Integrand size = 25, antiderivative size = 140 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {b (d-e) x}{2 c}-\frac {b e x}{c}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {b (d-e) \text {arctanh}(c x)}{2 c^2}+\frac {b e \text {arctanh}(c x)}{c^2}+\frac {b e x \log \left (1-c^2 x^2\right )}{2 c}-\frac {e \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{2 c^2} \]
1/2*b*(d-e)*x/c-b*e*x/c+1/2*d*x^2*(a+b*arccoth(c*x))-1/2*e*x^2*(a+b*arccot h(c*x))-1/2*b*(d-e)*arctanh(c*x)/c^2+b*e*arctanh(c*x)/c^2+1/2*b*e*x*ln(-c^ 2*x^2+1)/c-1/2*e*(-c^2*x^2+1)*(a+b*arccoth(c*x))*ln(-c^2*x^2+1)/c^2
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {2 b c (d-3 e) x+2 a c^2 (d-e) x^2+2 b c^2 (d-e) x^2 \coth ^{-1}(c x)+(b (d-3 e)-2 a e) \log (1-c x)-(b (d-3 e)+2 a e) \log (1+c x)+2 e \left (c x (b+a c x)+b \left (-1+c^2 x^2\right ) \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{4 c^2} \]
(2*b*c*(d - 3*e)*x + 2*a*c^2*(d - e)*x^2 + 2*b*c^2*(d - e)*x^2*ArcCoth[c*x ] + (b*(d - 3*e) - 2*a*e)*Log[1 - c*x] - (b*(d - 3*e) + 2*a*e)*Log[1 + c*x ] + 2*e*(c*x*(b + a*c*x) + b*(-1 + c^2*x^2)*ArcCoth[c*x])*Log[1 - c^2*x^2] )/(4*c^2)
Time = 0.36 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6646, 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.
\(\displaystyle \int x \left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right ) \, dx\) |
\(\Big \downarrow \) 6646 |
\(\displaystyle -b c \int \left (\frac {(d-e) x^2}{2 \left (1-c^2 x^2\right )}-\frac {e \log \left (1-c^2 x^2\right )}{2 c^2}\right )dx-\frac {e \left (1-c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {e \left (1-c^2 x^2\right ) \log \left (1-c^2 x^2\right ) \left (a+b \coth ^{-1}(c x)\right )}{2 c^2}+\frac {1}{2} d x^2 \left (a+b \coth ^{-1}(c x)\right )-\frac {1}{2} e x^2 \left (a+b \coth ^{-1}(c x)\right )-b c \left (\frac {(d-e) \text {arctanh}(c x)}{2 c^3}-\frac {e \text {arctanh}(c x)}{c^3}-\frac {x (d-e)}{2 c^2}-\frac {e x \log \left (1-c^2 x^2\right )}{2 c^2}+\frac {e x}{c^2}\right )\) |
(d*x^2*(a + b*ArcCoth[c*x]))/2 - (e*x^2*(a + b*ArcCoth[c*x]))/2 - (e*(1 - c^2*x^2)*(a + b*ArcCoth[c*x])*Log[1 - c^2*x^2])/(2*c^2) - b*c*(-1/2*((d - e)*x)/c^2 + (e*x)/c^2 + ((d - e)*ArcTanh[c*x])/(2*c^3) - (e*ArcTanh[c*x])/ c^3 - (e*x*Log[1 - c^2*x^2])/(2*c^2))
3.3.68.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]* (e_.))*(x_)^(m_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*Log[f + g*x^2] ), x]}, Simp[(a + b*ArcCoth[c*x]) u, x] - Simp[b*c Int[ExpandIntegrand[ u/(1 - c^2*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[(m + 1)/2, 0]
Time = 1.63 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.24
method | result | size |
parallelrisch | \(\frac {\ln \left (-c^{2} x^{2}+1\right ) \operatorname {arccoth}\left (c x \right ) b \,c^{2} e \,x^{2}+\operatorname {arccoth}\left (c x \right ) b \,c^{2} d \,x^{2}-\operatorname {arccoth}\left (c x \right ) b \,c^{2} e \,x^{2}+\ln \left (-c^{2} x^{2}+1\right ) a \,c^{2} e \,x^{2}+a \,c^{2} d \,x^{2}-a \,c^{2} e \,x^{2}+\ln \left (-c^{2} x^{2}+1\right ) b c e x +b c d x -3 b x e c -\operatorname {arccoth}\left (c x \right ) \ln \left (-c^{2} x^{2}+1\right ) b e -\operatorname {arccoth}\left (c x \right ) b d +3 \,\operatorname {arccoth}\left (c x \right ) b e -\ln \left (-c^{2} x^{2}+1\right ) a e}{2 c^{2}}\) | \(174\) |
