Optimal. Leaf size=83 \[ \frac {3 i \, _2F_1\left (\frac {1}{6},\frac {4}{3};\frac {7}{6};\frac {1}{2} (1-i \tan (c+d x))\right ) \sqrt [3]{e \sec (c+d x)} \sqrt [3]{1+i \tan (c+d x)}}{\sqrt [3]{2} d \sqrt {a+i a \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3586, 3604, 72,
71} \begin {gather*} \frac {3 i \sqrt [3]{1+i \tan (c+d x)} \sqrt [3]{e \sec (c+d x)} \, _2F_1\left (\frac {1}{6},\frac {4}{3};\frac {7}{6};\frac {1}{2} (1-i \tan (c+d x))\right )}{\sqrt [3]{2} d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {\sqrt [3]{e \sec (c+d x)} \int \frac {\sqrt [6]{a-i a \tan (c+d x)}}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{\sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}}\\ &=\frac {\left (a^2 \sqrt [3]{e \sec (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(a-i a x)^{5/6} (a+i a x)^{4/3}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [6]{a-i a \tan (c+d x)} \sqrt [6]{a+i a \tan (c+d x)}}\\ &=\frac {\left (a \sqrt [3]{e \sec (c+d x)} \sqrt [3]{\frac {a+i a \tan (c+d x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{4/3} (a-i a x)^{5/6}} \, dx,x,\tan (c+d x)\right )}{2 \sqrt [3]{2} d \sqrt [6]{a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {3 i \, _2F_1\left (\frac {1}{6},\frac {4}{3};\frac {7}{6};\frac {1}{2} (1-i \tan (c+d x))\right ) \sqrt [3]{e \sec (c+d x)} \sqrt [3]{1+i \tan (c+d x)}}{\sqrt [3]{2} d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.61, size = 95, normalized size = 1.14 \begin {gather*} \frac {3 \left (8 i-\frac {2 i e^{2 i (c+d x)} \, _2F_1\left (\frac {2}{3},\frac {5}{6};\frac {5}{3};-e^{2 i (c+d x)}\right )}{\sqrt [6]{1+e^{2 i (c+d x)}}}\right ) \sqrt [3]{e \sec (c+d x)}}{16 d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.86, size = 0, normalized size = 0.00 \[\int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {1}{3}}}{\sqrt {a +i a \tan \left (d x +c \right )}}\, dx\]
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 [3]{e \sec {\left (c + d x \right )}}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\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 = -0.01 \begin {gather*} \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{1/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________