Optimal. Leaf size=128 \[ -\frac {4 b (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{-1-\frac {i n}{2}} \, _2F_1\left (2,1+\frac {i n}{2};2+\frac {i n}{2};\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{(i+a)^2 (2 i-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5203, 133}
\begin {gather*} -\frac {4 b (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{-1-\frac {i n}{2}} \, _2F_1\left (2,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {(i-a) (-i a-i b x+1)}{(a+i) (i a+i b x+1)}\right )}{(a+i)^2 (-n+2 i)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 5203
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {(1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}}}{x^2} \, dx\\ &=-\frac {4 b (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{-1-\frac {i n}{2}} \, _2F_1\left (2,1+\frac {i n}{2};2+\frac {i n}{2};\frac {(i-a) (1-i a-i b x)}{(i+a) (1+i a+i b x)}\right )}{(i+a)^2 (2 i-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 125, normalized size = 0.98 \begin {gather*} -\frac {4 i b (1+i a+i b x)^{-\frac {i n}{2}} (-i (i+a+b x))^{1+\frac {i n}{2}} \, _2F_1\left (2,1+\frac {i n}{2};2+\frac {i n}{2};\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )}{(i+a)^2 (-2 i+n) (-i+a+b x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (b x +a \right )}}{x^{2}}\, dx\]
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 {e^{n \operatorname {atan}{\left (a + b x \right )}}}{x^{2}}\, 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 {{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________