default | \(\text {Expression too large to display}\) | \(2474\) |
parts | \(\text {Expression too large to display}\) | \(2474\) |
risch | \(\text {Expression too large to display}\) | \(6696\) |
1/2*(ln(-c^2*x^2+1)*arccoth(c*x)*b*c^2*e*x^2+arccoth(c*x)*b*c^2*d*x^2-arcc oth(c*x)*b*c^2*e*x^2+ln(-c^2*x^2+1)*a*c^2*e*x^2+a*c^2*d*x^2-a*c^2*e*x^2+ln (-c^2*x^2+1)*b*c*e*x+b*c*d*x-3*b*x*e*c-arccoth(c*x)*ln(-c^2*x^2+1)*b*e-arc coth(c*x)*b*d+3*arccoth(c*x)*b*e-ln(-c^2*x^2+1)*a*e)/c^2
Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {2 \, {\left (a c^{2} d - a c^{2} e\right )} x^{2} + 2 \, {\left (b c d - 3 \, b c e\right )} x + 2 \, {\left (a c^{2} e x^{2} + b c e x - a e\right )} \log \left (-c^{2} x^{2} + 1\right ) + {\left ({\left (b c^{2} d - b c^{2} e\right )} x^{2} - b d + 3 \, b e + {\left (b c^{2} e x^{2} - b e\right )} \log \left (-c^{2} x^{2} + 1\right )\right )} \log \left (\frac {c x + 1}{c x - 1}\right )}{4 \, c^{2}} \]
1/4*(2*(a*c^2*d - a*c^2*e)*x^2 + 2*(b*c*d - 3*b*c*e)*x + 2*(a*c^2*e*x^2 + b*c*e*x - a*e)*log(-c^2*x^2 + 1) + ((b*c^2*d - b*c^2*e)*x^2 - b*d + 3*b*e + (b*c^2*e*x^2 - b*e)*log(-c^2*x^2 + 1))*log((c*x + 1)/(c*x - 1)))/c^2
Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.49 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{2}}{2} + \frac {a e x^{2} \log {\left (- c^{2} x^{2} + 1 \right )}}{2} - \frac {a e x^{2}}{2} - \frac {a e \log {\left (- c^{2} x^{2} + 1 \right )}}{2 c^{2}} + \frac {b d x^{2} \operatorname {acoth}{\left (c x \right )}}{2} + \frac {b e x^{2} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{2} - \frac {b e x^{2} \operatorname {acoth}{\left (c x \right )}}{2} + \frac {b d x}{2 c} + \frac {b e x \log {\left (- c^{2} x^{2} + 1 \right )}}{2 c} - \frac {3 b e x}{2 c} - \frac {b d \operatorname {acoth}{\left (c x \right )}}{2 c^{2}} - \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {acoth}{\left (c x \right )}}{2 c^{2}} + \frac {3 b e \operatorname {acoth}{\left (c x \right )}}{2 c^{2}} & \text {for}\: c \neq 0 \\\frac {d x^{2} \left (a + \frac {i \pi b}{2}\right )}{2} & \text {otherwise} \end {cases} \]
Piecewise((a*d*x**2/2 + a*e*x**2*log(-c**2*x**2 + 1)/2 - a*e*x**2/2 - a*e* log(-c**2*x**2 + 1)/(2*c**2) + b*d*x**2*acoth(c*x)/2 + b*e*x**2*log(-c**2* x**2 + 1)*acoth(c*x)/2 - b*e*x**2*acoth(c*x)/2 + b*d*x/(2*c) + b*e*x*log(- c**2*x**2 + 1)/(2*c) - 3*b*e*x/(2*c) - b*d*acoth(c*x)/(2*c**2) - b*e*log(- c**2*x**2 + 1)*acoth(c*x)/(2*c**2) + 3*b*e*acoth(c*x)/(2*c**2), Ne(c, 0)), (d*x**2*(a + I*pi*b/2)/2, True))
Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.22 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b d - \frac {{\left (c^{2} x^{2} - {\left (c^{2} x^{2} - 1\right )} \log \left (-c^{2} x^{2} + 1\right ) - 1\right )} b e \operatorname {arcoth}\left (c x\right )}{2 \, c^{2}} - \frac {{\left (c^{2} x^{2} - {\left (c^{2} x^{2} - 1\right )} \log \left (-c^{2} x^{2} + 1\right ) - 1\right )} a e}{2 \, c^{2}} - \frac {{\left (3 \, c x - {\left (c x + 1\right )} \log \left (c x + 1\right ) - {\left (c x - 1\right )} \log \left (-c x + 1\right )\right )} b e}{2 \, c^{2}} \]
1/2*a*d*x^2 + 1/4*(2*x^2*arccoth(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + lo g(c*x - 1)/c^3))*b*d - 1/2*(c^2*x^2 - (c^2*x^2 - 1)*log(-c^2*x^2 + 1) - 1) *b*e*arccoth(c*x)/c^2 - 1/2*(c^2*x^2 - (c^2*x^2 - 1)*log(-c^2*x^2 + 1) - 1 )*a*e/c^2 - 1/2*(3*c*x - (c*x + 1)*log(c*x + 1) - (c*x - 1)*log(-c*x + 1)) *b*e/c^2
