\(\int f^{a+b x^n} x^{-1-2 n} \, dx\) [188]

Optimal result

Integrand size = 17, antiderivative size = 71 \[ \int f^{a+b x^n} x^{-1-2 n} \, dx=-\frac {f^{a+b x^n} x^{-2 n}}{2 n}-\frac {b f^{a+b x^n} x^{-n} \log (f)}{2 n}+\frac {b^2 f^a \operatorname {ExpIntegralEi}\left (b x^n \log (f)\right ) \log ^2(f)}{2 n} \]

[Out]

-1/2*f^(a+b*x^n)/n/(x^(2*n))-1/2*b*f^(a+b*x^n)*ln(f)/n/(x^n)+1/2*b^2*f^a*Ei(b*x^n*ln(f))*ln(f)^2/n

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2246, 2241} \[ \int f^{a+b x^n} x^{-1-2 n} \, dx=\frac {b^2 f^a \log ^2(f) \operatorname {ExpIntegralEi}\left (b x^n \log (f)\right )}{2 n}-\frac {x^{-2 n} f^{a+b x^n}}{2 n}-\frac {b \log (f) x^{-n} f^{a+b x^n}}{2 n} \]

[In]

Int[f^(a + b*x^n)*x^(-1 - 2*n),x]

[Out]

-1/2*f^(a + b*x^n)/(n*x^(2*n)) - (b*f^(a + b*x^n)*Log[f])/(2*n*x^n) + (b^2*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log
[f]^2)/(2*n)

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2246

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^Simplify[m + n]*F^(a +
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && IntegerQ[2*Simplify[(m + 1)/n]] && LtQ[-4, Simpl
ify[(m + 1)/n], 5] &&  !RationalQ[m] && SumSimplerQ[m, n]

Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+b x^n} x^{-2 n}}{2 n}+\frac {1}{2} (b \log (f)) \int f^{a+b x^n} x^{-1-n} \, dx \\ & = -\frac {f^{a+b x^n} x^{-2 n}}{2 n}-\frac {b f^{a+b x^n} x^{-n} \log (f)}{2 n}+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^n}}{x} \, dx \\ & = -\frac {f^{a+b x^n} x^{-2 n}}{2 n}-\frac {b f^{a+b x^n} x^{-n} \log (f)}{2 n}+\frac {b^2 f^a \text {Ei}\left (b x^n \log (f)\right ) \log ^2(f)}{2 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.35 \[ \int f^{a+b x^n} x^{-1-2 n} \, dx=-\frac {b^2 f^a \Gamma \left (-2,-b x^n \log (f)\right ) \log ^2(f)}{n} \]

[In]

Integrate[f^(a + b*x^n)*x^(-1 - 2*n),x]

[Out]

