Integrand size = 19, antiderivative size = 107 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=-\frac {b^2}{2 c d^2 (1+c x)}+\frac {b^2 \text {arctanh}(c x)}{2 c d^2}-\frac {b (a+b \text {arctanh}(c x))}{c d^2 (1+c x)}+\frac {(a+b \text {arctanh}(c x))^2}{2 c d^2}-\frac {(a+b \text {arctanh}(c x))^2}{c d^2 (1+c x)} \] Output:
-1/2*b^2/c/d^2/(c*x+1)+1/2*b^2*arctanh(c*x)/c/d^2-b*(a+b*arctanh(c*x))/c/d ^2/(c*x+1)+1/2*(a+b*arctanh(c*x))^2/c/d^2-(a+b*arctanh(c*x))^2/c/d^2/(c*x+ 1)
Time = 0.32 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {-4 a^2-4 a b-2 b^2-4 b (2 a+b) \text {arctanh}(c x)+2 b^2 (-1+c x) \text {arctanh}(c x)^2-b (2 a+b) (1+c x) \log (1-c x)+2 a b \log (1+c x)+b^2 \log (1+c x)+2 a b c x \log (1+c x)+b^2 c x \log (1+c x)}{4 c d^2 (1+c x)} \] Input:
Integrate[(a + b*ArcTanh[c*x])^2/(d + c*d*x)^2,x]
Output:
(-4*a^2 - 4*a*b - 2*b^2 - 4*b*(2*a + b)*ArcTanh[c*x] + 2*b^2*(-1 + c*x)*Ar cTanh[c*x]^2 - b*(2*a + b)*(1 + c*x)*Log[1 - c*x] + 2*a*b*Log[1 + c*x] + b ^2*Log[1 + c*x] + 2*a*b*c*x*Log[1 + c*x] + b^2*c*x*Log[1 + c*x])/(4*c*d^2* (1 + c*x))
Time = 0.36 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6480, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{(c d x+d)^2} \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {2 b \int \left (\frac {a+b \text {arctanh}(c x)}{2 d \left (1-c^2 x^2\right )}+\frac {a+b \text {arctanh}(c x)}{2 d (c x+1)^2}\right )dx}{d}-\frac {(a+b \text {arctanh}(c x))^2}{c d^2 (c x+1)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b \left (\frac {(a+b \text {arctanh}(c x))^2}{4 b c d}-\frac {a+b \text {arctanh}(c x)}{2 c d (c x+1)}+\frac {b \text {arctanh}(c x)}{4 c d}-\frac {b}{4 c d (c x+1)}\right )}{d}-\frac {(a+b \text {arctanh}(c x))^2}{c d^2 (c x+1)}\) |
Input:
Int[(a + b*ArcTanh[c*x])^2/(d + c*d*x)^2,x]
Output:
-((a + b*ArcTanh[c*x])^2/(c*d^2*(1 + c*x))) + (2*b*(-1/4*b/(c*d*(1 + c*x)) + (b*ArcTanh[c*x])/(4*c*d) - (a + b*ArcTanh[c*x])/(2*c*d*(1 + c*x)) + (a + b*ArcTanh[c*x])^2/(4*b*c*d)))/d
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^(p - 1 ), (d + e*x)^(q + 1)/(1 - c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && NeQ[q, -1]
Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.90
| method | result | size |
| parallelrisch | \(\frac {b^{2} c x \operatorname {arctanh}\left (c x \right )^{2}+2 \,\operatorname {arctanh}\left (c x \right ) a b c x +c \,b^{2} \operatorname {arctanh}\left (c x \right ) x +2 a^{2} c x +2 a b c x +b^{2} c x -b^{2} \operatorname {arctanh}\left (c x \right )^{2}-2 \,\operatorname {arctanh}\left (c x \right ) a b -b^{2} \operatorname {arctanh}\left (c x \right )}{2 d^{2} \left (c x +1\right ) c}\) | \(96\) |
| derivativedivides | \(\frac {-\frac {a^{2}}{d^{2} \left (c x +1\right )}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c x -1\right )^{2}}{8}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}-\frac {\ln \left (c x +1\right )^{2}}{8}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}\right )}{d^{2}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}}{c}\) | \(212\) |
