Optimal. Leaf size=58 \[ \frac {3 \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {5}{2};\frac {4}{3};\sin ^2(e+f x)\right )}{2 b f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2577} \[ \frac {3 \sqrt {\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {5}{2};\frac {4}{3};\sin ^2(e+f x)\right )}{2 b f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2577
Rubi steps
\begin {align*} \int \frac {\sec ^4(e+f x)}{\sqrt [3]{b \sin (e+f x)}} \, dx &=\frac {3 \sqrt {\cos ^2(e+f x)} \, _2F_1\left (\frac {1}{3},\frac {5}{2};\frac {4}{3};\sin ^2(e+f x)\right ) \sec (e+f x) (b \sin (e+f x))^{2/3}}{2 b f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 55, normalized size = 0.95 \[ \frac {3 \sqrt {\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left (\frac {1}{3},\frac {5}{2};\frac {4}{3};\sin ^2(e+f x)\right )}{2 f \sqrt [3]{b \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (b \sin \left (f x + e\right )\right )^{\frac {2}{3}} \sec \left (f x + e\right )^{4}}{b \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (f x + e\right )^{4}}{\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}\left (f x +e \right )}{\left (b \sin \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (f x + e\right )^{4}}{\left (b \sin \left (f x + e\right )\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\cos \left (e+f\,x\right )}^4\,{\left (b\,\sin \left (e+f\,x\right )\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}{\left (e + f x \right )}}{\sqrt [3]{b \sin {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________