3.2.25 \(\int x \log (d+e (f^{c (a+b x)})^n) \, dx\) [125]

Optimal. Leaf size=118 \[ \frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)} \]

[Out]

1/2*x^2*ln(d+e*(f^(c*(b*x+a)))^n)-1/2*x^2*ln(1+e*(f^(c*(b*x+a)))^n/d)-x*polylog(2,-e*(f^(c*(b*x+a)))^n/d)/b/c/
n/ln(f)+polylog(3,-e*(f^(c*(b*x+a)))^n/d)/b^2/c^2/n^2/ln(f)^2

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Rubi [A]
time = 0.04, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2612, 2611, 2320, 6724} \begin {gather*} \frac {\text {PolyLog}\left (3,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}-\frac {x \text {PolyLog}\left (2,-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {1}{2} x^2 \log \left (e \left (f^{c (a+b x)}\right )^n+d\right )-\frac {1}{2} x^2 \log \left (\frac {e \left (f^{c (a+b x)}\right )^n}{d}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Log[d + e*(f^(c*(a + b*x)))^n],x]

[Out]

(x^2*Log[d + e*(f^(c*(a + b*x)))^n])/2 - (x^2*Log[1 + (e*(f^(c*(a + b*x)))^n)/d])/2 - (x*PolyLog[2, -((e*(f^(c
*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)]/(b^2*c^2*n^2*Log[f]^2)

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2612

Int[Log[(d_) + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[
(f + g*x)^(m + 1)*(Log[d + e*(F^(c*(a + b*x)))^n]/(g*(m + 1))), x] + (Int[(f + g*x)^m*Log[1 + (e/d)*(F^(c*(a +
b*x)))^n], x] - Simp[(f + g*x)^(m + 1)*(Log[1 + (e/d)*(F^(c*(a + b*x)))^n]/(g*(m + 1))), x]) /; FreeQ[{F, a,
b, c, d, e, f, g, n}, x] && GtQ[m, 0] && NeQ[d, 1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int x \log \left (d+e \left (f^{c (a+b x)}\right )^n\right ) \, dx &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )+\int x \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx\\ &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\int \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right ) \, dx}{b c n \log (f)}\\ &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {e x^n}{d}\right )}{x} \, dx,x,f^{c (a+b x)}\right )}{b^2 c^2 n \log ^2(f)}\\ &=\frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 118, normalized size = 1.00 \begin {gather*} \frac {1}{2} x^2 \log \left (d+e \left (f^{c (a+b x)}\right )^n\right )-\frac {1}{2} x^2 \log \left (1+\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )-\frac {x \text {Li}_2\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (f)}+\frac {\text {Li}_3\left (-\frac {e \left (f^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Log[d + e*(f^(c*(a + b*x)))^n],x]

[Out]

(x^2*Log[d + e*(f^(c*(a + b*x)))^n])/2 - (x^2*Log[1 + (e*(f^(c*(a + b*x)))^n)/d])/2 - (x*PolyLog[2, -((e*(f^(c
*(a + b*x)))^n)/d)])/(b*c*n*Log[f]) + PolyLog[3, -((e*(f^(c*(a + b*x)))^n)/d)]/(b^2*c^2*n^2*Log[f]^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(557\) vs. \(2(114)=228\).
time = 0.02, size = 558, normalized size = 4.73

method	result	size
risch	\(\frac {x^{2} \ln \left (d +e \left (f^{c \left (b x +a \right )}\right )^{n}\right )}{2}+\frac {\polylog \left (3, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}}{d}\right )}{c^{2} b^{2} \ln \left (f \right )^{2} n^{2}}+\frac {\ln \left (d +e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}\right ) \ln \left (f^{c \left (b x +a \right )}\right ) x}{c b \ln \left (f \right )}-\frac {\ln \left (\frac {d +e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}}{d}\right ) x \ln \left (f^{c \left (b x +a \right )}\right )}{c b \ln \left (f \right )}-\frac {\dilog \left (\frac {d +e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}}{d}\right ) x}{c b \ln \left (f \right ) n}+\frac {\dilog \left (\frac {d +e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}}{d}\right ) \ln \left (f^{c \left (b x +a \right )}\right )}{c^{2} b^{2} \ln \left (f \right )^{2} n}+\frac {\ln \left (\frac {d +e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}}{d}\right ) \ln \left (f^{c \left (b x +a \right )}\right )^{2}}{c^{2} b^{2} \ln \left (f \right )^{2}}-\frac {\ln \left (1+\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}}{d}\right ) \ln \left (f^{c \left (b x +a \right )}\right )^{2}}{2 c^{2} b^{2} \ln \left (f \right )^{2}}-\frac {\polylog \left (2, -\frac {e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}}{d}\right ) \ln \left (f^{c \left (b x +a \right )}\right )}{c^{2} b^{2} \ln \left (f \right )^{2} n}-\frac {\ln \left (d +e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}\right ) x^{2}}{2}-\frac {\ln \left (d +e \,f^{b c n x} f^{-b c n x} \left (f^{c \left (b x +a \right )}\right )^{n}\right ) \ln \left (f^{c \left (b x +a \right )}\right )^{2}}{2 c^{2} b^{2} \ln \left (f \right )^{2}}\)	\(558\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(d+e*(f^(c*(b*x+a)))^n),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2*ln(d+e*(f^(c*(b*x+a)))^n)+1/c^2/b^2/ln(f)^2/n^2*polylog(3,-e*f^(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^
n/d)+1/c/b/ln(f)*ln(d+e*f^(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^n)*ln(f^(c*(b*x+a)))*x-1/c/b/ln(f)*ln((d+e*f^
(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^n)/d)*x*ln(f^(c*(b*x+a)))-1/c/b/ln(f)/n*dilog((d+e*f^(b*c*n*x)*f^(-b*c*
n*x)*(f^(c*(b*x+a)))^n)/d)*x+1/c^2/b^2/ln(f)^2/n*dilog((d+e*f^(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^n)/d)*ln(
f^(c*(b*x+a)))+1/c^2/b^2/ln(f)^2*ln((d+e*f^(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^n)/d)*ln(f^(c*(b*x+a)))^2-1/
2/c^2/b^2/ln(f)^2*ln(1+e*f^(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^n/d)*ln(f^(c*(b*x+a)))^2-1/c^2/b^2/ln(f)^2/n
*polylog(2,-e*f^(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^n/d)*ln(f^(c*(b*x+a)))-1/2*ln(d+e*f^(b*c*n*x)*f^(-b*c*n
*x)*(f^(c*(b*x+a)))^n)*x^2-1/2/c^2/b^2/ln(f)^2*ln(d+e*f^(b*c*n*x)*f^(-b*c*n*x)*(f^(c*(b*x+a)))^n)*ln(f^(c*(b*x
+a)))^2

