Optimal. Leaf size=95 \[ -\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{6 f} \]
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Rubi [A] time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2625, 3771, 2641} \[ -\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{6 f} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx &=-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5}{6} \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5}{12} \int \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {1}{12} \left (5 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=-\frac {5 b \csc (e+f x)}{6 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^3(e+f x)}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{6 f}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 63, normalized size = 0.66 \[ \frac {\sqrt {b \sec (e+f x)} \left (5 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )-\cot (e+f x) \left (2 \csc ^2(e+f x)+5\right )\right )}{6 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4}, x\right ) \]
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 \[ \int \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 335, normalized size = 3.53 \[ -\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (5 i \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right )+5 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-5 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )-5 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-5 \left (\cos ^{3}\left (f x +e \right )\right )+7 \cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{6 f \sin \left (f x +e \right )^{7}} \]
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 \[ \int \sqrt {b \sec \left (f x + e\right )} \csc \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec {\left (e + f x \right )}} \csc ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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