Integrand size = 27, antiderivative size = 263 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {2 a e x}{5 c^4}-\frac {77 b e x^2}{300 c^3}-\frac {9 b e x^4}{200 c}-\frac {2 b e x \text {arctanh}(c x)}{5 c^4}-\frac {2 e x^3 (a+b \text {arctanh}(c x))}{15 c^2}-\frac {2}{25} e x^5 (a+b \text {arctanh}(c x))+\frac {e (a+b \text {arctanh}(c x))^2}{5 b c^5}-\frac {137 b e \log \left (1-c^2 x^2\right )}{300 c^5}-\frac {b e \log ^2\left (1-c^2 x^2\right )}{20 c^5}+\frac {b x^2 \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^3}+\frac {b x^4 \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 c}+\frac {1}{5} x^5 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right )+\frac {b \log \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 c^5} \] Output:
-2/5*a*e*x/c^4-77/300*b*e*x^2/c^3-9/200*b*e*x^4/c-2/5*b*e*x*arctanh(c*x)/c ^4-2/15*e*x^3*(a+b*arctanh(c*x))/c^2-2/25*e*x^5*(a+b*arctanh(c*x))+1/5*e*( a+b*arctanh(c*x))^2/b/c^5-137/300*b*e*ln(-c^2*x^2+1)/c^5-1/20*b*e*ln(-c^2* x^2+1)^2/c^5+1/10*b*x^2*(d+e*ln(-c^2*x^2+1))/c^3+1/20*b*x^4*(d+e*ln(-c^2*x ^2+1))/c+1/5*x^5*(a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1))+1/10*b*ln(-c^2*x^ 2+1)*(d+e*ln(-c^2*x^2+1))/c^5
Time = 0.09 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.90 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {-240 a c e x+2 b c^2 (30 d-77 e) x^2-80 a c^3 e x^3+3 b c^4 (10 d-9 e) x^4+24 a c^5 (5 d-2 e) x^5-8 b c x \left (-15 c^4 d x^4+2 e \left (15+5 c^2 x^2+3 c^4 x^4\right )\right ) \text {arctanh}(c x)+120 b e \text {arctanh}(c x)^2+2 (30 b d-60 a e-137 b e) \log (1-c x)+2 (30 b d+60 a e-137 b e) \log (1+c x)+30 c^2 e x^2 \left (4 a c^3 x^3+b \left (2+c^2 x^2\right )+4 b c^3 x^3 \text {arctanh}(c x)\right ) \log \left (1-c^2 x^2\right )+30 b e \log ^2\left (1-c^2 x^2\right )}{600 c^5} \] Input:
Integrate[x^4*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]
Output:
(-240*a*c*e*x + 2*b*c^2*(30*d - 77*e)*x^2 - 80*a*c^3*e*x^3 + 3*b*c^4*(10*d - 9*e)*x^4 + 24*a*c^5*(5*d - 2*e)*x^5 - 8*b*c*x*(-15*c^4*d*x^4 + 2*e*(15 + 5*c^2*x^2 + 3*c^4*x^4))*ArcTanh[c*x] + 120*b*e*ArcTanh[c*x]^2 + 2*(30*b* d - 60*a*e - 137*b*e)*Log[1 - c*x] + 2*(30*b*d + 60*a*e - 137*b*e)*Log[1 + c*x] + 30*c^2*e*x^2*(4*a*c^3*x^3 + b*(2 + c^2*x^2) + 4*b*c^3*x^3*ArcTanh[ c*x])*Log[1 - c^2*x^2] + 30*b*e*Log[1 - c^2*x^2]^2)/(600*c^5)
Time = 1.14 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6647, 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.
