Optimal. Leaf size=57 \[ \frac {\cos ^2(e+f x)^{7/3} \, _2F_1\left (\frac {7}{3},\frac {5}{2};\frac {7}{2};\sin ^2(e+f x)\right ) \tan ^5(e+f x)}{5 f \sqrt [3]{d \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 57, 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 = {2697}
\begin {gather*} \frac {\cos ^2(e+f x)^{7/3} \tan ^5(e+f x) \, _2F_1\left (\frac {7}{3},\frac {5}{2};\frac {7}{2};\sin ^2(e+f x)\right )}{5 f \sqrt [3]{d \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2697
Rubi steps
\begin {align*} \int \frac {\tan ^4(e+f x)}{\sqrt [3]{d \sec (e+f x)}} \, dx &=\frac {\cos ^2(e+f x)^{7/3} \, _2F_1\left (\frac {7}{3},\frac {5}{2};\frac {7}{2};\sin ^2(e+f x)\right ) \tan ^5(e+f x)}{5 f \sqrt [3]{d \sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 69, normalized size = 1.21 \begin {gather*} \frac {3 \left (-11+9 \sqrt [3]{\cos ^2(e+f x)} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\sin ^2(e+f x)\right )+2 \sec ^2(e+f x)\right ) \tan (e+f x)}{16 f \sqrt [3]{d \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\tan ^{4}\left (f x +e \right )}{\left (d \sec \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \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 \frac {\tan ^{4}{\left (e + f x \right )}}{\sqrt [3]{d \sec {\left (e + f x \right )}}}\, 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 \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (\frac {d}{\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]
________________________________________________________________________________________