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

Optimal. Leaf size=95 \[ \frac{\sqrt{\pi } x \left (c x^n\right )^{-1/n} e^{-\frac{4 a b n \log (F)+1}{4 b^2 n^2 \log (F)}} \text{Erfi}\left (\frac{2 a b \log (F)+2 b^2 \log (F) \log \left (c x^n\right )+\frac{1}{n}}{2 b \sqrt{\log (F)}}\right )}{2 b n \sqrt{\log (F)}} \]

[Out]

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

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Rubi [A]  time = 0.220295, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{\sqrt{\pi } x \left (c x^n\right )^{-1/n} e^{-\frac{4 a b n \log (F)+1}{4 b^2 n^2 \log (F)}} \text{Erfi}\left (\frac{2 a b \log (F)+2 b^2 \log (F) \log \left (c x^n\right )+\frac{1}{n}}{2 b \sqrt{\log (F)}}\right )}{2 b n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

[In]  Int[F^(a + b*Log[c*x^n])^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.147054, size = 93, normalized size = 0.98 \[ \frac{\sqrt{\pi } x \left (c x^n\right )^{-1/n} e^{-\frac{4 a b n \log (F)+1}{4 b^2 n^2 \log (F)}} \text{Erfi}\left (\frac{2 b n \log (F) \left (a+b \log \left (c x^n\right )\right )+1}{2 b n \sqrt{\log (F)}}\right )}{2 b n \sqrt{\log (F)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{F}^{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(F^((a+b*ln(c*x^n))^2),x)

[Out]

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

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Maxima [A]  time = 0.890416, size = 155, normalized size = 1.63 \[ \frac{\sqrt{\pi } F^{b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}} \operatorname{erf}\left (b n \sqrt{-\log \left (F\right )} \log \left (x\right ) - \frac{2 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (F\right ) + 1}{2 \, b n \sqrt{-\log \left (F\right )}}\right ) e^{\left (-\frac{{\left (2 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (F\right ) + 1\right )}^{2}}{4 \, b^{2} n^{2} \log \left (F\right )}\right )}}{2 \, b n \sqrt{-\log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.268176, size = 147, normalized size = 1.55 \[ \frac{\sqrt{\pi } b n \operatorname{erf}\left (\frac{{\left (2 \, b^{2} n^{2} \log \left (F\right ) \log \left (x\right ) + 2 \, b^{2} n \log \left (F\right ) \log \left (c\right ) + 2 \, a b n \log \left (F\right ) + 1\right )} \sqrt{-b^{2} n^{2} \log \left (F\right )}}{2 \, b^{2} n^{2} \log \left (F\right )}\right ) e^{\left (-\frac{4 \, b^{2} n \log \left (F\right ) \log \left (c\right ) + 4 \, a b n \log \left (F\right ) + 1}{4 \, b^{2} n^{2} \log \left (F\right )}\right )} \log \left (F\right )}{2 \, \sqrt{-b^{2} n^{2} \log \left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.381137, size = 130, normalized size = 1.37 \[ \frac{\sqrt{\pi } \operatorname{erf}\left (b n \sqrt{-{\rm ln}\left (F\right )}{\rm ln}\left (x\right ) + b \sqrt{-{\rm ln}\left (F\right )}{\rm ln}\left (c\right ) + a \sqrt{-{\rm ln}\left (F\right )} + \frac{\sqrt{-{\rm ln}\left (F\right )}}{2 \, b n{\rm ln}\left (F\right )}\right ) e^{\left (-\frac{{\rm ln}\left (c\right )}{n} - \frac{a}{b n} - \frac{1}{4 \, b^{2} n^{2}{\rm ln}\left (F\right )}\right )}}{2 \, b n \sqrt{-{\rm ln}\left (F\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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