Optimal. Leaf size=95 \[ -\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{7 f} \]
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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 = {2627, 3771, 2641} \[ -\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 2641
Rule 3771
Rubi steps
\begin {align*} \int \sqrt {b \sec (e+f x)} \sin ^4(e+f x) \, dx &=-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {6}{7} \int \sqrt {b \sec (e+f x)} \sin ^2(e+f x) \, dx\\ &=-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {4}{7} \int \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}+\frac {1}{7} \left (4 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=\frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{7 f}-\frac {4 b \sin (e+f x)}{7 f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{7 f \sqrt {b \sec (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 61, normalized size = 0.64 \[ \frac {\sqrt {b \sec (e+f x)} \left (-10 \sin (2 (e+f x))+\sin (4 (e+f x))+32 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right )}{28 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b \sec \left (f x + e\right )}, x\right ) \]
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 \[ \int \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 143, normalized size = 1.51 \[ \frac {2 \left (-1+\cos \left (f x +e \right )\right ) \left (-4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+\cos ^{4}\left (f x +e \right )-\left (\cos ^{3}\left (f x +e \right )\right )-3 \left (\cos ^{2}\left (f x +e \right )\right )+3 \cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{7 f \sin \left (f x +e \right )^{3}} \]
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 \[ \int \sqrt {b \sec \left (f x + e\right )} \sin \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int {\sin \left (e+f\,x\right )}^4\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
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 \[ \int \sqrt {b \sec {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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