Optimal. Leaf size=123 \[ -\frac {5 b^3 \csc (e+f x)}{2 f \sqrt {b \sec (e+f x)}}+\frac {5 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{2 f}-\frac {b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f} \]
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Rubi [A] time = 0.15, antiderivative size = 123, 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 = {2625, 2626, 3771, 2641} \[ -\frac {5 b^3 \csc (e+f x)}{2 f \sqrt {b \sec (e+f x)}}+\frac {5 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{2 f}-\frac {b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 2626
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \csc ^4(e+f x) (b \sec (e+f x))^{5/2} \, dx &=-\frac {b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {3}{2} \int \csc ^2(e+f x) (b \sec (e+f x))^{5/2} \, dx\\ &=\frac {b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac {b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {1}{2} \left (5 b^2\right ) \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {5 b^3 \csc (e+f x)}{2 f \sqrt {b \sec (e+f x)}}+\frac {b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac {b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {1}{4} \left (5 b^2\right ) \int \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {5 b^3 \csc (e+f x)}{2 f \sqrt {b \sec (e+f x)}}+\frac {b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac {b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac {1}{4} \left (5 b^2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=-\frac {5 b^3 \csc (e+f x)}{2 f \sqrt {b \sec (e+f x)}}+\frac {5 b^2 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{2 f}+\frac {b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}-\frac {b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 79, normalized size = 0.64 \[ \frac {b \sin (e+f x) (b \sec (e+f x))^{3/2} \left (-\left (\cot ^2(e+f x) \left (2 \csc ^2(e+f x)+11\right )\right )+15 \cos ^{\frac {3}{2}}(e+f x) \csc (e+f x) F\left (\left .\frac {1}{2} (e+f x)\right |2\right )+4\right )}{6 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (f x + e\right )} b^{2} \csc \left (f x + e\right )^{4} \sec \left (f x + e\right )^{2}, x\right ) \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}} \csc \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 352, normalized size = 2.86 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (15 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+15 i \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )-15 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-15 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-15 \left (\cos ^{4}\left (f x +e \right )\right )+21 \left (\cos ^{2}\left (f x +e \right )\right )-4\right ) \cos \left (f x +e \right ) \left (\cos \left (f x +e \right )+1\right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{6 f \sin \left (f x +e \right )^{7}} \]
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 \[ \int \left (b \sec \left (f x + e\right )\right )^{\frac {5}{2}} \csc \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\sin \left (e+f\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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