Optimal. Leaf size=120 \[ -\frac {3\ 2^{\frac {2}{3}-\frac {i n}{2}} \sqrt [3]{a^2 x^2+1} (1-i a x)^{\frac {1}{6} (4+3 i n)} \, _2F_1\left (\frac {1}{6} (3 i n+2),\frac {1}{6} (3 i n+4);\frac {1}{6} (3 i n+10);\frac {1}{2} (1-i a x)\right )}{a (-3 n+4 i) \sqrt [3]{a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5076, 5073, 69} \[ -\frac {3\ 2^{\frac {2}{3}-\frac {i n}{2}} \sqrt [3]{a^2 x^2+1} (1-i a x)^{\frac {1}{6} (4+3 i n)} \, _2F_1\left (\frac {1}{6} (3 i n+2),\frac {1}{6} (3 i n+4);\frac {1}{6} (3 i n+10);\frac {1}{2} (1-i a x)\right )}{a (-3 n+4 i) \sqrt [3]{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{\sqrt [3]{c+a^2 c x^2}} \, dx &=\frac {\sqrt [3]{1+a^2 x^2} \int \frac {e^{n \tan ^{-1}(a x)}}{\sqrt [3]{1+a^2 x^2}} \, dx}{\sqrt [3]{c+a^2 c x^2}}\\ &=\frac {\sqrt [3]{1+a^2 x^2} \int (1-i a x)^{-\frac {1}{3}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{3}-\frac {i n}{2}} \, dx}{\sqrt [3]{c+a^2 c x^2}}\\ &=-\frac {3\ 2^{\frac {2}{3}-\frac {i n}{2}} (1-i a x)^{\frac {1}{6} (4+3 i n)} \sqrt [3]{1+a^2 x^2} \, _2F_1\left (\frac {1}{6} (2+3 i n),\frac {1}{6} (4+3 i n);\frac {1}{6} (10+3 i n);\frac {1}{2} (1-i a x)\right )}{a (4 i-3 n) \sqrt [3]{c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 120, normalized size = 1.00 \[ \frac {3\ 2^{\frac {2}{3}-\frac {i n}{2}} \sqrt [3]{a^2 x^2+1} (1-i a x)^{\frac {2}{3}+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+\frac {1}{3},\frac {i n}{2}+\frac {2}{3};\frac {i n}{2}+\frac {5}{3};\frac {1}{2}-\frac {i a x}{2}\right )}{a (3 n-4 i) \sqrt [3]{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt [3]{c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________