| default | \(\frac {-\frac {a^{2}}{d^{2} \left (c x +1\right )}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c x -1\right )^{2}}{8}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}-\frac {\ln \left (c x +1\right )^{2}}{8}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}\right )}{d^{2}}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2}}}{c}\) | \(212\) |
| parts | \(-\frac {a^{2}}{d^{2} \left (c x +1\right ) c}+\frac {b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{c x +1}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{2}-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}+\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{2}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{4}-\frac {\ln \left (c x -1\right )^{2}}{8}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}-\frac {\ln \left (c x +1\right )^{2}}{8}+\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{4}\right )}{d^{2} c}+\frac {2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {\ln \left (c x -1\right )}{4}-\frac {1}{2 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{4}\right )}{d^{2} c}\) | \(217\) |
| risch | \(\frac {b^{2} \left (c x -1\right ) \ln \left (c x +1\right )^{2}}{8 d^{2} \left (c x +1\right ) c}-\frac {b \left (b c x \ln \left (-c x +1\right )-b \ln \left (-c x +1\right )+4 a +2 b \right ) \ln \left (c x +1\right )}{4 d^{2} \left (c x +1\right ) c}-\frac {-b^{2} c x \ln \left (-c x +1\right )^{2}+4 a b c \ln \left (c x -1\right ) x +2 \ln \left (c x -1\right ) b^{2} c x -4 \ln \left (-c x -1\right ) a b c x -2 \ln \left (-c x -1\right ) b^{2} c x +\ln \left (-c x +1\right )^{2} b^{2}+4 a b \ln \left (c x -1\right )+2 b^{2} \ln \left (c x -1\right )-4 \ln \left (-c x -1\right ) a b -2 b^{2} \ln \left (-c x -1\right )-8 \ln \left (-c x +1\right ) a b -4 b^{2} \ln \left (-c x +1\right )+8 a^{2}+8 b a +4 b^{2}}{8 d^{2} \left (c x +1\right ) c}\) | \(261\) |
| orering | \(-\frac {\left (4 x^{3} c^{3}-3 c^{2} x^{2}-4 c x +3\right ) \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2}}{2 c \left (c d x +d \right )^{2}}-\frac {\left (c x +1\right )^{2} \left (c x -1\right ) \left (7 c x -5\right ) \left (\frac {2 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b c}{\left (c d x +d \right )^{2} \left (-c^{2} x^{2}+1\right )}-\frac {2 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} c d}{\left (c d x +d \right )^{3}}\right )}{4 c^{2}}-\frac {\left (c x +1\right )^{3} \left (c x -1\right )^{2} \left (\frac {2 b^{2} c^{2}}{\left (-c^{2} x^{2}+1\right )^{2} \left (c d x +d \right )^{2}}-\frac {8 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b \,c^{2} d}{\left (c d x +d \right )^{3} \left (-c^{2} x^{2}+1\right )}+\frac {4 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right ) b \,c^{3} x}{\left (c d x +d \right )^{2} \left (-c^{2} x^{2}+1\right )^{2}}+\frac {6 \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )^{2} c^{2} d^{2}}{\left (c d x +d \right )^{4}}\right )}{4 c^{3}}\) | \(269\) |
