Optimal. Leaf size=123 \[ \frac {16 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{39 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}-\frac {20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2707, 3856,
2719} \begin {gather*} -\frac {2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}-\frac {20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac {8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}+\frac {16 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{39 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2707
Rule 2719
Rule 3856
Rubi steps
\begin {align*} \int \frac {\sin ^6(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx &=-\frac {2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac {10}{13} \int \frac {\sin ^4(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac {20}{39} \int \frac {\sin ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}-\frac {20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac {8}{39} \int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx\\ &=-\frac {8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}-\frac {20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}+\frac {8 \int \sqrt {\cos (e+f x)} \, dx}{39 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ &=\frac {16 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{39 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}-\frac {20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac {2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 73, normalized size = 0.59 \begin {gather*} \frac {\frac {768 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{\sqrt {\cos (e+f x)}}-317 \sin (2 (e+f x))+76 \sin (4 (e+f x))-9 \sin (6 (e+f x))}{1872 f \sqrt {b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.51, size = 338, normalized size = 2.75
method | result | size |
default | \(-\frac {2 \left (-9 \left (\cos ^{8}\left (f x +e \right )\right )+24 i \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-24 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )+37 \left (\cos ^{6}\left (f x +e \right )\right )+24 i \sin \left (f x +e \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-24 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-59 \left (\cos ^{4}\left (f x +e \right )\right )+55 \left (\cos ^{2}\left (f x +e \right )\right )-24 \cos \left (f x +e \right )\right ) \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{117 f \sin \left (f x +e \right ) b}\) | \(338\) |
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 = 126, normalized size = 1.02 \begin {gather*} -\frac {2 \, {\left ({\left (9 \, \cos \left (f x + e\right )^{6} - 28 \, \cos \left (f x + e\right )^{4} + 31 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 12 i \, \sqrt {2} \sqrt {b} {\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 ) + 12 i \, \sqrt {2} \sqrt {b} {\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 )\right )}}{117 \, b f} \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 \frac {\sin ^{6}{\left (e + f x \right )}}{\sqrt {b \sec {\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 {{\sin \left (e+f\,x\right )}^6}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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