Integrand size = 17, antiderivative size = 94 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{-\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {2 a b d^2-\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )}{x} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6538, 2314, 2308, 2266, 2235} \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {\left (c x^n\right )^{\frac {1}{n}} e^{\frac {a}{b n}-\frac {1}{4 b^2 d^2 n^2}} \text {erfi}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{2 b d}\right )}{x}-\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]
[In]
[Out]
Rule 2235
Rule 2266
Rule 2308
Rule 2314
Rule 6538
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {(2 b d n) \int \frac {e^{d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d n x^{-2 a b d^2 n} \left (c x^n\right )^{2 a b d^2}\right ) \int e^{a^2 d^2+b^2 d^2 \log ^2\left (c x^n\right )} x^{-2+2 a b d^2 n} \, dx}{\sqrt {\pi }} \\ & = -\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d \left (c x^n\right )^{2 a b d^2-\frac {-1+2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (a^2 d^2+\frac {\left (-1+2 a b d^2 n\right ) x}{n}+b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt {\pi } x} \\ & = -\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d e^{-\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{2 a b d^2-\frac {-1+2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (\frac {\left (\frac {-1+2 a b d^2 n}{n}+2 b^2 d^2 x\right )^2}{4 b^2 d^2}\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt {\pi } x} \\ & = -\frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{-\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {2 a b d^2-\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )}{x} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {-\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {-1+4 a b d^2 n}{4 b^2 d^2 n^2}} \left (c x^n\right )^{\frac {1}{n}} \text {erfi}\left (a d-\frac {1}{2 b d n}+b d \log \left (c x^n\right )\right )}{x} \]
[In]
[Out]
\[\int \frac {\operatorname {erfi}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.35 \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {\sqrt {-b^{2} d^{2} n^{2}} x \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n - 1\right )} \sqrt {-b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {4 \, b^{2} d^{2} n \log \left (c\right ) + 4 \, a b d^{2} n - 1}{4 \, b^{2} d^{2} n^{2}}\right )} + \operatorname {erfi}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x} \]
[In]
[Out]
\[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\operatorname {erfi}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {erfi}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {erfi}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]
[In]
[Out]