Optimal. Leaf size=126 \[ -\frac {e^{\frac {(1+m) \left (1+m-4 a b d^2 n\right )}{4 b^2 d^2 n^2}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {Erf}\left (\frac {1+m-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )}{2 b d n}\right )}{1+m}+\frac {(e x)^{1+m} \text {Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6537, 2314,
2308, 2266, 2236} \begin {gather*} \frac {(e x)^{m+1} \text {Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)}-\frac {x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (\frac {(m+1) \left (-4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text {Erf}\left (\frac {-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2236
Rule 2266
Rule 2308
Rule 2314
Rule 6537
Rubi steps
\begin {align*} \int (e x)^m \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac {(2 b d n) \int e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2} (e x)^m \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac {(2 b d n) \int \exp \left (-a^2 d^2-2 a b d^2 \log \left (c x^n\right )-b^2 d^2 \log ^2\left (c x^n\right )\right ) (e x)^m \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac {(2 b d n) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} (e x)^m \left (c x^n\right )^{-2 a b d^2} \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\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 a b d^2 n} (e x)^m \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac {\left (2 b d n x^{-m+2 a b d^2 n} (e x)^m \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^{m-2 a b d^2 n} \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac {\left (2 b d x (e x)^m \left (c x^n\right )^{-2 a b d^2-\frac {1+m-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (1+m-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 )}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac {\left (2 b d \exp \left (\frac {(1+m) \left (1+m-4 a b d^2 n\right )}{4 b^2 d^2 n^2}\right ) x (e x)^m \left (c x^n\right )^{-2 a b d^2-\frac {1+m-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {\left (\frac {1+m-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 )}{(1+m) \sqrt {\pi }}\\ &=-\frac {\exp \left (\frac {(1+m) \left (1+m-4 a b d^2 n\right )}{4 b^2 d^2 n^2}\right ) x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erf}\left (\frac {1+m-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )}{2 b d n}\right )}{1+m}+\frac {(e x)^{1+m} \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.37, size = 126, normalized size = 1.00 \begin {gather*} \frac {(e x)^m \left (e^{\frac {(1+m) \left (1+m-4 a b d^2 n+4 b^2 d^2 n^2 \log (x)-4 b^2 d^2 n \log \left (c x^n\right )\right )}{4 b^2 d^2 n^2}} x^{-m} \text {Erf}\left (a d-\frac {1+m-2 b^2 d^2 n \log \left (c x^n\right )}{2 b d n}\right )+x \text {Erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \mathrm {erfc}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 186, normalized size = 1.48 \begin {gather*} -\frac {x \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) e^{\left (m \log \left (x\right ) + m\right )} - \sqrt {b^{2} d^{2} n^{2}} \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 - m - 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} m n^{2} - 4 \, {\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n \log \left (c\right ) + m^{2} - 4 \, {\left (a b d^{2} m + a b d^{2}\right )} n + 2 \, m + 1}{4 \, b^{2} d^{2} n^{2}}\right )} - x e^{\left (m \log \left (x\right ) + m\right )}}{m + 1} \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 \left (e x\right )^{m} \operatorname {erfc}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, 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 \mathrm {erfc}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________