Integrand size = 20, antiderivative size = 81 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx=-\frac {b c d^2}{6 x^2}-\frac {b c^2 d^2}{x}-\frac {d^2 (1+c x)^3 (a+b \text {arctanh}(c x))}{3 x^3}+\frac {4}{3} b c^3 d^2 \log (x)-\frac {4}{3} b c^3 d^2 \log (1-c x) \]
-1/6*b*c*d^2/x^2-b*c^2*d^2/x-1/3*d^2*(c*x+1)^3*(a+b*arctanh(c*x))/x^3+4/3* b*c^3*d^2*ln(x)-4/3*b*c^3*d^2*ln(-c*x+1)
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.27 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx=-\frac {d^2 \left (2 a+6 a c x+b c x+6 a c^2 x^2+6 b c^2 x^2+2 b \left (1+3 c x+3 c^2 x^2\right ) \text {arctanh}(c x)-8 b c^3 x^3 \log (x)+7 b c^3 x^3 \log (1-c x)+b c^3 x^3 \log (1+c x)\right )}{6 x^3} \]
-1/6*(d^2*(2*a + 6*a*c*x + b*c*x + 6*a*c^2*x^2 + 6*b*c^2*x^2 + 2*b*(1 + 3* c*x + 3*c^2*x^2)*ArcTanh[c*x] - 8*b*c^3*x^3*Log[x] + 7*b*c^3*x^3*Log[1 - c *x] + b*c^3*x^3*Log[1 + c*x]))/x^3
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6498, 27, 99, 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.
\(\displaystyle \int \frac {(c d x+d)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx\) |
\(\Big \downarrow \) 6498 |
\(\displaystyle -b c \int -\frac {d^2 (c x+1)^2}{3 x^3 (1-c x)}dx-\frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} b c d^2 \int \frac {(c x+1)^2}{x^3 (1-c x)}dx-\frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{3} b c d^2 \int \left (-\frac {4 c^3}{c x-1}+\frac {4 c^2}{x}+\frac {3 c}{x^2}+\frac {1}{x^3}\right )dx-\frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} b c d^2 \left (4 c^2 \log (x)-4 c^2 \log (1-c x)-\frac {3 c}{x}-\frac {1}{2 x^2}\right )-\frac {d^2 (c x+1)^3 (a+b \text {arctanh}(c x))}{3 x^3}\) |
-1/3*(d^2*(1 + c*x)^3*(a + b*ArcTanh[c*x]))/x^3 + (b*c*d^2*(-1/2*1/x^2 - ( 3*c)/x + 4*c^2*Log[x] - 4*c^2*Log[1 - c*x]))/3
3.1.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*( x_))^(q_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x)^q, x]}, Simp[( a + b*ArcTanh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(1 - c^2*x ^2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && Intege rQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0 ]))
Time = 1.01 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.36
method | result | size |
parts | \(a \,d^{2} \left (-\frac {c^{2}}{x}-\frac {1}{3 x^{3}}-\frac {c}{x^{2}}\right )+d^{2} b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}\right )\) | \(110\) |
derivativedivides | \(c^{3} \left (a \,d^{2} \left (-\frac {1}{c x}-\frac {1}{c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+d^{2} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}\right )\right )\) | \(116\) |
default | \(c^{3} \left (a \,d^{2} \left (-\frac {1}{c x}-\frac {1}{c^{2} x^{2}}-\frac {1}{3 c^{3} x^{3}}\right )+d^{2} b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-\frac {\operatorname {arctanh}\left (c x \right )}{c^{2} x^{2}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\ln \left (c x +1\right )}{6}-\frac {7 \ln \left (c x -1\right )}{6}-\frac {1}{6 c^{2} x^{2}}-\frac {1}{c x}+\frac {4 \ln \left (c x \right )}{3}\right )\right )\) | \(116\) |
