ) - d\right )}}\right ) \cos \left (f x + e\right ) + 3 \, b d^{3} \sqrt {\frac {d}{b}} \cos \left (f x + e\right ) \log \left (\frac {d \cos \left (f x + e\right )^{4} - 72 \, d \cos \left (f x + e\right )^{2} - 8 \, {\left (7 \, \cos \left (f x + e\right )^{3} + {\left (\cos \left (f x + e\right )^{3} - 8 \, \cos \left (f x + e\right )\right )} \sin \left (f x + e\right ) - 8 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{b}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} - 28 \, {\left (d \cos \left (f x + e\right )^{2} - 2 \, d\right )} \sin \left (f x + e\right ) + 72 \, d}{\cos \left (f x + e\right )^{4} - 8 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} - 2\right )} \sin \left (f x + e\right ) + 8}\right ) + 16 \, d^{3} \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{32 \, b f \cos \left (f x + e\right )}\right ] \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 \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}}{\sqrt {b\,\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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