Integrand size = 21, antiderivative size = 126 \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {8 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{77 b^2 f}-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}}+\frac {8 \sin (e+f x)}{77 b f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2707, 3854, 3856, 2720} \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {8 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {b \sec (e+f x)}}{77 b^2 f}-\frac {2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac {8 \sin (e+f x)}{77 b f \sqrt {b \sec (e+f x)}}-\frac {12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}} \]
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Rule 2707
Rule 2720
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\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)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),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*}
Time = 0.31 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {\sec ^2(e+f x) \left (128 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )-5 \sin (2 (e+f x))-24 \sin (4 (e+f x))+7 \sin (6 (e+f x))\right )}{1232 f (b \sec (e+f x))^{3/2}} \]
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Result contains complex when optimal does not.
Time = 1.16 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.40
method | result | size |
default | \(-\frac {2 \left (-7 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right )+4 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )+13 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )\right )}{77 f \sqrt {b \sec \left (f x +e \right )}\, b}\) | \(176\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.87 \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\frac {2 \, {\left ({\left (7 \, \cos \left (f x + e\right )^{5} - 13 \, \cos \left (f x + e\right )^{3} + 4 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 2 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 2 i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{77 \, b^{2} f} \]
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\[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {\sin ^{4}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\sin \left (f x + e\right )^{4}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {{\sin \left (e+f\,x\right )}^4}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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