Integrand size = 16, antiderivative size = 130 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx=\frac {b c}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a+b \text {arctanh}(c x)}{2 e (d+e x)^2}-\frac {b c^2 \log (1-c x)}{4 e (c d+e)^2}+\frac {b c^2 \log (1+c x)}{4 (c d-e)^2 e}-\frac {b c^3 d \log (d+e x)}{\left (c^2 d^2-e^2\right )^2} \] Output:
1/2*b*c/(c^2*d^2-e^2)/(e*x+d)-1/2*(a+b*arctanh(c*x))/e/(e*x+d)^2-1/4*b*c^2 *ln(-c*x+1)/e/(c*d+e)^2+1/4*b*c^2*ln(c*x+1)/(c*d-e)^2/e-b*c^3*d*ln(e*x+d)/ (c^2*d^2-e^2)^2
Time = 0.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx=\frac {1}{4} \left (-\frac {2 a}{e (d+e x)^2}+\frac {2 b c}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {2 b \text {arctanh}(c x)}{e (d+e x)^2}-\frac {b c^2 \log (1-c x)}{e (c d+e)^2}+\frac {b c^2 \log (1+c x)}{e (-c d+e)^2}-\frac {4 b c^3 d \log (d+e x)}{\left (-c^2 d^2+e^2\right )^2}\right ) \] Input:
Integrate[(a + b*ArcTanh[c*x])/(d + e*x)^3,x]
Output:
((-2*a)/(e*(d + e*x)^2) + (2*b*c)/((c^2*d^2 - e^2)*(d + e*x)) - (2*b*ArcTa nh[c*x])/(e*(d + e*x)^2) - (b*c^2*Log[1 - c*x])/(e*(c*d + e)^2) + (b*c^2*L og[1 + c*x])/(e*(-(c*d) + e)^2) - (4*b*c^3*d*Log[d + e*x])/(-(c^2*d^2) + e ^2)^2)/4
Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6478, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 6478 |
| \(\displaystyle \frac {b c \int \frac {1}{(d+e x)^2 \left (1-c^2 x^2\right )}dx}{2 e}-\frac {a+b \text {arctanh}(c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {b c \int \left (\frac {c^2}{2 (c d+e)^2 (1-c x)}+\frac {c^2}{2 (c d-e)^2 (c x+1)}-\frac {2 d e^2 c^2}{\left (c^2 d^2-e^2\right )^2 (d+e x)}-\frac {e^2}{\left (c^2 d^2-e^2\right ) (d+e x)^2}\right )dx}{2 e}-\frac {a+b \text {arctanh}(c x)}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c \left (\frac {e}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {2 c^2 d e \log (d+e x)}{\left (c^2 d^2-e^2\right )^2}-\frac {c \log (1-c x)}{2 (c d+e)^2}+\frac {c \log (c x+1)}{2 (c d-e)^2}\right )}{2 e}-\frac {a+b \text {arctanh}(c x)}{2 e (d+e x)^2}\) |
Input:
Int[(a + b*ArcTanh[c*x])/(d + e*x)^3,x]
Output:
-1/2*(a + b*ArcTanh[c*x])/(e*(d + e*x)^2) + (b*c*(e/((c^2*d^2 - e^2)*(d + e*x)) - (c*Log[1 - c*x])/(2*(c*d + e)^2) + (c*Log[1 + c*x])/(2*(c*d - e)^2 ) - (2*c^2*d*e*Log[d + e*x])/(c^2*d^2 - e^2)^2))/(2*e)
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_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol ] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])/(e*(q + 1))), x] - Simp[b *(c/(e*(q + 1))) Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[q, -1]
Time = 0.32 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09
| method | result | size |
| parts | \(-\frac {a}{2 \left (e x +d \right )^{2} e}+\frac {b \left (-\frac {c^{3} \operatorname {arctanh}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {c^{3} \left (\frac {e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}\right )}{2 e}\right )}{c}\) | \(142\) |
| derivativedivides | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\frac {e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}}{2 e}\right )}{c}\) | \(146\) |
