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]

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

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Rubi [A]  time = 0.0271898, antiderivative size = 40, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.024212, 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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Maple [A]  time = 0.006, size = 43, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ) }{na}}-{\frac{b}{na}\arctan \left ({a\ln \left ( c{x}^{n} \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{1}{2 \, a^{2} x \log \left (c\right ) \log \left (x^{n}\right ) + a^{2} x \log \left (x^{n}\right )^{2} +{\left (a^{2} \log \left (c\right )^{2} + a b\right )} x}\,{d x} + \frac{\log \left (x\right )}{a} \end{align*}

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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Fricas [A]  time = 1.9646, size = 366, normalized size = 9.15 \begin{align*} \left [\frac{2 \, n \log \left (x\right ) + \sqrt{-\frac{b}{a}} \log \left (\frac{a n^{2} \log \left (x\right )^{2} + 2 \, a n \log \left (c\right ) \log \left (x\right ) + a \log \left (c\right )^{2} - 2 \,{\left (a n \log \left (x\right ) + a \log \left (c\right )\right )} \sqrt{-\frac{b}{a}} - b}{a n^{2} \log \left (x\right )^{2} + 2 \, a n \log \left (c\right ) \log \left (x\right ) + a \log \left (c\right )^{2} + b}\right )}{2 \, a n}, \frac{n \log \left (x\right ) - \sqrt{\frac{b}{a}} \arctan \left (\frac{{\left (a n \log \left (x\right ) + a \log \left (c\right )\right )} \sqrt{\frac{b}{a}}}{b}\right )}{a n}\right ] \end{align*}

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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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.27586, size = 51, normalized size = 1.27 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{b \arctan \left (\frac{a n \log \left (x\right ) + a \log \left (c\right )}{\sqrt{a b}}\right )}{\sqrt{a b} a n} \end{align*}

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)