Integrand size = 20, antiderivative size = 218 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=-\frac {b c}{8 d^3 (1+c x)^2}-\frac {9 b c}{8 d^3 (1+c x)}+\frac {9 b c \text {arctanh}(c x)}{8 d^3}-\frac {a+b \text {arctanh}(c x)}{d^3 x}-\frac {c (a+b \text {arctanh}(c x))}{2 d^3 (1+c x)^2}-\frac {2 c (a+b \text {arctanh}(c x))}{d^3 (1+c x)}-\frac {3 a c \log (x)}{d^3}+\frac {b c \log (x)}{d^3}-\frac {3 c (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{d^3}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^3}+\frac {3 b c \operatorname {PolyLog}(2,-c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,c x)}{2 d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 d^3} \] Output:
-1/8*b*c/d^3/(c*x+1)^2-9/8*b*c/d^3/(c*x+1)+9/8*b*c*arctanh(c*x)/d^3-(a+b*a rctanh(c*x))/d^3/x-1/2*c*(a+b*arctanh(c*x))/d^3/(c*x+1)^2-2*c*(a+b*arctanh (c*x))/d^3/(c*x+1)-3*a*c*ln(x)/d^3+b*c*ln(x)/d^3-3*c*(a+b*arctanh(c*x))*ln (2/(c*x+1))/d^3-1/2*b*c*ln(-c^2*x^2+1)/d^3+3/2*b*c*polylog(2,-c*x)/d^3-3/2 *b*c*polylog(2,c*x)/d^3+3/2*b*c*polylog(2,1-2/(c*x+1))/d^3
Time = 0.80 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=\frac {-\frac {32 a}{x}-\frac {16 a c}{(1+c x)^2}-\frac {64 a c}{1+c x}-96 a c \log (x)+96 a c \log (1+c x)+b c \left (-20 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))+32 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )+48 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )+20 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))+4 \text {arctanh}(c x) \left (-\frac {8}{c x}-10 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-24 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )+10 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))\right )\right )}{32 d^3} \] Input:
Integrate[(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^3),x]
Output:
((-32*a)/x - (16*a*c)/(1 + c*x)^2 - (64*a*c)/(1 + c*x) - 96*a*c*Log[x] + 9 6*a*c*Log[1 + c*x] + b*c*(-20*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] + 32*Log[(c*x)/Sqrt[1 - c^2*x^2]] + 48*PolyLog[2, E^(-2*ArcTanh[c*x])] + 2 0*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-8/(c*x) - 10*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 24*Log[1 - E^(-2*ArcTanh [c*x])] + 10*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(32*d^3)
Time = 0.55 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6502, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{x^2 (c d x+d)^3} \, dx\) |
| \(\Big \downarrow \) 6502 |
| \(\displaystyle \int \left (\frac {3 c^2 (a+b \text {arctanh}(c x))}{d^3 (c x+1)}+\frac {2 c^2 (a+b \text {arctanh}(c x))}{d^3 (c x+1)^2}+\frac {c^2 (a+b \text {arctanh}(c x))}{d^3 (c x+1)^3}+\frac {a+b \text {arctanh}(c x)}{d^3 x^2}-\frac {3 c (a+b \text {arctanh}(c x))}{d^3 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c (a+b \text {arctanh}(c x))}{d^3 (c x+1)}-\frac {c (a+b \text {arctanh}(c x))}{2 d^3 (c x+1)^2}-\frac {a+b \text {arctanh}(c x)}{d^3 x}-\frac {3 c \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{d^3}-\frac {3 a c \log (x)}{d^3}+\frac {9 b c \text {arctanh}(c x)}{8 d^3}-\frac {b c \log \left (1-c^2 x^2\right )}{2 d^3}+\frac {3 b c \operatorname {PolyLog}(2,-c x)}{2 d^3}-\frac {3 b c \operatorname {PolyLog}(2,c x)}{2 d^3}+\frac {3 b c \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 d^3}-\frac {9 b c}{8 d^3 (c x+1)}-\frac {b c}{8 d^3 (c x+1)^2}+\frac {b c \log (x)}{d^3}\) |
