3.26 \(\int \frac {a x+2 b n \log (c x^n)}{a x^2+b x \log ^2(c x^n)} \, dx\)

Optimal. Leaf size=15 \[ \log \left (a x+b \log ^2\left (c x^n\right )\right ) \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2561, 2541} \[ \log \left (a x+b \log ^2\left (c x^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a*x + b*Log[c*x^n]^2]

Rule 2541

Int[(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x_)^(m_.))/((x_)*(Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_
)^(m_.))), x_Symbol] :> Simp[(e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q), x] /; FreeQ[{a, b, c, d, e, m, n, q, r},
x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {a x+2 b n \log \left (c x^n\right )}{a x^2+b x \log ^2\left (c x^n\right )} \, dx &=\int \frac {a x+2 b n \log \left (c x^n\right )}{x \left (a x+b \log ^2\left (c x^n\right )\right )} \, dx\\ &=\log \left (a x+b \log ^2\left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 15, normalized size = 1.00 \[ \log \left (a x+b \log ^2\left (c x^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a*x + b*Log[c*x^n]^2]

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fricas [A]  time = 0.46, size = 28, normalized size = 1.87 \[ \log \left (b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} + a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*n^2*log(x)^2 + 2*b*n*log(c)*log(x) + b*log(c)^2 + a*x)

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giac [A]  time = 0.16, size = 28, normalized size = 1.87 \[ \log \left (b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} + a x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(b*n^2*log(x)^2 + 2*b*n*log(c)*log(x) + b*log(c)^2 + a*x)

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maple [C]  time = 0.29, size = 432, normalized size = 28.80 \[ \ln \left (\ln \left (x^{n}\right )^{2}+\left (-i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+2 \ln \relax (c )\right ) \ln \left (x^{n}\right )+\frac {-\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+2 \pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-4 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}+2 \pi ^{2} b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}-\pi ^{2} b \mathrm {csgn}\left (i c \,x^{n}\right )^{6}-4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )+4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )+4 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )-4 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )+4 b \ln \relax (c )^{2}+4 a x}{4 b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(1/4*(-b*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4+2*b*Pi^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-b*Pi^2*csgn(I*x
^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2+2*b*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^5-4*b*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^4*cs
gn(I*c)+2*b*Pi^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2-b*Pi^2*csgn(I*c*x^n)^6+2*b*Pi^2*csgn(I*c*x^n)^5*csgn(
I*c)-b*Pi^2*csgn(I*c*x^n)^4*csgn(I*c)^2+4*I*b*ln(c)*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*b*ln(c)*Pi*csgn(I*x^n)*
csgn(I*c*x^n)*csgn(I*c)-4*I*b*ln(c)*Pi*csgn(I*c*x^n)^3+4*I*b*ln(c)*Pi*csgn(I*c*x^n)^2*csgn(I*c)+4*b*ln(c)^2+4*
a*x)/b+(I*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*csgn(I*c*x^n)^3+I*Pi*cs
gn(I*c*x^n)^2*csgn(I*c)+2*ln(c))*ln(x^n)+ln(x^n)^2)

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maxima [B]  time = 0.61, size = 32, normalized size = 2.13 \[ \log \left (\frac {b \log \relax (c)^{2} + 2 \, b \log \relax (c) \log \left (x^{n}\right ) + b \log \left (x^{n}\right )^{2} + a x}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.35, size = 16, normalized size = 1.07 \[ \ln \left ({\ln \left (c\,x^n\right )}^2+\frac {a\,x}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(log(c*x^n)^2 + (a*x)/b)

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sympy [A]  time = 2.33, size = 48, normalized size = 3.20 \[ \begin {cases} \log {\left (x + \frac {b n^{2} \log {\relax (x )}^{2}}{a} + \frac {2 b n \log {\relax (c )} \log {\relax (x )}}{a} + \frac {b \log {\relax (c )}^{2}}{a} \right )} & \text {for}\: a \neq 0 \\2 \log {\left (n \log {\relax (x )} + \log {\relax (c )} \right )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+2*b*n*ln(c*x**n))/(a*x**2+b*x*ln(c*x**n)**2),x)

[Out]

Piecewise((log(x + b*n**2*log(x)**2/a + 2*b*n*log(c)*log(x)/a + b*log(c)**2/a), Ne(a, 0)), (2*log(n*log(x) + l
og(c)), True))

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