risch | \(-\frac {d^{2} b \left (3 c^{2} x^{2}+3 c x +1\right ) \ln \left (c x +1\right )}{6 x^{3}}-\frac {d^{2} \left (b \,c^{3} \ln \left (c x +1\right ) x^{3}+7 b \,x^{3} \ln \left (-c x +1\right ) c^{3}-8 b \,c^{3} \ln \left (-x \right ) x^{3}-3 b \,x^{2} \ln \left (-c x +1\right ) c^{2}+6 a \,c^{2} x^{2}+6 b \,c^{2} x^{2}-3 b c x \ln \left (-c x +1\right )+6 c x a +b c x -b \ln \left (-c x +1\right )+2 a \right )}{6 x^{3}}\) | \(151\) |
parallelrisch | \(-\frac {8 \ln \left (c x -1\right ) x^{3} b \,c^{3} d^{2}-8 b \,c^{3} d^{2} \ln \left (x \right ) x^{3}+2 x^{3} \operatorname {arctanh}\left (c x \right ) b \,d^{2} c^{3}+6 a \,c^{3} d^{2} x^{3}+b \,c^{3} d^{2} x^{3}+6 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{2}+6 a \,c^{2} d^{2} x^{2}+6 b \,c^{2} d^{2} x^{2}+6 b c \,d^{2} x \,\operatorname {arctanh}\left (c x \right )+6 a \,d^{2} c x +b c \,d^{2} x +2 b \,d^{2} \operatorname {arctanh}\left (c x \right )+2 a \,d^{2}}{6 x^{3}}\) | \(161\) |
a*d^2*(-c^2/x-1/3/x^3-c/x^2)+d^2*b*c^3*(-1/c/x*arctanh(c*x)-1/c^2/x^2*arct anh(c*x)-1/3/c^3/x^3*arctanh(c*x)-1/6*ln(c*x+1)-7/6*ln(c*x-1)-1/6/c^2/x^2- 1/c/x+4/3*ln(c*x))
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.58 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx=-\frac {b c^{3} d^{2} x^{3} \log \left (c x + 1\right ) + 7 \, b c^{3} d^{2} x^{3} \log \left (c x - 1\right ) - 8 \, b c^{3} d^{2} x^{3} \log \left (x\right ) + 6 \, {\left (a + b\right )} c^{2} d^{2} x^{2} + {\left (6 \, a + b\right )} c d^{2} x + 2 \, a d^{2} + {\left (3 \, b c^{2} d^{2} x^{2} + 3 \, b c d^{2} x + b d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{6 \, x^{3}} \]
-1/6*(b*c^3*d^2*x^3*log(c*x + 1) + 7*b*c^3*d^2*x^3*log(c*x - 1) - 8*b*c^3* d^2*x^3*log(x) + 6*(a + b)*c^2*d^2*x^2 + (6*a + b)*c*d^2*x + 2*a*d^2 + (3* b*c^2*d^2*x^2 + 3*b*c*d^2*x + b*d^2)*log(-(c*x + 1)/(c*x - 1)))/x^3
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (78) = 156\).
Time = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.95 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx=\begin {cases} - \frac {a c^{2} d^{2}}{x} - \frac {a c d^{2}}{x^{2}} - \frac {a d^{2}}{3 x^{3}} + \frac {4 b c^{3} d^{2} \log {\left (x \right )}}{3} - \frac {4 b c^{3} d^{2} \log {\left (x - \frac {1}{c} \right )}}{3} - \frac {b c^{3} d^{2} \operatorname {atanh}{\left (c x \right )}}{3} - \frac {b c^{2} d^{2} \operatorname {atanh}{\left (c x \right )}}{x} - \frac {b c^{2} d^{2}}{x} - \frac {b c d^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}} - \frac {b c d^{2}}{6 x^{2}} - \frac {b d^{2} \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} & \text {for}\: c \neq 0 \\- \frac {a d^{2}}{3 x^{3}} & \text {otherwise} \end {cases} \]
Piecewise((-a*c**2*d**2/x - a*c*d**2/x**2 - a*d**2/(3*x**3) + 4*b*c**3*d** 2*log(x)/3 - 4*b*c**3*d**2*log(x - 1/c)/3 - b*c**3*d**2*atanh(c*x)/3 - b*c **2*d**2*atanh(c*x)/x - b*c**2*d**2/x - b*c*d**2*atanh(c*x)/x**2 - b*c*d** 2/(6*x**2) - b*d**2*atanh(c*x)/(3*x**3), Ne(c, 0)), (-a*d**2/(3*x**3), Tru e))
Leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (73) = 146\).
