Integrand size = 10, antiderivative size = 64 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=-\frac {\coth ^{-1}(a+b x)}{x}+\frac {b \log (x)}{1-a^2}-\frac {b \log (1-a-b x)}{2 (1-a)}-\frac {b \log (1+a+b x)}{2 (1+a)} \] Output:
-arccoth(b*x+a)/x+b*ln(x)/(-a^2+1)-b*ln(-b*x-a+1)/(2-2*a)-b*ln(b*x+a+1)/(2 +2*a)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.86 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=-\frac {\coth ^{-1}(a+b x)}{x}+\frac {b (-2 \log (x)+(1+a) \log (1-a-b x)-(-1+a) \log (1+a+b x))}{2 \left (-1+a^2\right )} \] Input:
Integrate[ArcCoth[a + b*x]/x^2,x]
Output:
-(ArcCoth[a + b*x]/x) + (b*(-2*Log[x] + (1 + a)*Log[1 - a - b*x] - (-1 + a )*Log[1 + a + b*x]))/(2*(-1 + a^2))
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6660, 896, 25, 477, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 6660 |
\(\displaystyle b \int \frac {1}{x \left (1-(a+b x)^2\right )}dx-\frac {\coth ^{-1}(a+b x)}{x}\) |
\(\Big \downarrow \) 896 |
\(\displaystyle b \int \frac {1}{b x \left (1-(a+b x)^2\right )}d(a+b x)-\frac {\coth ^{-1}(a+b x)}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -b \int -\frac {1}{b x \left (1-(a+b x)^2\right )}d(a+b x)-\frac {\coth ^{-1}(a+b x)}{x}\) |
\(\Big \downarrow \) 477 |
\(\displaystyle -b \int \left (-\frac {1}{2 (1-a) (-a-b x+1)}+\frac {1}{2 (a+1) (a+b x+1)}-\frac {1}{\left (1-a^2\right ) b x}\right )d(a+b x)-\frac {\coth ^{-1}(a+b x)}{x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle b \left (\frac {\log (-b x)}{1-a^2}-\frac {\log (-a-b x+1)}{2 (1-a)}-\frac {\log (a+b x+1)}{2 (a+1)}\right )-\frac {\coth ^{-1}(a+b x)}{x}\) |
Input:
Int[ArcCoth[a + b*x]/x^2,x]
Output:
-(ArcCoth[a + b*x]/x) + b*(Log[-(b*x)]/(1 - a^2) - Log[1 - a - b*x]/(2*(1 - a)) - Log[1 + a + b*x]/(2*(1 + a)))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_), x_Symbol] :> Simp[(e + f*x)^(m + 1)*((a + b*ArcCoth[c + d*x])^p/(f*(m + 1))), x] - Simp[b*d*(p/(f*(m + 1))) Int[(e + f*x)^(m + 1)*((a + b*ArcCo th[c + d*x])^(p - 1)/(1 - (c + d*x)^2)), x], x] /; FreeQ[{a, b, c, d, e, f} , x] && IGtQ[p, 0] && ILtQ[m, -1]
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95
method | result | size |
parts | \(-\frac {\operatorname {arccoth}\left (b x +a \right )}{x}-b \left (-\frac {\ln \left (b x +a -1\right )}{2 a -2}+\frac {\ln \left (x \right )}{\left (1+a \right ) \left (a -1\right )}+\frac {\ln \left (b x +a +1\right )}{2+2 a}\right )\) | \(61\) |
derivativedivides | \(b \left (-\frac {\operatorname {arccoth}\left (b x +a \right )}{b x}-\frac {\ln \left (b x +a +1\right )}{2+2 a}-\frac {\ln \left (-b x \right )}{\left (a -1\right ) \left (1+a \right )}+\frac {\ln \left (b x +a -1\right )}{2 a -2}\right )\) | \(66\) |
default | \(b \left (-\frac {\operatorname {arccoth}\left (b x +a \right )}{b x}-\frac {\ln \left (b x +a +1\right )}{2+2 a}-\frac {\ln \left (-b x \right )}{\left (a -1\right ) \left (1+a \right )}+\frac {\ln \left (b x +a -1\right )}{2 a -2}\right )\) | \(66\) |
parallelrisch | \(-\frac {x \,\operatorname {arccoth}\left (b x +a \right ) a \,b^{3}+\ln \left (x \right ) x \,b^{3}-\ln \left (b x +a -1\right ) x \,b^{3}-x \,\operatorname {arccoth}\left (b x +a \right ) b^{3}+\operatorname {arccoth}\left (b x +a \right ) a^{2} b^{2}-\operatorname {arccoth}\left (b x +a \right ) b^{2}}{\left (a^{2}-1\right ) x \,b^{2}}\) | \(85\) |