| default | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}+b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {\frac {e}{\left (c d +e \right ) \left (c d -e \right ) \left (c e x +c d \right )}-\frac {2 e d c \ln \left (c e x +c d \right )}{\left (c d +e \right )^{2} \left (c d -e \right )^{2}}-\frac {\ln \left (c x -1\right )}{2 \left (c d +e \right )^{2}}+\frac {\ln \left (c x +1\right )}{2 \left (c d -e \right )^{2}}}{2 e}\right )}{c}\) | \(146\) |
| parallelrisch | \(\frac {-e^{3} a -x^{2} a \,c^{2} e^{3}-2 \ln \left (e x +d \right ) b \,c^{3} d^{3}+x^{2} a \,c^{4} d^{2} e +2 b \,d^{3} \operatorname {arctanh}\left (c x \right ) x \,c^{4}-x^{2} b \,c^{3} d \,e^{2}-x b \,c^{3} d^{2} e +3 \,\operatorname {arctanh}\left (c x \right ) b \,c^{2} d^{2} e -\operatorname {arctanh}\left (c x \right ) b \,e^{3}+x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{4} d^{2} e +4 x \,\operatorname {arctanh}\left (c x \right ) b \,c^{3} d^{2} e +2 x \,\operatorname {arctanh}\left (c x \right ) b \,c^{2} d \,e^{2}+2 \ln \left (c x -1\right ) x^{2} b \,c^{3} d \,e^{2}-2 \ln \left (e x +d \right ) x^{2} b \,c^{3} d \,e^{2}+4 \ln \left (c x -1\right ) x b \,c^{3} d^{2} e -4 \ln \left (e x +d \right ) x b \,c^{3} d^{2} e +2 x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{3} d \,e^{2}+x^{2} \operatorname {arctanh}\left (c x \right ) b \,c^{2} e^{3}-2 x a \,c^{2} d \,e^{2}+2 x a \,c^{4} d^{3}+2 \ln \left (c x -1\right ) b \,c^{3} d^{3}+2 \,\operatorname {arctanh}\left (c x \right ) b \,c^{3} d^{3}-x b c \,e^{3}-b d \,e^{2} c +a \,c^{2} d^{2} e}{2 \left (e x +d \right )^{2} \left (c^{2} d^{2}-2 c d e +e^{2}\right ) \left (c^{2} d^{2}+2 c d e +e^{2}\right )}\) | \(378\) |
| risch | \(-\frac {b \ln \left (c x +1\right )}{4 e \left (e x +d \right )^{2}}+\frac {\ln \left (-c x +1\right ) b \,e^{4}-b \,c^{4} d^{2} e^{2} x^{2} \ln \left (-c x +1\right )-2 b \,c^{4} d^{3} e x \ln \left (-c x +1\right )+\ln \left (-c x -1\right ) b \,c^{2} d^{2} e^{2}+\ln \left (-c x -1\right ) b \,c^{2} e^{4} x^{2}-\ln \left (-c x +1\right ) b \,c^{2} e^{4} x^{2}+2 \ln \left (-c x -1\right ) b \,c^{3} d^{3} e -4 \ln \left (e x +d \right ) b \,c^{3} d^{3} e -2 a \,e^{4}+4 a \,c^{2} d^{2} e^{2}+\ln \left (-c x -1\right ) b \,c^{4} d^{4}-2 a \,c^{4} d^{4}-2 b c \,e^{4} x +2 b \,c^{3} d^{3} e -2 b c d \,e^{3}+2 b \,c^{3} d^{2} e^{2} x +2 \ln \left (-c x +1\right ) b \,c^{3} d^{3} e -3 \ln \left (-c x +1\right ) b \,c^{2} d^{2} e^{2}+\ln \left (-c x -1\right ) b \,c^{4} d^{2} e^{2} x^{2}+2 \ln \left (-c x -1\right ) b \,c^{4} d^{3} e x +2 \ln \left (-c x -1\right ) b \,c^{3} d \,e^{3} x^{2}-4 \ln \left (e x +d \right ) b \,c^{3} d \,e^{3} x^{2}+2 \ln \left (-c x +1\right ) b \,c^{3} d \,e^{3} x^{2}+4 \ln \left (-c x -1\right ) b \,c^{3} d^{2} e^{2} x -8 \ln \left (e x +d \right ) b \,c^{3} d^{2} e^{2} x +4 \ln \left (-c x +1\right ) b \,c^{3} d^{2} e^{2} x +2 \ln \left (-c x -1\right ) b \,c^{2} d \,e^{3} x -2 \ln \left (-c x +1\right ) b \,c^{2} d \,e^{3} x}{4 \left (c^{2} d^{2}+2 c d e +e^{2}\right ) \left (c^{2} d^{2}-2 c d e +e^{2}\right ) \left (e x +d \right )^{2} e}\) | \(520\) |
Input:
int((a+b*arctanh(c*x))/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*a/(e*x+d)^2/e+b/c*(-1/2*c^3/(c*e*x+c*d)^2/e*arctanh(c*x)+1/2*c^3/e*(e /(c*d+e)/(c*d-e)/(c*e*x+c*d)-2*e*d*c/(c*d+e)^2/(c*d-e)^2*ln(c*e*x+c*d)-1/2 /(c*d+e)^2*ln(c*x-1)+1/2/(c*d-e)^2*ln(c*x+1)))
Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (122) = 244\).
