Optimal. Leaf size=102 \[ \frac {\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2703, 2705,
3856, 2719} \begin {gather*} \frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}-\frac {\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac {\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2703
Rule 2705
Rule 2719
Rule 3856
Rubi steps
\begin {align*} \int \frac {\csc ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=-\frac {\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}-\frac {\int \frac {\csc ^2(e+f x)}{\sqrt {b \sec (e+f x)}} \, dx}{2 b^2}\\ &=\frac {\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac {\int \frac {1}{\sqrt {b \sec (e+f x)}} \, dx}{4 b^2}\\ &=\frac {\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac {\int \sqrt {\cos (e+f x)} \, dx}{4 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ &=\frac {\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}-\frac {\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac {E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{2 b^2 f \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 79, normalized size = 0.77 \begin {gather*} \frac {\left (-3+5 \csc ^2(e+f x)-2 \csc ^4(e+f x)+3 \sqrt {\cos (e+f x)} \csc (e+f x) E\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \sqrt {b \sec (e+f x)} \sin (e+f x)}{6 b^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.25, size = 623, normalized size = 6.11
method | result | size |
default | \(-\frac {\left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} \left (3 i \EllipticF \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}}\, \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 i \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\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}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+3 i \EllipticF \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}}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 i \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\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}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-3 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 )+3 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}}-3 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 )+3 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}}+3 \left (\cos ^{3}\left (f x +e \right )\right )+2 \left (\cos ^{2}\left (f x +e \right )\right )-3 \cos \left (f x +e \right )\right )}{6 f \cos \left (f x +e \right )^{3} \sin \left (f x +e \right )^{7} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}\) | \(623\) |
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 = 0.10, size = 175, normalized size = 1.72 \begin {gather*} -\frac {3 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \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 ) + 3 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \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 ) - 2 \, {\left (3 \, \cos \left (f x + e\right )^{4} - \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{12 \, {\left (b^{3} f \cos \left (f x + e\right )^{2} - b^{3} f\right )} \sin \left (f x + e\right )} \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 {\csc ^{4}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, 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 {1}{{\sin \left (e+f\,x\right )}^4\,{\left (\frac {b}{\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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