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Maxima [A]
time = 0.30, size = 133, normalized size = 1.13 \begin {gather*} \frac {1}{2} \, x^{2} \log \left (f^{{\left (b x + a\right )} c n} e + d\right ) - \frac {b^{2} c^{2} n^{2} x^{2} \log \left (f\right )^{2} \log \left (\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d} + 1\right ) + 2 \, b c n x {\rm Li}_2\left (-\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d}\right ) \log \left (f\right ) - 2 \, {\rm Li}_{3}(-\frac {f^{a c n} e^{\left (b c n x \log \left (f\right ) + 1\right )}}{d})}{2 \, b^{2} c^{2} n^{2} \log \left (f\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(d+e*(f^(c*(b*x+a)))^n),x, algorithm="maxima")

[Out]

1/2*x^2*log(f^((b*x + a)*c*n)*e + d) - 1/2*(b^2*c^2*n^2*x^2*log(f)^2*log(f^(a*c*n)*e^(b*c*n*x*log(f) + 1)/d +
1) + 2*b*c*n*x*dilog(-f^(a*c*n)*e^(b*c*n*x*log(f) + 1)/d)*log(f) - 2*polylog(3, -f^(a*c*n)*e^(b*c*n*x*log(f) +
1)/d))/(b^2*c^2*n^2*log(f)^2)

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Fricas [A]
time = 0.37, size = 173, normalized size = 1.47 \begin {gather*} -\frac {2 \, b c n x {\rm Li}_2\left (-\frac {f^{b c n x + a c n} e + d}{d} + 1\right ) \log \left (f\right ) - {\left (b^{2} c^{2} n^{2} x^{2} - a^{2} c^{2} n^{2}\right )} \log \left (f^{b c n x + a c n} e + d\right ) \log \left (f\right )^{2} + {\left (b^{2} c^{2} n^{2} x^{2} - a^{2} c^{2} n^{2}\right )} \log \left (f\right )^{2} \log \left (\frac {f^{b c n x + a c n} e + d}{d}\right ) - 2 \, {\rm polylog}\left (3, -\frac {f^{b c n x + a c n} e}{d}\right )}{2 \, b^{2} c^{2} n^{2} \log \left (f\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(d+e*(f^(c*(b*x+a)))^n),x, algorithm="fricas")

[Out]

-1/2*(2*b*c*n*x*dilog(-(f^(b*c*n*x + a*c*n)*e + d)/d + 1)*log(f) - (b^2*c^2*n^2*x^2 - a^2*c^2*n^2)*log(f^(b*c*
n*x + a*c*n)*e + d)*log(f)^2 + (b^2*c^2*n^2*x^2 - a^2*c^2*n^2)*log(f)^2*log((f^(b*c*n*x + a*c*n)*e + d)/d) - 2
*polylog(3, -f^(b*c*n*x + a*c*n)*e/d))/(b^2*c^2*n^2*log(f)^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {b c e n e^{a c n \log {\left (f \right )}} \log {\left (f \right )} \int \frac {x^{2} e^{b c n x \log {\left (f \right )}}}{d + e e^{a c n \log {\left (f \right )}} e^{b c n x \log {\left (f \right )}}}\, dx}{2} + \frac {x^{2} \log {\left (d + e \left (f^{c \left (a + b x\right )}\right )^{n} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(d+e*(f**(c*(b*x+a)))**n),x)

[Out]

-b*c*e*n*exp(a*c*n*log(f))*log(f)*Integral(x**2*exp(b*c*n*x*log(f))/(d + e*exp(a*c*n*log(f))*exp(b*c*n*x*log(f
))), x)/2 + x**2*log(d + e*(f**(c*(a + b*x)))**n)/2

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(d+e*(f^(c*(b*x+a)))^n),x, algorithm="giac")

[Out]

integrate(x*log((f^((b*x + a)*c))^n*e + d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\ln \left (d+e\,{\left (f^{c\,\left (a+b\,x\right )}\right )}^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(d + e*(f^(c*(a + b*x)))^n),x)

[Out]

int(x*log(d + e*(f^(c*(a + b*x)))^n), x)

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