Optimal. Leaf size=64 \[ \frac {2 \cos ^2(e+f x)^{17/12} \, _2F_1\left (\frac {5}{4},\frac {17}{12};\frac {9}{4};\sin ^2(e+f x)\right ) \sqrt [3]{b \sec (e+f x)} (d \tan (e+f x))^{5/2}}{5 d f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {2697}
\begin {gather*} \frac {2 \cos ^2(e+f x)^{17/12} \sqrt [3]{b \sec (e+f x)} (d \tan (e+f x))^{5/2} \, _2F_1\left (\frac {5}{4},\frac {17}{12};\frac {9}{4};\sin ^2(e+f x)\right )}{5 d f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2697
Rubi steps
\begin {align*} \int \sqrt [3]{b \sec (e+f x)} (d \tan (e+f x))^{3/2} \, dx &=\frac {2 \cos ^2(e+f x)^{17/12} \, _2F_1\left (\frac {5}{4},\frac {17}{12};\frac {9}{4};\sin ^2(e+f x)\right ) \sqrt [3]{b \sec (e+f x)} (d \tan (e+f x))^{5/2}}{5 d f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.15, size = 62, normalized size = 0.97 \begin {gather*} \frac {3 d \, _2F_1\left (-\frac {1}{4},\frac {1}{6};\frac {7}{6};\sec ^2(e+f x)\right ) \sqrt [3]{b \sec (e+f x)} \sqrt {d \tan (e+f x)}}{f \sqrt [4]{-\tan ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (b \sec \left (f x +e \right )\right )^{\frac {1}{3}} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{b \sec {\left (e + f x \right )}} \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________