\(\int \frac {1}{a x+\frac {b x}{\log ^2(c x^n)}} \, dx\) [255]

Optimal result

Integrand size = 17, antiderivative size = 40 \[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \log \left (c x^n\right )}{\sqrt {b}}\right )}{a^{3/2} n}+\frac {\log (x)}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {327, 211} \[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=\frac {\log (x)}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \log \left (c x^n\right )}{\sqrt {b}}\right )}{a^{3/2} n} \]

[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 211

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

Rule 327

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*} \text {integral}& = \frac {\text {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 \text {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] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \log \left (c x^n\right )}{\sqrt {b}}\right )}{a^{3/2} n}+\frac {\log (x)}{a} \]

[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[x]/a

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02

method	result	size
default	\(\frac {\frac {\ln \left (c \,x^{n}\right )}{a}-\frac {b \arctan \left (\frac {a \ln \left (c \,x^{n}\right )}{\sqrt {a b}}\right )}{a \sqrt {a b}}}{n}\)	\(41\)
risch	\(\frac {\ln \left (x \right )}{a}+\frac {\sqrt {-a b}\, \ln \left (\ln \left (x^{n}\right )-\frac {i \pi a \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-i \pi a \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi a \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi a \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 \ln \left (c \right ) a +2 \sqrt {-a b}}{2 a}\right )}{2 a^{2} n}-\frac {\sqrt {-a b}\, \ln \left (\ln \left (x^{n}\right )-\frac {i \pi a \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )-i \pi a \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i \pi a \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i \pi a \operatorname {csgn}\left (i c \,x^{n}\right )^{3}-2 \ln \left (c \right ) a -2 \sqrt {-a b}}{2 a}\right )}{2 a^{2} n}\)	\(250\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.58 \[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=\left [\frac {2 \, n \log \left (x\right ) + \sqrt {-\frac {b}{a}} \log \left (\frac {a n^{2} \log \left (x\right )^{2} + 2 \, a n \log \left (c\right ) \log \left (x\right ) + a \log \left (c\right )^{2} - 2 \, {\left (a n \log \left (x\right ) + a \log \left (c\right )\right )} \sqrt {-\frac {b}{a}} - b}{a n^{2} \log \left (x\right )^{2} + 2 \, a n \log \left (c\right ) \log \left (x\right ) + a \log \left (c\right )^{2} + b}\right )}{2 \, a n}, \frac {n \log \left (x\right ) - \sqrt {\frac {b}{a}} \arctan \left (\frac {{\left (a n \log \left (x\right ) + a \log \left (c\right )\right )} \sqrt {\frac {b}{a}}}{b}\right )}{a n}\right ] \]

[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)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (34) = 68\).

Time = 3.63 (sec) , antiderivative size = 204, normalized size of antiderivative = 5.10 \[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=\begin {cases} \tilde {\infty } \log {\left (c \right )}^{2} \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac {\begin {cases} - \frac {\log {\left (\frac {x^{- n}}{c} \right )}^{3}}{3 n} + \frac {\log {\left (c x^{n} \right )}^{3}}{3 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \wedge \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (c x^{n} \right )}^{3}}{3 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\- \frac {\log {\left (\frac {x^{- n}}{c} \right )}^{3}}{3 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\- \frac {2 {G_{4, 4}^{4, 0}\left (\begin {matrix} & 1, 1, 1, 1 \\0, 0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {2 {G_{4, 4}^{0, 4}\left (\begin {matrix} 1, 1, 1, 1 & \\ & 0, 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}}{b} & \text {for}\: a = 0 \\\frac {\log {\left (c \right )}^{2} \log {\left (x \right )}}{a \log {\left (c \right )}^{2} + b} & \text {for}\: n = 0 \\\frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (c x^{n} \right )}}{a n} - \frac {b \log {\left (- \sqrt {- \frac {b}{a}} + \log {\left (c x^{n} \right )} \right )}}{2 a^{2} n \sqrt {- \frac {b}{a}}} + \frac {b \log {\left (\sqrt {- \frac {b}{a}} + \log {\left (c x^{n} \right )} \right )}}{2 a^{2} n \sqrt {- \frac {b}{a}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*log(c)**2*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (Piecewise((-log(1/(c*x**n))**3/(3*n) + log(
c*x**n)**3/(3*n), (Abs(c*x**n) < 1) & (1/Abs(c*x**n) < 1)), (log(c*x**n)**3/(3*n), Abs(c*x**n) < 1), (-log(1/(
c*x**n))**3/(3*n), 1/Abs(c*x**n) < 1), (-2*meijerg(((), (1, 1, 1, 1)), ((0, 0, 0, 0), ()), c*x**n)/n + 2*meije
rg(((1, 1, 1, 1), ()), ((), (0, 0, 0, 0)), c*x**n)/n, True))/b, Eq(a, 0)), (log(c)**2*log(x)/(a*log(c)**2 + b)
, Eq(n, 0)), (log(x)/a, Eq(b, 0)), (log(c*x**n)/(a*n) - b*log(-sqrt(-b/a) + log(c*x**n))/(2*a**2*n*sqrt(-b/a))
+ b*log(sqrt(-b/a) + log(c*x**n))/(2*a**2*n*sqrt(-b/a)), True))

Maxima [F]

\[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=\int { \frac {1}{a x + \frac {b x}{\log \left (c x^{n}\right )^{2}}} \,d x } \]

[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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.95 \[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=\frac {\log \left (x\right )}{a} - \frac {b \arctan \left (\frac {a n \log \left (x\right ) + a \log \left (c\right )}{\sqrt {a b}}\right )}{\sqrt {a b} a n} \]

[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)

Mupad [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \frac {1}{a x+\frac {b x}{\log ^2\left (c x^n\right )}} \, dx=\frac {\ln \left (x\right )}{a}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {a^2\,n\,\ln \left (c\,x^n\right )}{\sqrt {b}\,\sqrt {a^3\,n^2}}\right )}{\sqrt {a^3\,n^2}} \]

[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)