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

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Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 32, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \log \left (c x^n\right )}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} n} \]

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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fricas [A]  time = 0.44, size = 121, normalized size = 3.78 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} - 2 \, \sqrt {-a b} {\left (n \log \relax (x) + \log \relax (c)\right )} - a}{b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + b \log \relax (c)^{2} + a}\right )}{2 \, a b n}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} {\left (n \log \relax (x) + \log \relax (c)\right )}}{a}\right )}{a b n}\right ] \]

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*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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giac [A]  time = 0.16, size = 26, normalized size = 0.81 \[ \frac {\arctan \left (\frac {b n \log \relax (x) + b \log \relax (c)}{\sqrt {a b}}\right )}{\sqrt {a b} n} \]

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]

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

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maple [A]  time = 0.07, size = 24, normalized size = 0.75 \[ \frac {\arctan \left (\frac {b \ln \left (c \,x^{n}\right )}{\sqrt {a b}}\right )}{\sqrt {a b}\, n} \]

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

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

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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mupad [B]  time = 0.47, size = 71, normalized size = 2.22 \[ -\frac {\ln \left (\frac {1}{b\,x}+\frac {\ln \left (c\,x^n\right )}{\sqrt {-a}\,\sqrt {b}\,x}\right )-\ln \left (\frac {1}{b\,x}-\frac {\ln \left (c\,x^n\right )}{\sqrt {-a}\,\sqrt {b}\,x}\right )}{2\,\sqrt {-a}\,\sqrt {b}\,n} \]

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

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sympy [A]  time = 5.98, size = 126, normalized size = 3.94 \[ \begin {cases} \frac {\tilde {\infty } \log {\relax (x )}}{\log {\relax (c )}^{2}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\log {\relax (x )}}{a + b \log {\relax (c )}^{2}} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{a} & \text {for}\: b = 0 \\- \frac {1}{b \left (n^{2} \log {\relax (x )} + n \log {\relax (c )}\right )} & \text {for}\: a = 0 \\- \frac {i \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{2 \sqrt {a} b n \sqrt {\frac {1}{b}}} + \frac {i \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + n \log {\relax (x )} + \log {\relax (c )} \right )}}{2 \sqrt {a} b n \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]

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((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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