Input:
Int[(a + b*ArcTanh[c*x])/(x^2*(d + c*d*x)^3),x]
Output:
-1/8*(b*c)/(d^3*(1 + c*x)^2) - (9*b*c)/(8*d^3*(1 + c*x)) + (9*b*c*ArcTanh[ c*x])/(8*d^3) - (a + b*ArcTanh[c*x])/(d^3*x) - (c*(a + b*ArcTanh[c*x]))/(2 *d^3*(1 + c*x)^2) - (2*c*(a + b*ArcTanh[c*x]))/(d^3*(1 + c*x)) - (3*a*c*Lo g[x])/d^3 + (b*c*Log[x])/d^3 - (3*c*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)]) /d^3 - (b*c*Log[1 - c^2*x^2])/(2*d^3) + (3*b*c*PolyLog[2, -(c*x)])/(2*d^3) - (3*b*c*PolyLog[2, c*x])/(2*d^3) + (3*b*c*PolyLog[2, 1 - 2/(1 + c*x)])/( 2*d^3)
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
Time = 0.49 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.02
| method | result | size |
| parts | \(\frac {a \left (-\frac {c}{2 \left (c x +1\right )^{2}}-\frac {2 c}{c x +1}+3 c \ln \left (c x +1\right )-\frac {1}{x}-3 c \ln \left (x \right )\right )}{d^{3}}+\frac {b c \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x +1}+3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {17 \ln \left (c x -1\right )}{16}+\ln \left (c x \right )-\frac {1}{8 \left (c x +1\right )^{2}}-\frac {9}{8 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}+\frac {3 \operatorname {dilog}\left (c x \right )}{2}+\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}+\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {3 \ln \left (c x +1\right )^{2}}{4}+\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{d^{3}}\) | \(222\) |
| derivativedivides | \(c \left (\frac {a \left (-\frac {1}{c x}-3 \ln \left (c x \right )-\frac {1}{2 \left (c x +1\right )^{2}}-\frac {2}{c x +1}+3 \ln \left (c x +1\right )\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x +1}+3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {17 \ln \left (c x -1\right )}{16}+\ln \left (c x \right )-\frac {1}{8 \left (c x +1\right )^{2}}-\frac {9}{8 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}+\frac {3 \operatorname {dilog}\left (c x \right )}{2}+\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}+\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {3 \ln \left (c x +1\right )^{2}}{4}+\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{d^{3}}\right )\) | \(224\) |
| default | \(c \left (\frac {a \left (-\frac {1}{c x}-3 \ln \left (c x \right )-\frac {1}{2 \left (c x +1\right )^{2}}-\frac {2}{c x +1}+3 \ln \left (c x +1\right )\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x}-3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x \right )-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}-\frac {2 \,\operatorname {arctanh}\left (c x \right )}{c x +1}+3 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )-\frac {17 \ln \left (c x -1\right )}{16}+\ln \left (c x \right )-\frac {1}{8 \left (c x +1\right )^{2}}-\frac {9}{8 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{16}+\frac {3 \operatorname {dilog}\left (c x \right )}{2}+\frac {3 \operatorname {dilog}\left (c x +1\right )}{2}+\frac {3 \ln \left (c x \right ) \ln \left (c x +1\right )}{2}-\frac {3 \ln \left (c x +1\right )^{2}}{4}+\frac {3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2}-\frac {3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{2}\right )}{d^{3}}\right )\) | \(224\) |
