Optimal. Leaf size=65 \[ \frac {i e^{n \tan ^{-1}(a x)}}{c n}-\frac {2 i e^{n \tan ^{-1}(a x)} \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};e^{2 i \tan ^{-1}(a x)}\right )}{c n} \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 0.10, antiderivative size = 132, normalized size of antiderivative = 2.03, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5082, 96, 131} \[ \frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {2 (1-i a x)^{1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, _2F_1\left (1,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {1-i a x}{i a x+1}\right )}{c (2+i n)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 96
Rule 131
Rule 5082
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{x \left (c+a^2 c x^2\right )} \, dx &=\frac {\int \frac {(1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c}\\ &=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}+\frac {\int \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{c}\\ &=\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{c n}-\frac {2 (1-i a x)^{1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, _2F_1\left (1,1+\frac {i n}{2};2+\frac {i n}{2};\frac {1-i a x}{1+i a x}\right )}{c (2+i n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 120, normalized size = 1.85 \[ \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (2 (n-i a n x) \, _2F_1\left (1,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {a x+i}{i-a x}\right )+(2+i n) (a x-i)\right )}{c n (n-2 i) (a x-i)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{3} + c x}, x\right ) \]
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 = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x \left (a^{2} c \,x^{2}+c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x\,\left (c\,a^2\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{3} + x}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________