Time = 0.14 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.49 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx=-\frac {2 \, a c^{4} d^{4} - 2 \, b c^{3} d^{3} e - 4 \, a c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + 2 \, a e^{4} - 2 \, {\left (b c^{3} d^{2} e^{2} - b c e^{4}\right )} x - {\left (b c^{4} d^{4} + 2 \, b c^{3} d^{3} e + b c^{2} d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{3} d e^{3} + b c^{2} e^{4}\right )} x^{2} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{3} d^{2} e^{2} + b c^{2} d e^{3}\right )} x\right )} \log \left (c x + 1\right ) + {\left (b c^{4} d^{4} - 2 \, b c^{3} d^{3} e + b c^{2} d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} - 2 \, b c^{3} d e^{3} + b c^{2} e^{4}\right )} x^{2} + 2 \, {\left (b c^{4} d^{3} e - 2 \, b c^{3} d^{2} e^{2} + b c^{2} d e^{3}\right )} x\right )} \log \left (c x - 1\right ) + 4 \, {\left (b c^{3} d e^{3} x^{2} + 2 \, b c^{3} d^{2} e^{2} x + b c^{3} d^{3} e\right )} \log \left (e x + d\right ) + {\left (b c^{4} d^{4} - 2 \, b c^{2} d^{2} e^{2} + b e^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{4 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{4} e^{3} + d^{2} e^{5} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{2} e^{5} + e^{7}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{3} e^{4} + d e^{6}\right )} x\right )}} \] Input:
integrate((a+b*arctanh(c*x))/(e*x+d)^3,x, algorithm="fricas")
Output:
-1/4*(2*a*c^4*d^4 - 2*b*c^3*d^3*e - 4*a*c^2*d^2*e^2 + 2*b*c*d*e^3 + 2*a*e^ 4 - 2*(b*c^3*d^2*e^2 - b*c*e^4)*x - (b*c^4*d^4 + 2*b*c^3*d^3*e + b*c^2*d^2 *e^2 + (b*c^4*d^2*e^2 + 2*b*c^3*d*e^3 + b*c^2*e^4)*x^2 + 2*(b*c^4*d^3*e + 2*b*c^3*d^2*e^2 + b*c^2*d*e^3)*x)*log(c*x + 1) + (b*c^4*d^4 - 2*b*c^3*d^3* e + b*c^2*d^2*e^2 + (b*c^4*d^2*e^2 - 2*b*c^3*d*e^3 + b*c^2*e^4)*x^2 + 2*(b *c^4*d^3*e - 2*b*c^3*d^2*e^2 + b*c^2*d*e^3)*x)*log(c*x - 1) + 4*(b*c^3*d*e ^3*x^2 + 2*b*c^3*d^2*e^2*x + b*c^3*d^3*e)*log(e*x + d) + (b*c^4*d^4 - 2*b* c^2*d^2*e^2 + b*e^4)*log(-(c*x + 1)/(c*x - 1)))/(c^4*d^6*e - 2*c^2*d^4*e^3 + d^2*e^5 + (c^4*d^4*e^3 - 2*c^2*d^2*e^5 + e^7)*x^2 + 2*(c^4*d^5*e^2 - 2* c^2*d^3*e^4 + d*e^6)*x)
Leaf count of result is larger than twice the leaf count of optimal. 3216 vs. \(2 (109) = 218\).
