Optimal. Leaf size=85 \[ \frac {3 i \sqrt [6]{1+i \tan (c+d x)} (e \sec (c+d x))^{2/3} \, _2F_1\left (\frac {1}{3},\frac {7}{6};\frac {4}{3};\frac {1}{2} (1-i \tan (c+d x))\right )}{2 \sqrt [6]{2} d \sqrt {a+i a \tan (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 85, 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 = {3505, 3523, 70, 69} \[ \frac {3 i \sqrt [6]{1+i \tan (c+d x)} (e \sec (c+d x))^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {7}{6},\frac {4}{3},\frac {1}{2} (1-i \tan (c+d x))\right )}{2 \sqrt [6]{2} d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \frac {(e \sec (c+d x))^{2/3}}{\sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {(e \sec (c+d x))^{2/3} \int \frac {\sqrt [3]{a-i a \tan (c+d x)}}{\sqrt [6]{a+i a \tan (c+d x)}} \, dx}{\sqrt [3]{a-i a \tan (c+d x)} \sqrt [3]{a+i a \tan (c+d x)}}\\ &=\frac {\left (a^2 (e \sec (c+d x))^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{(a-i a x)^{2/3} (a+i a x)^{7/6}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [3]{a-i a \tan (c+d x)} \sqrt [3]{a+i a \tan (c+d x)}}\\ &=\frac {\left (a (e \sec (c+d x))^{2/3} \sqrt [6]{\frac {a+i a \tan (c+d x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {i x}{2}\right )^{7/6} (a-i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{2 \sqrt [6]{2} d \sqrt [3]{a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {3 i \, _2F_1\left (\frac {1}{3},\frac {7}{6};\frac {4}{3};\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{2/3} \sqrt [6]{1+i \tan (c+d x)}}{2 \sqrt [6]{2} d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.46, size = 116, normalized size = 1.36 \[ \frac {3 i \sqrt [6]{2} \sqrt [6]{1+e^{2 i (c+d x)}} \left (\frac {e e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {1}{6};\frac {5}{6};-e^{2 i (c+d x)}\right )}{d \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ \frac {2^{\frac {1}{6}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (3 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 6 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + {\left (a d e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}\right )} {\rm integral}\left (\frac {2^{\frac {1}{6}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \left (\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{\left (i \, d x + i \, c\right )} + 4 i\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}}{a d e^{\left (4 i \, d x + 4 i \, c\right )} - 3 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} + 3 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} - a d e^{\left (i \, d x + i \, c\right )}}, x\right )}{a d e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d e^{\left (i \, d x + i \, c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.40, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec \left (d x +c \right )\right )^{\frac {2}{3}}}{\sqrt {a +i a \tan \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {2}{3}}}{\sqrt {i \, a \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2/3}}{\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sec {\left (c + d x \right )}\right )^{\frac {2}{3}}}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________