| risch | \(\frac {c b}{8 d^{3} \left (-c x -1\right )}-\frac {a}{d^{3} x}-\frac {c a}{2 d^{3} \left (-c x -1\right )^{2}}+\frac {2 c a}{d^{3} \left (-c x -1\right )}-\frac {3 c b \operatorname {dilog}\left (-c x +1\right )}{2 d^{3}}+\frac {3 c b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}+\frac {c b \ln \left (-c x \right )}{2 d^{3}}-\frac {c b \ln \left (-c x +1\right )}{2 d^{3}}+\frac {b \ln \left (-c x +1\right )}{2 d^{3} x}-\frac {3 c a \ln \left (-c x \right )}{d^{3}}+\frac {3 c a \ln \left (-c x -1\right )}{d^{3}}+\frac {3 b c \ln \left (c x +1\right )^{2}}{4 d^{3}}+\frac {3 b c \operatorname {dilog}\left (c x +1\right )}{2 d^{3}}+\frac {b c \ln \left (c x \right )}{2 d^{3}}-\frac {b c \ln \left (c x +1\right )}{2 d^{3}}-\frac {b \ln \left (c x +1\right )}{2 d^{3} x}+\frac {c^{2} b \ln \left (-c x +1\right ) x}{2 d^{3} \left (-c x -1\right )}-\frac {c^{3} b \ln \left (-c x +1\right ) x^{2}}{16 d^{3} \left (-c x -1\right )^{2}}-\frac {c^{2} b \ln \left (-c x +1\right ) x}{8 d^{3} \left (-c x -1\right )^{2}}+\frac {9 b c \ln \left (-c x -1\right )}{16 d^{3}}-\frac {b c}{8 d^{3} \left (c x +1\right )^{2}}-\frac {b c}{d^{3} \left (c x +1\right )}-\frac {b c \ln \left (c x +1\right )}{d^{3} \left (c x +1\right )}-\frac {b c \ln \left (c x +1\right )}{4 d^{3} \left (c x +1\right )^{2}}-\frac {c b \ln \left (-c x +1\right )}{2 d^{3} \left (-c x -1\right )}+\frac {3 c b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{2 d^{3}}-\frac {3 c b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{2 d^{3}}+\frac {3 c b \ln \left (-c x +1\right )}{16 d^{3} \left (-c x -1\right )^{2}}\) | \(464\) |
Input:
int((a+b*arctanh(c*x))/x^2/(c*d*x+d)^3,x,method=_RETURNVERBOSE)
Output:
a/d^3*(-1/2/(c*x+1)^2*c-2*c/(c*x+1)+3*c*ln(c*x+1)-1/x-3*c*ln(x))+b/d^3*c*( -arctanh(c*x)/c/x-3*arctanh(c*x)*ln(c*x)-1/2/(c*x+1)^2*arctanh(c*x)-2/(c*x +1)*arctanh(c*x)+3*arctanh(c*x)*ln(c*x+1)-17/16*ln(c*x-1)+ln(c*x)-1/8/(c*x +1)^2-9/8/(c*x+1)+1/16*ln(c*x+1)+3/2*dilog(c*x)+3/2*dilog(c*x+1)+3/2*ln(c* x)*ln(c*x+1)-3/4*ln(c*x+1)^2+3/2*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(-1/2*c*x+1 /2)-3/2*dilog(1/2*c*x+1/2))
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(c*d*x+d)^3,x, algorithm="fricas")
Output:
integral((b*arctanh(c*x) + a)/(c^3*d^3*x^5 + 3*c^2*d^3*x^4 + 3*c*d^3*x^3 + d^3*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=\frac {\int \frac {a}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx + \int \frac {b \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{5} + 3 c^{2} x^{4} + 3 c x^{3} + x^{2}}\, dx}{d^{3}} \] Input:
integrate((a+b*atanh(c*x))/x**2/(c*d*x+d)**3,x)
Output:
(Integral(a/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x) + Integral(b*a tanh(c*x)/(c**3*x**5 + 3*c**2*x**4 + 3*c*x**3 + x**2), x))/d**3
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(c*d*x+d)^3,x, algorithm="maxima")
Output:
