3.255 \(\int \frac {1}{a x+\frac {b x}{\log ^2(c x^n)}} \, dx\)

Optimal. Leaf size=40 \[ \frac {\log (x)}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \log \left (c x^n\right )}{\sqrt {b}}\right )}{a^{3/2} n} \]

[Out]

ln(x)/a-arctan(ln(c*x^n)*a^(1/2)/b^(1/2))*b^(1/2)/a^(3/2)/n

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Rubi [A]  time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {321, 205} \[ \frac {\log (x)}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \log \left (c x^n\right )}{\sqrt {b}}\right )}{a^{3/2} n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + (b*x)/Log[c*x^n]^2)^(-1),x]

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[a]*Log[c*x^n])/Sqrt[b]])/(a^(3/2)*n)) + Log[x]/a

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log (x)}{a}-\frac {b \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\log \left (c x^n\right )\right )}{a n}\\ &=-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \log \left (c x^n\right )}{\sqrt {b}}\right )}{a^{3/2} n}+\frac {\log (x)}{a}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 47, normalized size = 1.18 \[ \frac {\log \left (c x^n\right )}{a n}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} \log \left (c x^n\right )}{\sqrt {b}}\right )}{a^{3/2} n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[b]*ArcTan[(Sqrt[a]*Log[c*x^n])/Sqrt[b]])/(a^(3/2)*n)) + Log[c*x^n]/(a*n)

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fricas [A]  time = 0.49, size = 143, normalized size = 3.58 \[ \left [\frac {2 \, n \log \relax (x) + \sqrt {-\frac {b}{a}} \log \left (\frac {a n^{2} \log \relax (x)^{2} + 2 \, a n \log \relax (c) \log \relax (x) + a \log \relax (c)^{2} - 2 \, {\left (a n \log \relax (x) + a \log \relax (c)\right )} \sqrt {-\frac {b}{a}} - b}{a n^{2} \log \relax (x)^{2} + 2 \, a n \log \relax (c) \log \relax (x) + a \log \relax (c)^{2} + b}\right )}{2 \, a n}, \frac {n \log \relax (x) - \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a n \log \relax (x) + a \log \relax (c)\right )} \sqrt {\frac {b}{a}}}{b}\right )}{a n}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x/log(c*x^n)^2),x, algorithm="fricas")

[Out]

[1/2*(2*n*log(x) + sqrt(-b/a)*log((a*n^2*log(x)^2 + 2*a*n*log(c)*log(x) + a*log(c)^2 - 2*(a*n*log(x) + a*log(c
))*sqrt(-b/a) - b)/(a*n^2*log(x)^2 + 2*a*n*log(c)*log(x) + a*log(c)^2 + b)))/(a*n), (n*log(x) - sqrt(b/a)*arct
an((a*n*log(x) + a*log(c))*sqrt(b/a)/b))/(a*n)]

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giac [A]  time = 0.16, size = 38, normalized size = 0.95 \[ \frac {\log \relax (x)}{a} - \frac {b \arctan \left (\frac {a n \log \relax (x) + a \log \relax (c)}{\sqrt {a b}}\right )}{\sqrt {a b} a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x/log(c*x^n)^2),x, algorithm="giac")

[Out]

log(x)/a - b*arctan((a*n*log(x) + a*log(c))/sqrt(a*b))/(sqrt(a*b)*a*n)

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maple [A]  time = 0.07, size = 43, normalized size = 1.08 \[ -\frac {b \arctan \left (\frac {a \ln \left (c \,x^{n}\right )}{\sqrt {a b}}\right )}{\sqrt {a b}\, a n}+\frac {\ln \left (c \,x^{n}\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x+b*x/ln(c*x^n)^2),x)

[Out]

1/a/n*ln(c*x^n)-1/n/a*b/(a*b)^(1/2)*arctan(a*ln(c*x^n)/(a*b)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -b \int \frac {1}{2 \, a^{2} x \log \relax (c) \log \left (x^{n}\right ) + a^{2} x \log \left (x^{n}\right )^{2} + {\left (a^{2} \log \relax (c)^{2} + a b\right )} x}\,{d x} + \frac {\log \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x/log(c*x^n)^2),x, algorithm="maxima")

[Out]

-b*integrate(1/(2*a^2*x*log(c)*log(x^n) + a^2*x*log(x^n)^2 + (a^2*log(c)^2 + a*b)*x), x) + log(x)/a

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mupad [B]  time = 0.38, size = 45, normalized size = 1.12 \[ \frac {\ln \relax (x)}{a}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {a^2\,n\,\ln \left (c\,x^n\right )}{\sqrt {b}\,\sqrt {a^3\,n^2}}\right )}{\sqrt {a^3\,n^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x + (b*x)/log(c*x^n)^2),x)

[Out]

log(x)/a - (b^(1/2)*atan((a^2*n*log(c*x^n))/(b^(1/2)*(a^3*n^2)^(1/2))))/(a^3*n^2)^(1/2)

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sympy [A]  time = 7.89, size = 177, normalized size = 4.42 \[ \begin {cases} \frac {\log {\relax (c )}^{2} \log {\relax (x )}}{b} & \text {for}\: a = 0 \wedge n = 0 \\\frac {\log {\relax (c )}^{2} \log {\relax (x )}}{a \log {\relax (c )}^{2} + b} & \text {for}\: n = 0 \\\frac {\begin {cases} \frac {\log {\left (c x^{n} \right )}^{3}}{3 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\- \frac {\log {\left (\frac {x^{- n}}{c} \right )}^{3}}{3 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\- \frac {2 {G_{4, 4}^{4, 0}\left (\begin {matrix} & 1, 1, 1, 1 \\0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {2 {G_{4, 4}^{0, 4}\left (\begin {matrix} 1, 1, 1, 1 & \\ & 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}}{b} & \text {for}\: a = 0 \\\frac {\log {\relax (x )}}{a} + \frac {i \sqrt {b} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{2 a^{2} n \sqrt {\frac {1}{a}}} - \frac {i \sqrt {b} \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{2 a^{2} n \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*x+b*x/ln(c*x**n)**2),x)

[Out]

Piecewise((log(c)**2*log(x)/b, Eq(a, 0) & Eq(n, 0)), (log(c)**2*log(x)/(a*log(c)**2 + b), Eq(n, 0)), (Piecewis
e((log(c*x**n)**3/(3*n), Abs(c*x**n) < 1), (-log(x**(-n)/c)**3/(3*n), 1/Abs(c*x**n) < 1), (-2*meijerg(((), (1,
1, 1, 1)), ((0, 0, 0, 0), ()), c*x**n)/n + 2*meijerg(((1, 1, 1, 1), ()), ((), (0, 0, 0, 0)), c*x**n)/n, True)
)/b, Eq(a, 0)), (log(x)/a + I*sqrt(b)*log(-I*sqrt(b)*sqrt(1/a) + n*log(x) + log(c))/(2*a**2*n*sqrt(1/a)) - I*s
qrt(b)*log(I*sqrt(b)*sqrt(1/a) + n*log(x) + log(c))/(2*a**2*n*sqrt(1/a)), True))

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