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {1}{4} \, b e x^{2} \log \left (-c x + 1\right )^{2} - \frac {1}{4} \, {\left (-i \, \pi b d + i \, \pi b e - 2 \, a d + 2 \, a e\right )} x^{2} + \frac {1}{4} \, {\left (b e x^{2} - \frac {b e}{c^{2}}\right )} \log \left (c x + 1\right )^{2} - \frac {1}{4} \, {\left ({\left (-i \, \pi b e - b d - 2 \, a e + b e\right )} x^{2} - \frac {2 \, b e x}{c}\right )} \log \left (c x + 1\right ) - \frac {b e \log \left (c x - 1\right )^{2}}{4 \, c^{2}} - \frac {1}{4} \, {\left ({\left (-i \, \pi b e + b d - 2 \, a e - b e\right )} x^{2} - \frac {2 \, b e x}{c} - \frac {2 \, b e \log \left (c x - 1\right )}{c^{2}}\right )} \log \left (-c x + 1\right ) + \frac {{\left (b d - 3 \, b e\right )} x}{2 \, c} + \frac {{\left (-i \, \pi b e - b d - 2 \, a e + 3 \, b e\right )} \log \left (c x + 1\right )}{4 \, c^{2}} + \frac {{\left (-i \, \pi b e + b d - 2 \, a e - 3 \, b e\right )} \log \left (c x - 1\right )}{4 \, c^{2}} \]
-1/4*b*e*x^2*log(-c*x + 1)^2 - 1/4*(-I*pi*b*d + I*pi*b*e - 2*a*d + 2*a*e)* x^2 + 1/4*(b*e*x^2 - b*e/c^2)*log(c*x + 1)^2 - 1/4*((-I*pi*b*e - b*d - 2*a *e + b*e)*x^2 - 2*b*e*x/c)*log(c*x + 1) - 1/4*b*e*log(c*x - 1)^2/c^2 - 1/4 *((-I*pi*b*e + b*d - 2*a*e - b*e)*x^2 - 2*b*e*x/c - 2*b*e*log(c*x - 1)/c^2 )*log(-c*x + 1) + 1/2*(b*d - 3*b*e)*x/c + 1/4*(-I*pi*b*e - b*d - 2*a*e + 3 *b*e)*log(c*x + 1)/c^2 + 1/4*(-I*pi*b*e + b*d - 2*a*e - 3*b*e)*log(c*x - 1 )/c^2
Time = 5.60 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.35 \[ \int x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\ln \left (1-\frac {1}{c\,x}\right )\,\left (\frac {\frac {b\,d\,x^3}{2}-\frac {b\,c^2\,d\,x^5}{2}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {\frac {b\,e\,x^3}{2}-\frac {b\,c^2\,e\,x^5}{2}}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}+\frac {\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e\,x^3}{2}-\frac {b\,c^2\,e\,x^5}{2}\right )}{2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}-\frac {b\,e\,\ln \left (1-c^2\,x^2\right )\,\left (x-c^2\,x^3\right )}{4\,c^2\,\left (c\,x^2+x\right )\,\left (c\,x-1\right )}\right )+\ln \left (1-c^2\,x^2\right )\,\left (\frac {a\,e\,x^2}{2}+\frac {b\,e\,x}{2\,c}\right )-\ln \left (\frac {1}{c\,x}+1\right )\,\left (\ln \left (1-c^2\,x^2\right )\,\left (\frac {b\,e}{4\,c^2}-\frac {b\,e\,x^2}{4}\right )-\frac {b\,d\,x^2}{4}+\frac {b\,e\,x^2}{4}\right )+\frac {a\,x^2\,\left (d-e\right )}{2}-\frac {\ln \left (c\,x+1\right )\,\left (2\,a\,e+b\,d-3\,b\,e\right )}{4\,c^2}-\frac {\ln \left (c\,x-1\right )\,\left (2\,a\,e-b\,d+3\,b\,e\right )}{4\,c^2}+\frac {b\,x\,\left (d-3\,e\right )}{2\,c} \]
log(1 - 1/(c*x))*(((b*d*x^3)/2 - (b*c^2*d*x^5)/2)/(2*(x + c*x^2)*(c*x - 1) ) - ((b*e*x^3)/2 - (b*c^2*e*x^5)/2)/(2*(x + c*x^2)*(c*x - 1)) + (log(1 - c ^2*x^2)*((b*e*x^3)/2 - (b*c^2*e*x^5)/2))/(2*(x + c*x^2)*(c*x - 1)) - (b*e* log(1 - c^2*x^2)*(x - c^2*x^3))/(4*c^2*(x + c*x^2)*(c*x - 1))) + log(1 - c ^2*x^2)*((a*e*x^2)/2 + (b*e*x)/(2*c)) - log(1/(c*x) + 1)*(log(1 - c^2*x^2) *((b*e)/(4*c^2) - (b*e*x^2)/4) - (b*d*x^2)/4 + (b*e*x^2)/4) + (a*x^2*(d - e))/2 - (log(c*x + 1)*(2*a*e + b*d - 3*b*e))/(4*c^2) - (log(c*x - 1)*(2*a* e - b*d + 3*b*e))/(4*c^2) + (b*x*(d - 3*e))/(2*c)