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

Optimal. Leaf size=32 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \log \left (c x^n\right )}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} n} \]

[Out]

ArcTan[(Sqrt[b]*Log[c*x^n])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*n)

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Rubi [A]  time = 0.0182284, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \log \left (c x^n\right )}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} n} \]

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x*Log[c*x^n]^2)^(-1),x]

[Out]

ArcTan[(Sqrt[b]*Log[c*x^n])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*n)

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

Mathematica [A]  time = 0.0239537, size = 32, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \log \left (c x^n\right )}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[b]*Log[c*x^n])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*n)

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Maple [A]  time = 0.007, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{n}\arctan \left ({b\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*b)^(1/2)*arctan(ln(c*x^n)*b/(a*b)^(1/2))

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

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

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Fricas [A]  time = 1.90145, size = 324, normalized size = 10.12 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} - 2 \, \sqrt{-a b}{\left (n \log \left (x\right ) + \log \left (c\right )\right )} - a}{b n^{2} \log \left (x\right )^{2} + 2 \, b n \log \left (c\right ) \log \left (x\right ) + b \log \left (c\right )^{2} + a}\right )}{2 \, a b n}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}}{a}\right )}{a b 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*sqrt(-a*b)*log((b*n^2*log(x)^2 + 2*b*n*log(c)*log(x) + b*log(c)^2 - 2*sqrt(-a*b)*(n*log(x) + log(c)) - a
)/(b*n^2*log(x)^2 + 2*b*n*log(c)*log(x) + b*log(c)^2 + a))/(a*b*n), sqrt(a*b)*arctan(sqrt(a*b)*(n*log(x) + log
(c))/a)/(a*b*n)]

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Sympy [A]  time = 8.80349, size = 126, normalized size = 3.94 \begin{align*} \begin{cases} \frac{\tilde{\infty } \log{\left (x \right )}}{\log{\left (c \right )}^{2}} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a + b \log{\left (c \right )}^{2}} & \text{for}\: n = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\- \frac{1}{b \left (n^{2} \log{\left (x \right )} + n \log{\left (c \right )}\right )} & \text{for}\: a = 0 \\- \frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{2 \sqrt{a} b n \sqrt{\frac{1}{b}}} + \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{2 \sqrt{a} b n \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \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]

Piecewise((zoo*log(x)/log(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (log(x)/(a + b*log(c)**2), Eq(n, 0)), (log(x
)/a, Eq(b, 0)), (-1/(b*(n**2*log(x) + n*log(c))), Eq(a, 0)), (-I*log(-I*sqrt(a)*sqrt(1/b) + n*log(x) + log(c))
/(2*sqrt(a)*b*n*sqrt(1/b)) + I*log(I*sqrt(a)*sqrt(1/b) + n*log(x) + log(c))/(2*sqrt(a)*b*n*sqrt(1/b)), True))

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Giac [A]  time = 1.3592, size = 35, normalized size = 1.09 \begin{align*} \frac{\arctan \left (\frac{b n \log \left (x\right ) + b \log \left (c\right )}{\sqrt{a b}}\right )}{\sqrt{a b} 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]

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