Optimal. Leaf size=64 \[ \frac {2 \cos ^2(e+f x)^{7/12} \, _2F_1\left (\frac {7}{12},\frac {3}{4};\frac {7}{4};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{3 d f \sqrt [3]{b \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, 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)^{7/12} (d \tan (e+f x))^{3/2} \, _2F_1\left (\frac {7}{12},\frac {3}{4};\frac {7}{4};\sin ^2(e+f x)\right )}{3 d f \sqrt [3]{b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2697
Rubi steps
\begin {align*} \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx &=\frac {2 \cos ^2(e+f x)^{7/12} \, _2F_1\left (\frac {7}{12},\frac {3}{4};\frac {7}{4};\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{3 d f \sqrt [3]{b \sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 62, normalized size = 0.97 \begin {gather*} -\frac {3 d \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{6};\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}}{f \sqrt [3]{b \sec (e+f x)} \sqrt {d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {d \tan \left (f x +e \right )}}{\left (b \sec \left (f x +e \right )\right )^{\frac {1}{3}}}\, dx\]
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 {\sqrt {d \tan {\left (e + f x \right )}}}{\sqrt [3]{b \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 {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{{\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]
________________________________________________________________________________________