-((b^2*f^a*Gamma[-2, -(b*x^n*Log[f])]*Log[f]^2)/n)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {f^{b \,x^{n}} f^{a} x^{-2 n}}{2 n}-\frac {\ln \left (f \right ) b \,f^{b \,x^{n}} f^{a} x^{-n}}{2 n}-\frac {\ln \left (f \right )^{2} b^{2} f^{a} \operatorname {Ei}_{1}\left (-b \,x^{n} \ln \left (f \right )\right )}{2 n}\)	\(70\)
meijerg	\(\frac {f^{a} \ln \left (f \right )^{2} b^{2} \left (-\frac {x^{\left (\frac {-1-2 n}{n}+\frac {1}{n}\right ) n} \left (-b \right )^{\frac {-1-2 n}{n}+\frac {1}{n}} \ln \left (f \right )^{\frac {-1-2 n}{n}+\frac {1}{n}}}{2}+x^{\left (1+\frac {-1-2 n}{n}+\frac {1}{n}\right ) n} \left (-b \right )^{1+\frac {-1-2 n}{n}+\frac {1}{n}} \ln \left (f \right )^{1+\frac {-1-2 n}{n}+\frac {1}{n}}+\frac {\left (-1\right )^{-\frac {-1-2 n}{n}-\frac {1}{n}} \left (-\Psi \left (1-\frac {-1-2 n}{n}-\frac {1}{n}\right )+n \ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )\right )}{\Gamma \left (1-\frac {-1-2 n}{n}-\frac {1}{n}\right )}+\frac {\left (-1\right )^{-\frac {-1-2 n}{n}-\frac {1}{n}} x^{-2 n} \left (9 b^{2} x^{2 n} \ln \left (f \right )^{2}+12 b \,x^{n} \ln \left (f \right )+6\right )}{2 \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right ) b^{2} \ln \left (f \right )^{2}}-\frac {\left (-1\right )^{-\frac {-1-2 n}{n}-\frac {1}{n}} x^{-2 n} \left (3+3 b \,x^{n} \ln \left (f \right )\right ) {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{\Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right ) b^{2} \ln \left (f \right )^{2}}+\frac {3 \left (-1\right )^{-\frac {-1-2 n}{n}-\frac {1}{n}} \left (-\gamma -\ln \left (-b \,x^{n} \ln \left (f \right )\right )-\operatorname {Ei}_{1}\left (-b \,x^{n} \ln \left (f \right )\right )\right )}{\Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}\right )}{n}\)	\(406\)

[In]

int(f^(a+b*x^n)*x^(-1-2*n),x,method=_RETURNVERBOSE)

[Out]

-1/2/n*f^(b*x^n)*f^a/(x^n)^2-1/2/n*ln(f)*b*f^(b*x^n)*f^a/(x^n)-1/2/n*ln(f)^2*b^2*f^a*Ei(1,-b*x^n*ln(f))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int f^{a+b x^n} x^{-1-2 n} \, dx=\frac {b^{2} f^{a} x^{2 \, n} {\rm Ei}\left (b x^{n} \log \left (f\right )\right ) \log \left (f\right )^{2} - {\left (b x^{n} \log \left (f\right ) + 1\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{2 \, n x^{2 \, n}} \]

[In]

integrate(f^(a+b*x^n)*x^(-1-2*n),x, algorithm="fricas")

[Out]

1/2*(b^2*f^a*x^(2*n)*Ei(b*x^n*log(f))*log(f)^2 - (b*x^n*log(f) + 1)*e^(b*x^n*log(f) + a*log(f)))/(n*x^(2*n))

Sympy [F]

\[ \int f^{a+b x^n} x^{-1-2 n} \, dx=\int f^{a + b x^{n}} x^{- 2 n - 1}\, dx \]

[In]

integrate(f**(a+b*x**n)*x**(-1-2*n),x)

[Out]

Integral(f**(a + b*x**n)*x**(-2*n - 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.35 \[ \int f^{a+b x^n} x^{-1-2 n} \, dx=-\frac {b^{2} f^{a} \Gamma \left (-2, -b x^{n} \log \left (f\right )\right ) \log \left (f\right )^{2}}{n} \]

[In]

integrate(f^(a+b*x^n)*x^(-1-2*n),x, algorithm="maxima")

[Out]

-b^2*f^a*gamma(-2, -b*x^n*log(f))*log(f)^2/n

Giac [F]

\[ \int f^{a+b x^n} x^{-1-2 n} \, dx=\int { f^{b x^{n} + a} x^{-2 \, n - 1} \,d x } \]

[In]

integrate(f^(a+b*x^n)*x^(-1-2*n),x, algorithm="giac")

[Out]

integrate(f^(b*x^n + a)*x^(-2*n - 1), x)

Mupad [F(-1)]

Timed out. \[ \int f^{a+b x^n} x^{-1-2 n} \, dx=\int \frac {f^{a+b\,x^n}}{x^{2\,n+1}} \,d x \]

[In]

int(f^(a + b*x^n)/x^(2*n + 1),x)

[Out]

int(f^(a + b*x^n)/x^(2*n + 1), x)