Input:
int((a+b*arctanh(c*x))^2/(c*d*x+d)^2,x,method=_RETURNVERBOSE)
Output:
1/2*(b^2*c*x*arctanh(c*x)^2+2*arctanh(c*x)*a*b*c*x+c*b^2*arctanh(c*x)*x+2* a^2*c*x+2*a*b*c*x+b^2*c*x-b^2*arctanh(c*x)^2-2*arctanh(c*x)*a*b-b^2*arctan h(c*x))/d^2/(c*x+1)/c
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {{\left (b^{2} c x - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 8 \, a^{2} - 8 \, a b - 4 \, b^{2} + 2 \, {\left ({\left (2 \, a b + b^{2}\right )} c x - 2 \, a b - b^{2}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{8 \, {\left (c^{2} d^{2} x + c d^{2}\right )}} \] Input:
integrate((a+b*arctanh(c*x))^2/(c*d*x+d)^2,x, algorithm="fricas")
Output:
1/8*((b^2*c*x - b^2)*log(-(c*x + 1)/(c*x - 1))^2 - 8*a^2 - 8*a*b - 4*b^2 + 2*((2*a*b + b^2)*c*x - 2*a*b - b^2)*log(-(c*x + 1)/(c*x - 1)))/(c^2*d^2*x + c*d^2)
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{c^{2} x^{2} + 2 c x + 1}\, dx}{d^{2}} \] Input:
integrate((a+b*atanh(c*x))**2/(c*d*x+d)**2,x)
Output:
(Integral(a**2/(c**2*x**2 + 2*c*x + 1), x) + Integral(b**2*atanh(c*x)**2/( c**2*x**2 + 2*c*x + 1), x) + Integral(2*a*b*atanh(c*x)/(c**2*x**2 + 2*c*x + 1), x))/d**2
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (101) = 202\).
Time = 0.05 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.59 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=-\frac {1}{2} \, {\left (c {\left (\frac {2}{c^{3} d^{2} x + c^{2} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{2}}\right )} + \frac {4 \, \operatorname {artanh}\left (c x\right )}{c^{2} d^{2} x + c d^{2}}\right )} a b - \frac {1}{8} \, {\left (4 \, c {\left (\frac {2}{c^{3} d^{2} x + c^{2} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{2} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{2} d^{2}}\right )} \operatorname {artanh}\left (c x\right ) + \frac {{\left ({\left (c x + 1\right )} \log \left (c x + 1\right )^{2} + {\left (c x + 1\right )} \log \left (c x - 1\right )^{2} - 2 \, {\left (c x + {\left (c x + 1\right )} \log \left (c x - 1\right ) + 1\right )} \log \left (c x + 1\right ) + 2 \, {\left (c x + 1\right )} \log \left (c x - 1\right ) + 4\right )} c^{2}}{c^{4} d^{2} x + c^{3} d^{2}}\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x\right )^{2}}{c^{2} d^{2} x + c d^{2}} - \frac {a^{2}}{c^{2} d^{2} x + c d^{2}} \] Input:
integrate((a+b*arctanh(c*x))^2/(c*d*x+d)^2,x, algorithm="maxima")
Output:
