\(\int f^{a+b x^n} x^{-1-n} \, dx\) [187]

Optimal result

Integrand size = 17, antiderivative size = 38 \[ \int f^{a+b x^n} x^{-1-n} \, dx=-\frac {f^{a+b x^n} x^{-n}}{n}+\frac {b f^a \operatorname {ExpIntegralEi}\left (b x^n \log (f)\right ) \log (f)}{n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2246, 2241} \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^a \log (f) \operatorname {ExpIntegralEi}\left (b x^n \log (f)\right )}{n}-\frac {x^{-n} f^{a+b x^n}}{n} \]

[In]

Int[f^(a + b*x^n)*x^(-1 - n),x]

[Out]

-(f^(a + b*x^n)/(n*x^n)) + (b*f^a*ExpIntegralEi[b*x^n*Log[f]]*Log[f])/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^{-n}}{n}+(b \log (f)) \int \frac {f^{a+b x^n}}{x} \, dx \\ & = -\frac {f^{a+b x^n} x^{-n}}{n}+\frac {b f^a \text {Ei}\left (b x^n \log (f)\right ) \log (f)}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53 \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^a \Gamma \left (-1,-b x^n \log (f)\right ) \log (f)}{n} \]

[In]

Integrate[f^(a + b*x^n)*x^(-1 - n),x]

[Out]

(b*f^a*Gamma[-1, -(b*x^n*Log[f])]*Log[f])/n

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {f^{b \,x^{n}} f^{a} x^{-n}}{n}-\frac {\ln \left (f \right ) b \,f^{a} \operatorname {Ei}_{1}\left (-b \,x^{n} \ln \left (f \right )\right )}{n}\)	\(43\)
meijerg	\(-\frac {f^{a} \ln \left (f \right ) b \left (-x^{\left (\frac {-1-n}{n}+\frac {1}{n}\right ) n} \left (-b \right )^{\frac {-1-n}{n}+\frac {1}{n}} \ln \left (f \right )^{\frac {-1-n}{n}+\frac {1}{n}}+\frac {\left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} \left (-\Psi \left (1-\frac {-1-n}{n}-\frac {1}{n}\right )+n \ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )\right )}{\Gamma \left (1-\frac {-1-n}{n}-\frac {1}{n}\right )}+\frac {\left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} x^{-n} \left (2+2 b \,x^{n} \ln \left (f \right )\right )}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right ) b \ln \left (f \right )}-\frac {2 \left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} x^{-n} {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right ) b \ln \left (f \right )}+\frac {2 \left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} \left (-\gamma -\ln \left (-b \,x^{n} \ln \left (f \right )\right )-\operatorname {Ei}_{1}\left (-b \,x^{n} \ln \left (f \right )\right )\right )}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right )}\right )}{n}\)	\(324\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.13 \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^{a} x^{n} {\rm Ei}\left (b x^{n} \log \left (f\right )\right ) \log \left (f\right ) - e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{n x^{n}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53 \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^{a} \Gamma \left (-1, -b x^{n} \log \left (f\right )\right ) \log \left (f\right )}{n} \]

[In]

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

[Out]

b*f^a*gamma(-1, -b*x^n*log(f))*log(f)/n

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(f^(a + b*x^n)/x^(n + 1),x)

[Out]

int(f^(a + b*x^n)/x^(n + 1), x)