risch | \(-\frac {\ln \left (b x +a +1\right )}{2 x}-\frac {-\ln \left (-b x -a +1\right ) a b x +\ln \left (b x +a +1\right ) a b x +2 \ln \left (-x \right ) b x -\ln \left (-b x -a +1\right ) b x -\ln \left (b x +a +1\right ) b x -\ln \left (b x +a -1\right ) a^{2}+\ln \left (b x +a -1\right )}{2 x \left (1+a \right ) \left (a -1\right )}\) | \(108\) |
Input:
int(arccoth(b*x+a)/x^2,x,method=_RETURNVERBOSE)
Output:
-arccoth(b*x+a)/x-b*(-1/(2*a-2)*ln(b*x+a-1)+1/(1+a)/(a-1)*ln(x)+1/(2+2*a)* ln(b*x+a+1))
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.06 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=-\frac {{\left (a - 1\right )} b x \log \left (b x + a + 1\right ) - {\left (a + 1\right )} b x \log \left (b x + a - 1\right ) + 2 \, b x \log \left (x\right ) + {\left (a^{2} - 1\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{2 \, {\left (a^{2} - 1\right )} x} \] Input:
integrate(arccoth(b*x+a)/x^2,x, algorithm="fricas")
Output:
-1/2*((a - 1)*b*x*log(b*x + a + 1) - (a + 1)*b*x*log(b*x + a - 1) + 2*b*x* log(x) + (a^2 - 1)*log((b*x + a + 1)/(b*x + a - 1)))/((a^2 - 1)*x)
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (48) = 96\).
Time = 0.67 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.25 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=\begin {cases} \frac {b \operatorname {acoth}{\left (b x - 1 \right )}}{2} - \frac {\operatorname {acoth}{\left (b x - 1 \right )}}{x} - \frac {1}{2 x} & \text {for}\: a = -1 \\- \frac {b \operatorname {acoth}{\left (b x + 1 \right )}}{2} - \frac {\operatorname {acoth}{\left (b x + 1 \right )}}{x} + \frac {1}{2 x} & \text {for}\: a = 1 \\- \frac {a^{2} \operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} - \frac {a b x \operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} - \frac {b x \log {\left (x \right )}}{a^{2} x - x} + \frac {b x \log {\left (a + b x + 1 \right )}}{a^{2} x - x} - \frac {b x \operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} + \frac {\operatorname {acoth}{\left (a + b x \right )}}{a^{2} x - x} & \text {otherwise} \end {cases} \] Input:
integrate(acoth(b*x+a)/x**2,x)
Output:
Piecewise((b*acoth(b*x - 1)/2 - acoth(b*x - 1)/x - 1/(2*x), Eq(a, -1)), (- b*acoth(b*x + 1)/2 - acoth(b*x + 1)/x + 1/(2*x), Eq(a, 1)), (-a**2*acoth(a + b*x)/(a**2*x - x) - a*b*x*acoth(a + b*x)/(a**2*x - x) - b*x*log(x)/(a** 2*x - x) + b*x*log(a + b*x + 1)/(a**2*x - x) - b*x*acoth(a + b*x)/(a**2*x - x) + acoth(a + b*x)/(a**2*x - x), True))
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.84 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=-\frac {1}{2} \, b {\left (\frac {\log \left (b x + a + 1\right )}{a + 1} - \frac {\log \left (b x + a - 1\right )}{a - 1} + \frac {2 \, \log \left (x\right )}{a^{2} - 1}\right )} - \frac {\operatorname {arcoth}\left (b x + a\right )}{x} \] Input:
integrate(arccoth(b*x+a)/x^2,x, algorithm="maxima")
Output:
-1/2*b*(log(b*x + a + 1)/(a + 1) - log(b*x + a - 1)/(a - 1) + 2*log(x)/(a^ 2 - 1)) - arccoth(b*x + a)/x
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (57) = 114\).