Time = 0.18 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.94 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx=-\frac {1}{2} \, {\left (c {\left (\log \left (c^{2} x^{2} - 1\right ) - \log \left (x^{2}\right )\right )} + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x}\right )} b c^{2} d^{2} + \frac {1}{2} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b c d^{2} - \frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b d^{2} - \frac {a c^{2} d^{2}}{x} - \frac {a c d^{2}}{x^{2}} - \frac {a d^{2}}{3 \, x^{3}} \]
-1/2*(c*(log(c^2*x^2 - 1) - log(x^2)) + 2*arctanh(c*x)/x)*b*c^2*d^2 + 1/2* ((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arctanh(c*x)/x^2)*b*c*d^2 - 1/6*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3 )*b*d^2 - a*c^2*d^2/x - a*c*d^2/x^2 - 1/3*a*d^2/x^3
Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (73) = 146\).
Time = 0.28 (sec) , antiderivative size = 330, normalized size of antiderivative = 4.07 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx=\frac {2}{3} \, {\left (2 \, b c^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 2 \, b c^{2} d^{2} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {2 \, {\left (\frac {3 \, {\left (c x + 1\right )}^{2} b c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )} b c^{2} d^{2}}{c x - 1} + b c^{2} d^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {12 \, {\left (c x + 1\right )}^{2} a c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {12 \, {\left (c x + 1\right )} a c^{2} d^{2}}{c x - 1} + 4 \, a c^{2} d^{2} + \frac {4 \, {\left (c x + 1\right )}^{2} b c^{2} d^{2}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )} b c^{2} d^{2}}{c x - 1} + 3 \, b c^{2} d^{2}}{\frac {{\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {3 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {3 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \]
2/3*(2*b*c^2*d^2*log(-(c*x + 1)/(c*x - 1) - 1) - 2*b*c^2*d^2*log(-(c*x + 1 )/(c*x - 1)) + 2*(3*(c*x + 1)^2*b*c^2*d^2/(c*x - 1)^2 + 3*(c*x + 1)*b*c^2* d^2/(c*x - 1) + b*c^2*d^2)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^3/(c*x - 1 )^3 + 3*(c*x + 1)^2/(c*x - 1)^2 + 3*(c*x + 1)/(c*x - 1) + 1) + (12*(c*x + 1)^2*a*c^2*d^2/(c*x - 1)^2 + 12*(c*x + 1)*a*c^2*d^2/(c*x - 1) + 4*a*c^2*d^ 2 + 4*(c*x + 1)^2*b*c^2*d^2/(c*x - 1)^2 + 7*(c*x + 1)*b*c^2*d^2/(c*x - 1) + 3*b*c^2*d^2)/((c*x + 1)^3/(c*x - 1)^3 + 3*(c*x + 1)^2/(c*x - 1)^2 + 3*(c *x + 1)/(c*x - 1) + 1))*c
Time = 3.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.43 \[ \int \frac {(d+c d x)^2 (a+b \text {arctanh}(c x))}{x^4} \, dx=\frac {d^2\,\left (6\,b\,c^3\,\mathrm {atanh}\left (c\,x\right )-4\,b\,c^3\,\ln \left (c^2\,x^2-1\right )+8\,b\,c^3\,\ln \left (x\right )\right )}{6}-\frac {\frac {d^2\,\left (2\,a+2\,b\,\mathrm {atanh}\left (c\,x\right )\right )}{6}+\frac {d^2\,x\,\left (6\,a\,c+b\,c+6\,b\,c\,\mathrm {atanh}\left (c\,x\right )\right )}{6}+\frac {d^2\,x^2\,\left (6\,a\,c^2+6\,b\,c^2+6\,b\,c^2\,\mathrm {atanh}\left (c\,x\right )\right )}{6}}{x^3} \]