Time = 3.31 (sec) , antiderivative size = 3216, normalized size of antiderivative = 24.74 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:
integrate((a+b*atanh(c*x))/(e*x+d)**3,x)
Output:
Piecewise((a*x/d**3, Eq(c, 0) & Eq(e, 0)), (-a/(2*d**2*e + 4*d*e**2*x + 2* e**3*x**2), Eq(c, 0)), ((a*x + b*x*atanh(c*x) + b*log(x - 1/c)/c + b*atanh (c*x)/c)/d**3, Eq(e, 0)), (-4*a*d**2/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e **3*x**2) + 3*b*d**2*atanh(e*x/d)/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3 *x**2) + 2*b*d**2/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - 2*b*d*e *x*atanh(e*x/d)/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) + b*d*e*x/( 8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - b*e**2*x**2*atanh(e*x/d)/( 8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2), Eq(c, -e/d)), (-4*a*d**2/(8 *d**4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - 3*b*d**2*atanh(e*x/d)/(8*d* *4*e + 16*d**3*e**2*x + 8*d**2*e**3*x**2) - 2*b*d**2/(8*d**4*e + 16*d**3*e **2*x + 8*d**2*e**3*x**2) + 2*b*d*e*x*atanh(e*x/d)/(8*d**4*e + 16*d**3*e** 2*x + 8*d**2*e**3*x**2) - b*d*e*x/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3 *x**2) + b*e**2*x**2*atanh(e*x/d)/(8*d**4*e + 16*d**3*e**2*x + 8*d**2*e**3 *x**2), Eq(c, e/d)), (-a*c**4*d**4/(2*c**4*d**6*e + 4*c**4*d**5*e**2*x + 2 *c**4*d**4*e**3*x**2 - 4*c**2*d**4*e**3 - 8*c**2*d**3*e**4*x - 4*c**2*d**2 *e**5*x**2 + 2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*a*c**2*d**2*e**2/ (2*c**4*d**6*e + 4*c**4*d**5*e**2*x + 2*c**4*d**4*e**3*x**2 - 4*c**2*d**4* e**3 - 8*c**2*d**3*e**4*x - 4*c**2*d**2*e**5*x**2 + 2*d**2*e**5 + 4*d*e**6 *x + 2*e**7*x**2) - a*e**4/(2*c**4*d**6*e + 4*c**4*d**5*e**2*x + 2*c**4*d* *4*e**3*x**2 - 4*c**2*d**4*e**3 - 8*c**2*d**3*e**4*x - 4*c**2*d**2*e**5...
Time = 0.03 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.46 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx=-\frac {1}{4} \, {\left ({\left (\frac {4 \, c^{2} d \log \left (e x + d\right )}{c^{4} d^{4} - 2 \, c^{2} d^{2} e^{2} + e^{4}} - \frac {c \log \left (c x + 1\right )}{c^{2} d^{2} e - 2 \, c d e^{2} + e^{3}} + \frac {c \log \left (c x - 1\right )}{c^{2} d^{2} e + 2 \, c d e^{2} + e^{3}} - \frac {2}{c^{2} d^{3} - d e^{2} + {\left (c^{2} d^{2} e - e^{3}\right )} x}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e}\right )} b - \frac {a}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \] Input:
integrate((a+b*arctanh(c*x))/(e*x+d)^3,x, algorithm="maxima")
Output:
-1/4*((4*c^2*d*log(e*x + d)/(c^4*d^4 - 2*c^2*d^2*e^2 + e^4) - c*log(c*x + 1)/(c^2*d^2*e - 2*c*d*e^2 + e^3) + c*log(c*x - 1)/(c^2*d^2*e + 2*c*d*e^2 + e^3) - 2/(c^2*d^3 - d*e^2 + (c^2*d^2*e - e^3)*x))*c + 2*arctanh(c*x)/(e^3 *x^2 + 2*d*e^2*x + d^2*e))*b - 1/2*a/(e^3*x^2 + 2*d*e^2*x + d^2*e)
Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (122) = 244\).
