3.589 \(\int \frac{F^{a+b \log ^2\left (c x^n\right )}}{x} \, dx\)

Optimal. Leaf size=45 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} \log \left (c x^n\right )\right )}{2 \sqrt{b} n \sqrt{\log (F)}} \]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[Log[F]]*Log[c*x^n]])/(2*Sqrt[b]*n*Sqrt[Log[F]])

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Rubi [A]  time = 0.0722003, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} \log \left (c x^n\right )\right )}{2 \sqrt{b} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*Log[c*x^n]^2)/x,x]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[Log[F]]*Log[c*x^n]])/(2*Sqrt[b]*n*Sqrt[Log[F]])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + b \log{\left (c x^{n} \right )}^{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(F**(a+b*ln(c*x**n)**2)/x,x)

[Out]

Integral(F**(a + b*log(c*x**n)**2)/x, x)

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Mathematica [A]  time = 0.0089944, size = 45, normalized size = 1. \[ \frac{\sqrt{\pi } F^a \text{Erfi}\left (\sqrt{b} \sqrt{\log (F)} \log \left (c x^n\right )\right )}{2 \sqrt{b} n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Integrate[F^(a + b*Log[c*x^n]^2)/x,x]

[Out]

(F^a*Sqrt[Pi]*Erfi[Sqrt[b]*Sqrt[Log[F]]*Log[c*x^n]])/(2*Sqrt[b]*n*Sqrt[Log[F]])

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Maple [F]  time = 180., size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*ln(c*x^n)^2)/x,x)

[Out]

int(F^(a+b*ln(c*x^n)^2)/x,x)

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Maxima [A]  time = 0.849054, size = 84, normalized size = 1.87 \[ \frac{\sqrt{\pi } F^{b \log \left (c\right )^{2} + a} \operatorname{erf}\left (-\frac{b \log \left (F\right ) \log \left (c\right )}{\sqrt{-b \log \left (F\right )}} + \sqrt{-b \log \left (F\right )} n \log \left (x\right )\right )}{2 \, \sqrt{-b \log \left (F\right )} F^{b \log \left (c\right )^{2}} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(c*x^n)^2 + a)/x,x, algorithm="maxima")

[Out]

1/2*sqrt(pi)*F^(b*log(c)^2 + a)*erf(-b*log(F)*log(c)/sqrt(-b*log(F)) + sqrt(-b*l
og(F))*n*log(x))/(sqrt(-b*log(F))*F^(b*log(c)^2)*n)

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Fricas [A]  time = 0.263826, size = 59, normalized size = 1.31 \[ \frac{\sqrt{\pi } F^{a} b n \operatorname{erf}\left (\frac{\sqrt{-b n^{2} \log \left (F\right )}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}}{n}\right ) \log \left (F\right )}{2 \, \sqrt{-b n^{2} \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(c*x^n)^2 + a)/x,x, algorithm="fricas")

[Out]

1/2*sqrt(pi)*F^a*b*n*erf(sqrt(-b*n^2*log(F))*(n*log(x) + log(c))/n)*log(F)/sqrt(
-b*n^2*log(F))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{a + b \log{\left (c x^{n} \right )}^{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F**(a+b*ln(c*x**n)**2)/x,x)

[Out]

Integral(F**(a + b*log(c*x**n)**2)/x, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{F^{b \log \left (c x^{n}\right )^{2} + a}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(F^(b*log(c*x^n)^2 + a)/x,x, algorithm="giac")

[Out]

integrate(F^(b*log(c*x^n)^2 + a)/x, x)