-1/2*a*((6*c^2*x^2 + 9*c*x + 2)/(c^2*d^3*x^3 + 2*c*d^3*x^2 + d^3*x) - 6*c* log(c*x + 1)/d^3 + 6*c*log(x)/d^3) + 1/2*b*integrate((log(c*x + 1) - log(- c*x + 1))/(c^3*d^3*x^5 + 3*c^2*d^3*x^4 + 3*c*d^3*x^3 + d^3*x^2), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=\int { \frac {b \operatorname {artanh}\left (c x\right ) + a}{{\left (c d x + d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arctanh(c*x))/x^2/(c*d*x+d)^3,x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)/((c*d*x + d)^3*x^2), x)
Timed out. \[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{x^2\,{\left (d+c\,d\,x\right )}^3} \,d x \] Input:
int((a + b*atanh(c*x))/(x^2*(d + c*d*x)^3),x)
Output:
int((a + b*atanh(c*x))/(x^2*(d + c*d*x)^3), x)
\[ \int \frac {a+b \text {arctanh}(c x)}{x^2 (d+c d x)^3} \, dx=\frac {-64 a -12 \mathit {atanh} \left (c x \right )^{2} b \,c^{3} x^{3}+192 \,\mathrm {log}\left (c x +1\right ) a \,c^{3} x^{3}-32 \,\mathrm {log}\left (x \right ) b \,c^{3} x^{3}-384 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}-12 \mathit {atanh} \left (c x \right )^{2} b c x +36 \mathit {atanh} \left (c x \right ) b c x -192 a c x -12 \mathit {atanh} \left (c x \right ) b \,c^{3} x^{3}+96 a \,c^{3} x^{3}+3 b \,c^{3} x^{3}-64 \,\mathrm {log}\left (x \right ) b \,c^{2} x^{2}+7 \,\mathrm {log}\left (c x -1\right ) b c x +25 \,\mathrm {log}\left (c x +1\right ) b c x -5 b c x -32 \,\mathrm {log}\left (x \right ) b c x -192 \,\mathrm {log}\left (x \right ) a c x -24 \mathit {atanh} \left (c x \right )^{2} b \,c^{2} x^{2}+50 \,\mathrm {log}\left (c x +1\right ) b \,c^{2} x^{2}-192 \,\mathrm {log}\left (x \right ) a \,c^{3} x^{3}+32 \mathit {atanh} \left (c x \right ) b -96 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{4} x^{6}+2 c^{3} x^{5}-2 c \,x^{3}-x^{2}}d x \right ) b \,c^{2} x^{3}-192 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{4} x^{6}+2 c^{3} x^{5}-2 c \,x^{3}-x^{2}}d x \right ) b c \,x^{2}+7 \,\mathrm {log}\left (c x -1\right ) b \,c^{3} x^{3}+14 \,\mathrm {log}\left (c x -1\right ) b \,c^{2} x^{2}+25 \,\mathrm {log}\left (c x +1\right ) b \,c^{3} x^{3}-96 \left (\int \frac {\mathit {atanh} \left (c x \right )}{c^{4} x^{6}+2 c^{3} x^{5}-2 c \,x^{3}-x^{2}}d x \right ) b x +192 \,\mathrm {log}\left (c x +1\right ) a c x +384 \,\mathrm {log}\left (c x +1\right ) a \,c^{2} x^{2}}{64 d^{3} x \left (c^{2} x^{2}+2 c x +1\right )} \] Input:
int((a+b*atanh(c*x))/x^2/(c*d*x+d)^3,x)
Output:
( - 12*atanh(c*x)**2*b*c**3*x**3 - 24*atanh(c*x)**2*b*c**2*x**2 - 12*atanh (c*x)**2*b*c*x - 12*atanh(c*x)*b*c**3*x**3 + 36*atanh(c*x)*b*c*x + 32*atan h(c*x)*b - 96*int(atanh(c*x)/(c**4*x**6 + 2*c**3*x**5 - 2*c*x**3 - x**2),x )*b*c**2*x**3 - 192*int(atanh(c*x)/(c**4*x**6 + 2*c**3*x**5 - 2*c*x**3 - x **2),x)*b*c*x**2 - 96*int(atanh(c*x)/(c**4*x**6 + 2*c**3*x**5 - 2*c*x**3 - x**2),x)*b*x + 7*log(c*x - 1)*b*c**3*x**3 + 14*log(c*x - 1)*b*c**2*x**2 + 7*log(c*x - 1)*b*c*x + 192*log(c*x + 1)*a*c**3*x**3 + 384*log(c*x + 1)*a* c**2*x**2 + 192*log(c*x + 1)*a*c*x + 25*log(c*x + 1)*b*c**3*x**3 + 50*log( c*x + 1)*b*c**2*x**2 + 25*log(c*x + 1)*b*c*x - 192*log(x)*a*c**3*x**3 - 38 4*log(x)*a*c**2*x**2 - 192*log(x)*a*c*x - 32*log(x)*b*c**3*x**3 - 64*log(x )*b*c**2*x**2 - 32*log(x)*b*c*x + 96*a*c**3*x**3 - 192*a*c*x - 64*a + 3*b* c**3*x**3 - 5*b*c*x)/(64*d**3*x*(c**2*x**2 + 2*c*x + 1))