Time = 0.13 (sec) , antiderivative size = 809, normalized size of antiderivative = 6.22 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
integrate((a+b*arctanh(c*x))/(e*x+d)^3,x, algorithm="giac")
Output:
-(b*c^2*d*log(-(c*x + 1)*c*d/(c*x - 1) + c*d - (c*x + 1)*e/(c*x - 1) - e)/ (c^4*d^4 - 2*c^2*d^2*e^2 + e^4) - b*c^2*d*log(-(c*x + 1)/(c*x - 1))/(c^4*d ^4 - 2*c^2*d^2*e^2 + e^4) - ((c*x + 1)*b*c^2*d/(c*x - 1) - b*c^2*d + (c*x + 1)*b*c*e/(c*x - 1))*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^2*c^4*d^4/(c*x - 1)^2 - 2*(c*x + 1)*c^4*d^4/(c*x - 1) + c^4*d^4 + 4*(c*x + 1)^2*c^3*d^3*e /(c*x - 1)^2 - 4*(c*x + 1)*c^3*d^3*e/(c*x - 1) + 6*(c*x + 1)^2*c^2*d^2*e^2 /(c*x - 1)^2 - 2*c^2*d^2*e^2 + 4*(c*x + 1)^2*c*d*e^3/(c*x - 1)^2 + 4*(c*x + 1)*c*d*e^3/(c*x - 1) + (c*x + 1)^2*e^4/(c*x - 1)^2 + 2*(c*x + 1)*e^4/(c* x - 1) + e^4) - (2*(c*x + 1)*a*c^3*d^2/(c*x - 1) - 2*a*c^3*d^2 + 2*a*c^2*d *e - (c*x + 1)*b*c^2*d*e/(c*x - 1) + b*c^2*d*e - 2*(c*x + 1)*a*c*e^2/(c*x - 1) - (c*x + 1)*b*c*e^2/(c*x - 1) - b*c*e^2)/((c*x + 1)^2*c^5*d^5/(c*x - 1)^2 - 2*(c*x + 1)*c^5*d^5/(c*x - 1) + c^5*d^5 + 3*(c*x + 1)^2*c^4*d^4*e/( c*x - 1)^2 - 2*(c*x + 1)*c^4*d^4*e/(c*x - 1) - c^4*d^4*e + 2*(c*x + 1)^2*c ^3*d^3*e^2/(c*x - 1)^2 + 4*(c*x + 1)*c^3*d^3*e^2/(c*x - 1) - 2*c^3*d^3*e^2 - 2*(c*x + 1)^2*c^2*d^2*e^3/(c*x - 1)^2 + 4*(c*x + 1)*c^2*d^2*e^3/(c*x - 1) + 2*c^2*d^2*e^3 - 3*(c*x + 1)^2*c*d*e^4/(c*x - 1)^2 - 2*(c*x + 1)*c*d*e ^4/(c*x - 1) + c*d*e^4 - (c*x + 1)^2*e^5/(c*x - 1)^2 - 2*(c*x + 1)*e^5/(c* x - 1) - e^5))*c
Time = 6.71 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.28 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx=\frac {b\,c^3\,d\,\ln \left (c^2\,x^2-1\right )}{2\,\left (c^4\,d^4-2\,c^2\,d^2\,e^2+e^4\right )}-\frac {\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )\,\left (b\,c^5\,d^2+b\,c^3\,e^2\right )}{2\,e^5\,\sqrt {-c^2}-c^2\,\left (2\,d^4\,e\,{\left (-c^2\right )}^{3/2}+4\,d^2\,e^3\,\sqrt {-c^2}\right )}-\frac {b\,c^3\,d\,\ln \left (d+e\,x\right )}{c^4\,d^4-2\,c^2\,d^2\,e^2+e^4}-\frac {\frac {b\,\mathrm {atanh}\left (c\,x\right )}{2\,e}-\frac {x\,\left (-a\,c^2\,d^2+\frac {b\,c\,d\,e}{2}+a\,e^2\right )}{d\,\left (e^2-c^2\,d^2\right )}-\frac {x^2\,\left (-\frac {a\,c^2\,d^2\,e}{2}+\frac {b\,c\,d\,e^2}{2}+\frac {a\,e^3}{2}\right )}{d^2\,\left (e^2-c^2\,d^2\right )}+\frac {x^4\,\left (-\frac {a\,c^4\,d^2\,e}{2}+\frac {b\,c^3\,d\,e^2}{2}+\frac {a\,c^2\,e^3}{2}\right )}{d^2\,\left (e^2-c^2\,d^2\right )}+\frac {x^3\,\left (-a\,c^4\,d^2+\frac {b\,c^3\,d\,e}{2}+a\,c^2\,e^2\right )}{d\,\left (e^2-c^2\,d^2\right )}-\frac {b\,c^2\,x^2\,\mathrm {atanh}\left (c\,x\right )}{2\,e}}{-c^2\,d^2\,x^2-2\,c^2\,d\,e\,x^3-c^2\,e^2\,x^4+d^2+2\,d\,e\,x+e^2\,x^2} \] Input:
int((a + b*atanh(c*x))/(d + e*x)^3,x)
Output:
(b*c^3*d*log(c^2*x^2 - 1))/(2*(e^4 + c^4*d^4 - 2*c^2*d^2*e^2)) - (atan((c^ 2*x)/(-c^2)^(1/2))*(b*c^5*d^2 + b*c^3*e^2))/(2*e^5*(-c^2)^(1/2) - c^2*(2*d ^4*e*(-c^2)^(3/2) + 4*d^2*e^3*(-c^2)^(1/2))) - (b*c^3*d*log(d + e*x))/(e^4 + c^4*d^4 - 2*c^2*d^2*e^2) - ((b*atanh(c*x))/(2*e) - (x*(a*e^2 - a*c^2*d^ 2 + (b*c*d*e)/2))/(d*(e^2 - c^2*d^2)) - (x^2*((a*e^3)/2 + (b*c*d*e^2)/2 - (a*c^2*d^2*e)/2))/(d^2*(e^2 - c^2*d^2)) + (x^4*((a*c^2*e^3)/2 - (a*c^4*d^2 *e)/2 + (b*c^3*d*e^2)/2))/(d^2*(e^2 - c^2*d^2)) + (x^3*(a*c^2*e^2 - a*c^4* d^2 + (b*c^3*d*e)/2))/(d*(e^2 - c^2*d^2)) - (b*c^2*x^2*atanh(c*x))/(2*e))/ (d^2 + e^2*x^2 + 2*d*e*x - c^2*d^2*x^2 - c^2*e^2*x^4 - 2*c^2*d*e*x^3)
Time = 0.19 (sec) , antiderivative size = 693, normalized size of antiderivative = 5.33 \[ \int \frac {a+b \text {arctanh}(c x)}{(d+e x)^3} \, dx=\frac {-2 a \,c^{4} d^{6}-2 a \,d^{2} e^{4}-b \,c^{3} d^{3} e^{3} x^{2}+b c d \,e^{5} x^{2}+4 \mathit {atanh} \left (c x \right ) b \,c^{4} d^{5} e x +2 \mathit {atanh} \left (c x \right ) b \,c^{4} d^{4} e^{2} x^{2}+\mathrm {log}\left (c^{2} x -c \right ) b \,d^{2} e^{4}+\mathrm {log}\left (c^{2} x -c \right ) b \,e^{6} x^{2}-\mathrm {log}\left (c^{2} x +c \right ) b \,d^{2} e^{4}-\mathrm {log}\left (c^{2} x +c \right ) b \,e^{6} x^{2}+b \,c^{3} d^{5} e -b c \,d^{3} e^{3}+2 \mathit {atanh} \left (c x \right ) b \,e^{6} x^{2}+4 a \,c^{2} d^{4} e^{2}+4 \mathit {atanh} \left (c x \right ) b d \,e^{5} x +2 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{3} d^{5} e -3 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{2} d^{4} e^{2}+2 \,\mathrm {log}\left (c^{2} x -c \right ) b d \,e^{5} x +2 \,\mathrm {log}\left (c^{2} x +c \right ) b \,c^{3} d^{5} e +3 \,\mathrm {log}\left (c^{2} x +c \right ) b \,c^{2} d^{4} e^{2}-2 \,\mathrm {log}\left (c^{2} x +c \right ) b d \,e^{5} x -4 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{5} e -8 \mathit {atanh} \left (c x \right ) b \,c^{2} d^{3} e^{3} x -4 \mathit {atanh} \left (c x \right ) b \,c^{2} d^{2} e^{4} x^{2}+4 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{3} d^{4} e^{2} x +2 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{3} d^{3} e^{3} x^{2}-6 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{2} d^{3} e^{3} x -3 \,\mathrm {log}\left (c^{2} x -c \right ) b \,c^{2} d^{2} e^{4} x^{2}+4 \,\mathrm {log}\left (c^{2} x +c \right ) b \,c^{3} d^{4} e^{2} x +2 \,\mathrm {log}\left (c^{2} x +c \right ) b \,c^{3} d^{3} e^{3} x^{2}+6 \,\mathrm {log}\left (c^{2} x +c \right ) b \,c^{2} d^{3} e^{3} x +3 \,\mathrm {log}\left (c^{2} x +c \right ) b \,c^{2} d^{2} e^{4} x^{2}-8 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{4} e^{2} x -4 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{3} e^{3} x^{2}}{4 d^{2} e \left (c^{4} d^{4} e^{2} x^{2}+2 c^{4} d^{5} e x +c^{4} d^{6}-2 c^{2} d^{2} e^{4} x^{2}-4 c^{2} d^{3} e^{3} x -2 c^{2} d^{4} e^{2}+e^{6} x^{2}+2 d \,e^{5} x +d^{2} e^{4}\right )} \] Input:
int((a+b*atanh(c*x))/(e*x+d)^3,x)
Output:
(4*atanh(c*x)*b*c**4*d**5*e*x + 2*atanh(c*x)*b*c**4*d**4*e**2*x**2 - 8*ata nh(c*x)*b*c**2*d**3*e**3*x - 4*atanh(c*x)*b*c**2*d**2*e**4*x**2 + 4*atanh( c*x)*b*d*e**5*x + 2*atanh(c*x)*b*e**6*x**2 + 2*log(c**2*x - c)*b*c**3*d**5 *e + 4*log(c**2*x - c)*b*c**3*d**4*e**2*x + 2*log(c**2*x - c)*b*c**3*d**3* e**3*x**2 - 3*log(c**2*x - c)*b*c**2*d**4*e**2 - 6*log(c**2*x - c)*b*c**2* d**3*e**3*x - 3*log(c**2*x - c)*b*c**2*d**2*e**4*x**2 + log(c**2*x - c)*b* d**2*e**4 + 2*log(c**2*x - c)*b*d*e**5*x + log(c**2*x - c)*b*e**6*x**2 + 2 *log(c**2*x + c)*b*c**3*d**5*e + 4*log(c**2*x + c)*b*c**3*d**4*e**2*x + 2* log(c**2*x + c)*b*c**3*d**3*e**3*x**2 + 3*log(c**2*x + c)*b*c**2*d**4*e**2 + 6*log(c**2*x + c)*b*c**2*d**3*e**3*x + 3*log(c**2*x + c)*b*c**2*d**2*e* *4*x**2 - log(c**2*x + c)*b*d**2*e**4 - 2*log(c**2*x + c)*b*d*e**5*x - log (c**2*x + c)*b*e**6*x**2 - 4*log(d + e*x)*b*c**3*d**5*e - 8*log(d + e*x)*b *c**3*d**4*e**2*x - 4*log(d + e*x)*b*c**3*d**3*e**3*x**2 - 2*a*c**4*d**6 + 4*a*c**2*d**4*e**2 - 2*a*d**2*e**4 + b*c**3*d**5*e - b*c**3*d**3*e**3*x** 2 - b*c*d**3*e**3 + b*c*d*e**5*x**2)/(4*d**2*e*(c**4*d**6 + 2*c**4*d**5*e* x + c**4*d**4*e**2*x**2 - 2*c**2*d**4*e**2 - 4*c**2*d**3*e**3*x - 2*c**2*d **2*e**4*x**2 + d**2*e**4 + 2*d*e**5*x + e**6*x**2))