Optimal. Leaf size=91 \[ x \text {Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {1+4 a b d^2 n}{4 b^2 d^2 n^2}} x \left (c x^n\right )^{-1/n} \text {Erfi}\left (\frac {2 a b d^2+\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6534, 2312,
2308, 2266, 2235} \begin {gather*} x \text {Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{-\frac {4 a b d^2 n+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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2308
Rule 2312
Rule 6534
Rubi steps
\begin {align*} \int \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {(2 b d n) \int e^{d^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{\sqrt {\pi }}\\ &=x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\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 ) \, dx}{\sqrt {\pi }}\\ &=x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {(2 b d n) \int e^{a^2 d^2+b^2 d^2 \log ^2\left (c x^n\right )} \left (c x^n\right )^{2 a b d^2} \, dx}{\sqrt {\pi }}\\ &=x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\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} \, dx}{\sqrt {\pi }}\\ &=x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (2 b d x \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 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (2 b d e^{-\frac {1+4 a b d^2 n}{4 b^2 d^2 n^2}} x \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 \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}} x \left (c x^n\right )^{-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 )\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 78, normalized size = 0.86 \begin {gather*} x \text {Erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {\frac {\frac {1}{d^2}+4 a b n}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}} x \text {Erfi}\left (a d+\frac {1}{2 b d n}+b d \log \left (c x^n\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \erfi \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.36, size = 122, normalized size = 1.34 \begin {gather*} -\sqrt {b^{2} d^{2} n^{2}} \operatorname {erfi}\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 )} + x \operatorname {erfi}\left (b d \log \left (c x^{n}\right ) + a d\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {erfi}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {erfi}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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