-1/2*(c*(2/(c^3*d^2*x + c^2*d^2) - log(c*x + 1)/(c^2*d^2) + log(c*x - 1)/( c^2*d^2)) + 4*arctanh(c*x)/(c^2*d^2*x + c*d^2))*a*b - 1/8*(4*c*(2/(c^3*d^2 *x + c^2*d^2) - log(c*x + 1)/(c^2*d^2) + log(c*x - 1)/(c^2*d^2))*arctanh(c *x) + ((c*x + 1)*log(c*x + 1)^2 + (c*x + 1)*log(c*x - 1)^2 - 2*(c*x + (c*x + 1)*log(c*x - 1) + 1)*log(c*x + 1) + 2*(c*x + 1)*log(c*x - 1) + 4)*c^2/( c^4*d^2*x + c^3*d^2))*b^2 - b^2*arctanh(c*x)^2/(c^2*d^2*x + c*d^2) - a^2/( c^2*d^2*x + c*d^2)
Time = 0.12 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {1}{8} \, c {\left (\frac {{\left (c x - 1\right )} b^{2} \log \left (-\frac {c x + 1}{c x - 1}\right )^{2}}{{\left (c x + 1\right )} c^{2} d^{2}} + \frac {2 \, {\left (2 \, a b + b^{2}\right )} {\left (c x - 1\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{{\left (c x + 1\right )} c^{2} d^{2}} + \frac {2 \, {\left (2 \, a^{2} + 2 \, a b + b^{2}\right )} {\left (c x - 1\right )}}{{\left (c x + 1\right )} c^{2} d^{2}}\right )} \] Input:
integrate((a+b*arctanh(c*x))^2/(c*d*x+d)^2,x, algorithm="giac")
Output:
1/8*c*((c*x - 1)*b^2*log(-(c*x + 1)/(c*x - 1))^2/((c*x + 1)*c^2*d^2) + 2*( 2*a*b + b^2)*(c*x - 1)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)*c^2*d^2) + 2*( 2*a^2 + 2*a*b + b^2)*(c*x - 1)/((c*x + 1)*c^2*d^2))
Time = 3.62 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {b^2\,{\mathrm {atanh}\left (c\,x\right )}^2+b^2\,\mathrm {atanh}\left (c\,x\right )+2\,a\,b\,\mathrm {atanh}\left (c\,x\right )}{2\,c\,d^2}-\frac {2\,a^2+4\,a\,b\,\mathrm {atanh}\left (c\,x\right )+2\,a\,b+2\,b^2\,{\mathrm {atanh}\left (c\,x\right )}^2+2\,b^2\,\mathrm {atanh}\left (c\,x\right )+b^2}{2\,x\,c^2\,d^2+2\,c\,d^2} \] Input:
int((a + b*atanh(c*x))^2/(d + c*d*x)^2,x)
Output:
(b^2*atanh(c*x)^2 + b^2*atanh(c*x) + 2*a*b*atanh(c*x))/(2*c*d^2) - (2*b^2* atanh(c*x)^2 + 2*a*b + 2*b^2*atanh(c*x) + 2*a^2 + b^2 + 4*a*b*atanh(c*x))/ (2*c*d^2 + 2*c^2*d^2*x)
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.60 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{(d+c d x)^2} \, dx=\frac {2 \mathit {atanh} \left (c x \right )^{2} b^{2} c x -2 \mathit {atanh} \left (c x \right )^{2} b^{2}+8 \mathit {atanh} \left (c x \right ) a b c x +4 \mathit {atanh} \left (c x \right ) b^{2} c x +2 \,\mathrm {log}\left (c x -1\right ) a b c x +2 \,\mathrm {log}\left (c x -1\right ) a b +\mathrm {log}\left (c x -1\right ) b^{2} c x +\mathrm {log}\left (c x -1\right ) b^{2}-2 \,\mathrm {log}\left (c x +1\right ) a b c x -2 \,\mathrm {log}\left (c x +1\right ) a b -\mathrm {log}\left (c x +1\right ) b^{2} c x -\mathrm {log}\left (c x +1\right ) b^{2}+4 a^{2} c x +4 a b c x +2 b^{2} c x}{4 c \,d^{2} \left (c x +1\right )} \] Input:
int((a+b*atanh(c*x))^2/(c*d*x+d)^2,x)
Output:
(2*atanh(c*x)**2*b**2*c*x - 2*atanh(c*x)**2*b**2 + 8*atanh(c*x)*a*b*c*x + 4*atanh(c*x)*b**2*c*x + 2*log(c*x - 1)*a*b*c*x + 2*log(c*x - 1)*a*b + log( c*x - 1)*b**2*c*x + log(c*x - 1)*b**2 - 2*log(c*x + 1)*a*b*c*x - 2*log(c*x + 1)*a*b - log(c*x + 1)*b**2*c*x - log(c*x + 1)*b**2 + 4*a**2*c*x + 4*a*b *c*x + 2*b**2*c*x)/(4*c*d**2*(c*x + 1))