Optimal. Leaf size=126 \[ \frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{77 b^2 f}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac {8 \sin (e+f x)}{77 b f \sqrt {b \sec (e+f x)}}-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 126, 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 = {2627, 3769, 3771, 2641} \[ \frac {8 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{77 b^2 f}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac {8 \sin (e+f x)}{77 b f \sqrt {b \sec (e+f x)}}-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2627
Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx &=-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac {6}{11} \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx\\ &=-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac {12}{77} \int \frac {1}{(b \sec (e+f x))^{3/2}} \, dx\\ &=-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}}+\frac {8 \sin (e+f x)}{77 b f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac {4 \int \sqrt {b \sec (e+f x)} \, dx}{77 b^2}\\ &=-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}}+\frac {8 \sin (e+f x)}{77 b f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac {\left (4 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{77 b^2}\\ &=\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)}}{77 b^2 f}-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}}+\frac {8 \sin (e+f x)}{77 b f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 81, normalized size = 0.64 \[ \frac {\sec ^2(e+f x) \left (-5 \sin (2 (e+f x))-24 \sin (4 (e+f x))+7 \sin (6 (e+f x))+128 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right )}{1232 f (b \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \sqrt {b \sec \left (f x + e\right )}}{b^{2} \sec \left (f x + e\right )^{2}}, x\right ) \]
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 \[ \int \frac {\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 173, normalized size = 1.37 \[ -\frac {2 \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right ) \left (-7 \left (\cos ^{6}\left (f x +e \right )\right )+4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+7 \left (\cos ^{5}\left (f x +e \right )\right )+13 \left (\cos ^{4}\left (f x +e \right )\right )-13 \left (\cos ^{3}\left (f x +e \right )\right )-4 \left (\cos ^{2}\left (f x +e \right )\right )+4 \cos \left (f x +e \right )\right )}{77 f \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{3} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\sin \left (e+f\,x\right )}^4}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sin ^{4}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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