Time = 0.14 (sec) , antiderivative size = 259, normalized size of antiderivative = 4.05 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=-\frac {1}{2} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {{\left (a - 1\right )} \log \left ({\left | \frac {{\left (b x + a + 1\right )} a}{b x + a - 1} - a - \frac {b x + a + 1}{b x + a - 1} - 1 \right |}\right )}{a^{3} - a^{2} - a + 1} - \frac {\log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{a^{2} - 1} - \frac {\log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{{\left (\frac {{\left (b x + a + 1\right )} a}{b x + a - 1} - a - \frac {b x + a + 1}{b x + a - 1} - 1\right )} {\left (a - 1\right )}}\right )} \] Input:
integrate(arccoth(b*x+a)/x^2,x, algorithm="giac")
Output:
-1/2*((a + 1)*b - (a - 1)*b)*((a - 1)*log(abs((b*x + a + 1)*a/(b*x + a - 1 ) - a - (b*x + a + 1)/(b*x + a - 1) - 1))/(a^3 - a^2 - a + 1) - log(abs(b* x + a + 1)/abs(b*x + a - 1))/(a^2 - 1) - log(-(1/(a - ((b*x + a + 1)*(a - 1)/(b*x + a - 1) - a - 1)*b/((b*x + a + 1)*b/(b*x + a - 1) - b)) + 1)/(1/( a - ((b*x + a + 1)*(a - 1)/(b*x + a - 1) - a - 1)*b/((b*x + a + 1)*b/(b*x + a - 1) - b)) - 1))/(((b*x + a + 1)*a/(b*x + a - 1) - a - (b*x + a + 1)/( b*x + a - 1) - 1)*(a - 1)))
Time = 4.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=-\frac {\mathrm {acoth}\left (a+b\,x\right )}{x}-\frac {b\,x\,\ln \left (x\right )-\frac {b\,x\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2-1\right )}{2}+a\,b\,x\,\mathrm {acoth}\left (a+b\,x\right )}{x\,\left (a^2-1\right )} \] Input:
int(acoth(a + b*x)/x^2,x)
Output:
- acoth(a + b*x)/x - (b*x*log(x) - (b*x*log(a^2 + b^2*x^2 + 2*a*b*x - 1))/ 2 + a*b*x*acoth(a + b*x))/(x*(a^2 - 1))
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {\coth ^{-1}(a+b x)}{x^2} \, dx=\frac {-\mathit {acoth} \left (b x +a \right ) a^{2}-\mathit {acoth} \left (b x +a \right ) a b x +\mathit {acoth} \left (b x +a \right ) b x +\mathit {acoth} \left (b x +a \right )-\mathrm {log}\left (b x +a -1\right ) b x +\mathrm {log}\left (x \right ) b x}{x \left (a^{2}-1\right )} \] Input:
int(acoth(b*x+a)/x^2,x)
Output:
( - acoth(a + b*x)*a**2 - acoth(a + b*x)*a*b*x + acoth(a + b*x)*b*x + acot h(a + b*x) - log(a + b*x - 1)*b*x + log(x)*b*x)/(x*(a**2 - 1))