Optimal. Leaf size=64 \[ \frac {3 \cos ^2(e+f x)^{17/12} \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {13}{6};\sin ^2(e+f x)\right )}{7 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2617} \[ \frac {3 \cos ^2(e+f x)^{17/12} \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {13}{6};\sin ^2(e+f x)\right )}{7 d f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2617
Rubi steps
\begin {align*} \int \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{4/3} \, dx &=\frac {3 \cos ^2(e+f x)^{17/12} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {13}{6};\sin ^2(e+f x)\right ) \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{7/3}}{7 d f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 62, normalized size = 0.97 \[ \frac {2 d \sqrt {b \sec (e+f x)} \sqrt [3]{d \tan (e+f x)} \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{4};\sec ^2(e+f x)\right )}{f \sqrt [6]{-\tan ^2(e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {1}{3}} d \tan \left (f x + e\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x +e \right )}\, \left (d \tan \left (f x +e \right )\right )^{\frac {4}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sec \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{4/3}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________