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

Optimal. Leaf size=87 \[ -\frac{\sqrt{\pi } F^a \left (c x^n\right )^{\frac{1}{n}} e^{-\frac{1}{4 b n^2 \log (F)}} \text{Erfi}\left (\frac{1-2 b n \log (F) \log \left (c x^n\right )}{2 \sqrt{b} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} n x \sqrt{\log (F)}} \]

[Out]

-(F^a*Sqrt[Pi]*(c*x^n)^n^(-1)*Erfi[(1 - 2*b*n*Log[F]*Log[c*x^n])/(2*Sqrt[b]*n*Sq
rt[Log[F]])])/(2*Sqrt[b]*E^(1/(4*b*n^2*Log[F]))*n*x*Sqrt[Log[F]])

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Rubi [A]  time = 0.100257, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\sqrt{\pi } F^a \left (c x^n\right )^{\frac{1}{n}} e^{-\frac{1}{4 b n^2 \log (F)}} \text{Erfi}\left (\frac{1-2 b n \log (F) \log \left (c x^n\right )}{2 \sqrt{b} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} n x \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

-(F^a*Sqrt[Pi]*(c*x^n)^n^(-1)*Erfi[(1 - 2*b*n*Log[F]*Log[c*x^n])/(2*Sqrt[b]*n*Sq
rt[Log[F]])])/(2*Sqrt[b]*E^(1/(4*b*n^2*Log[F]))*n*x*Sqrt[Log[F]])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.111284, size = 87, normalized size = 1. \[ \frac{\sqrt{\pi } F^a \left (c x^n\right )^{\frac{1}{n}} e^{-\frac{1}{4 b n^2 \log (F)}} \text{Erfi}\left (\frac{2 b n \log (F) \log \left (c x^n\right )-1}{2 \sqrt{b} n \sqrt{\log (F)}}\right )}{2 \sqrt{b} n x \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*Sqrt[Pi]*(c*x^n)^n^(-1)*Erfi[(-1 + 2*b*n*Log[F]*Log[c*x^n])/(2*Sqrt[b]*n*Sq
rt[Log[F]])])/(2*Sqrt[b]*E^(1/(4*b*n^2*Log[F]))*n*x*Sqrt[Log[F]])

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Maple [F]  time = 0.184, size = 0, normalized size = 0. \[ \int{\frac{{F}^{a+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^(a+b*ln(c*x^n)^2)/x^2,x)

[Out]

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

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Maxima [A]  time = 0.890079, size = 115, normalized size = 1.32 \[ \frac{\sqrt{\pi } F^{b \log \left (c\right )^{2} + a} \operatorname{erf}\left (\sqrt{-b \log \left (F\right )} n \log \left (x\right ) - \frac{2 \, b n \log \left (F\right ) \log \left (c\right ) - 1}{2 \, \sqrt{-b \log \left (F\right )} n}\right ) e^{\left (-\frac{{\left (2 \, b n \log \left (F\right ) \log \left (c\right ) - 1\right )}^{2}}{4 \, b n^{2} \log \left (F\right )}\right )}}{2 \, \sqrt{-b \log \left (F\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*F^(b*log(c)^2 + a)*erf(sqrt(-b*log(F))*n*log(x) - 1/2*(2*b*n*log(F)
*log(c) - 1)/(sqrt(-b*log(F))*n))*e^(-1/4*(2*b*n*log(F)*log(c) - 1)^2/(b*n^2*log
(F)))/(sqrt(-b*log(F))*n)

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Fricas [A]  time = 0.252334, size = 130, normalized size = 1.49 \[ \frac{\sqrt{\pi } b n \operatorname{erf}\left (\frac{{\left (2 \, b n^{2} \log \left (F\right ) \log \left (x\right ) + 2 \, b n \log \left (F\right ) \log \left (c\right ) - 1\right )} \sqrt{-b n^{2} \log \left (F\right )}}{2 \, b n^{2} \log \left (F\right )}\right ) e^{\left (\frac{4 \, a b n^{2} \log \left (F\right )^{2} + 4 \, b n \log \left (F\right ) \log \left (c\right ) - 1}{4 \, b n^{2} \log \left (F\right )}\right )} \log \left (F\right )}{2 \, \sqrt{-b n^{2} \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(pi)*b*n*erf(1/2*(2*b*n^2*log(F)*log(x) + 2*b*n*log(F)*log(c) - 1)*sqrt(
-b*n^2*log(F))/(b*n^2*log(F)))*e^(1/4*(4*a*b*n^2*log(F)^2 + 4*b*n*log(F)*log(c)
- 1)/(b*n^2*log(F)))*log(F)/sqrt(-b*n^2*log(F))

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Sympy [A]  time = 88.4593, size = 192, normalized size = 2.21 \[ - \frac{2 F^{a} F^{b \log{\left (c \right )}^{2}} F^{b n^{2} \log{\left (x \right )}^{2}} F^{2 b n \log{\left (c \right )} \log{\left (x \right )}} b n^{2} \log{\left (F \right )} \log{\left (x \right )}}{x} - \frac{2 F^{a} F^{b \log{\left (c \right )}^{2}} F^{b n^{2} \log{\left (x \right )}^{2}} F^{2 b n \log{\left (c \right )} \log{\left (x \right )}} b n^{2} \log{\left (F \right )}}{x} - \frac{2 F^{a} F^{b \log{\left (c \right )}^{2}} F^{b n^{2} \log{\left (x \right )}^{2}} F^{2 b n \log{\left (c \right )} \log{\left (x \right )}} b n \log{\left (F \right )} \log{\left (c \right )}}{x} - \frac{F^{a} F^{b \log{\left (c \right )}^{2}} F^{b n^{2} \log{\left (x \right )}^{2}} F^{2 b n \log{\left (c \right )} \log{\left (x \right )}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*F**a*F**(b*log(c)**2)*F**(b*n**2*log(x)**2)*F**(2*b*n*log(c)*log(x))*b*n**2*l
og(F)*log(x)/x - 2*F**a*F**(b*log(c)**2)*F**(b*n**2*log(x)**2)*F**(2*b*n*log(c)*
log(x))*b*n**2*log(F)/x - 2*F**a*F**(b*log(c)**2)*F**(b*n**2*log(x)**2)*F**(2*b*
n*log(c)*log(x))*b*n*log(F)*log(c)/x - F**a*F**(b*log(c)**2)*F**(b*n**2*log(x)**
2)*F**(2*b*n*log(c)*log(x))/x

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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