| \(\displaystyle \int x^4 (a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right ) \, dx\) |
| \(\Big \downarrow \) 6647 |
| \(\displaystyle 2 c^2 e \int \left (\frac {4 a c^3 x^6+4 b c^3 \text {arctanh}(c x) x^6+b c^2 x^5+2 b x^3}{20 c^3 \left (1-c^2 x^2\right )}+\frac {b x \log \left (1-c^2 x^2\right )}{10 c^5 \left (1-c^2 x^2\right )}\right )dx+\frac {1}{5} x^5 (a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )+\frac {b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}+\frac {b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}+\frac {b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \text {arctanh}(c x)) \left (e \log \left (1-c^2 x^2\right )+d\right )+2 c^2 e \left (\frac {a \text {arctanh}(c x)}{5 c^7}-\frac {a x}{5 c^6}-\frac {a x^3}{15 c^4}-\frac {a x^5}{25 c^2}+\frac {b \text {arctanh}(c x)^2}{10 c^7}-\frac {b x \text {arctanh}(c x)}{5 c^6}-\frac {b x^3 \text {arctanh}(c x)}{15 c^4}-\frac {b x^5 \text {arctanh}(c x)}{25 c^2}-\frac {77 b x^2}{600 c^5}-\frac {9 b x^4}{400 c^3}-\frac {b \log ^2\left (1-c^2 x^2\right )}{40 c^7}-\frac {137 b \log \left (1-c^2 x^2\right )}{600 c^7}\right )+\frac {b x^4 \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 c}+\frac {b \log \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^5}+\frac {b x^2 \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 c^3}\) |
Input:
Int[x^4*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]),x]
Output:
(b*x^2*(d + e*Log[1 - c^2*x^2]))/(10*c^3) + (b*x^4*(d + e*Log[1 - c^2*x^2] ))/(20*c) + (x^5*(a + b*ArcTanh[c*x])*(d + e*Log[1 - c^2*x^2]))/5 + (b*Log [1 - c^2*x^2]*(d + e*Log[1 - c^2*x^2]))/(10*c^5) + 2*c^2*e*(-1/5*(a*x)/c^6 - (77*b*x^2)/(600*c^5) - (a*x^3)/(15*c^4) - (9*b*x^4)/(400*c^3) - (a*x^5) /(25*c^2) + (a*ArcTanh[c*x])/(5*c^7) - (b*x*ArcTanh[c*x])/(5*c^6) - (b*x^3 *ArcTanh[c*x])/(15*c^4) - (b*x^5*ArcTanh[c*x])/(25*c^2) + (b*ArcTanh[c*x]^ 2)/(10*c^7) - (137*b*Log[1 - c^2*x^2])/(600*c^7) - (b*Log[1 - c^2*x^2]^2)/ (40*c^7))
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]* (e_.))*(x_)^(m_.), x_Symbol] :> With[{u = IntHide[x^m*(a + b*ArcTanh[c*x]), x]}, Simp[(d + e*Log[f + g*x^2]) u, x] - Simp[2*e*g Int[ExpandIntegran d[x*(u/(f + g*x^2)), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Inte gerQ[m] && NeQ[m, -1]
Time = 6.27 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.13
| method | result | size |
| parallelrisch | \(\frac {-154 b e +60 d b +60 b \,c^{2} d \,x^{2}+60 b d \ln \left (-c^{2} x^{2}+1\right )+120 c^{5} d \,x^{5} a +30 d \,x^{4} c^{4} b +120 b \,c^{5} d \,\operatorname {arctanh}\left (c x \right ) x^{5}-240 a c e x -80 a \,c^{3} e \,x^{3}-27 b \,c^{4} e \,x^{4}-154 b \,c^{2} e \,x^{2}-48 a \,c^{5} e \,x^{5}+120 b e \ln \left (-c^{2} x^{2}+1\right ) \operatorname {arctanh}\left (c x \right ) x^{5} c^{5}-274 \ln \left (-c^{2} x^{2}+1\right ) b e +240 \,\operatorname {arctanh}\left (c x \right ) a e +120 e b \operatorname {arctanh}\left (c x \right )^{2}+30 e b \ln \left (-c^{2} x^{2}+1\right )^{2}-80 e b \,\operatorname {arctanh}\left (c x \right ) x^{3} c^{3}-240 e b \,\operatorname {arctanh}\left (c x \right ) x c +30 b e \ln \left (-c^{2} x^{2}+1\right ) x^{4} c^{4}+120 a e \ln \left (-c^{2} x^{2}+1\right ) x^{5} c^{5}-48 x^{5} \operatorname {arctanh}\left (c x \right ) b \,c^{5} e +60 x^{2} \ln \left (-c^{2} x^{2}+1\right ) b \,c^{2} e}{600 c^{5}}\) | \(296\) |
| risch | \(\text {Expression too large to display}\) | \(4341\) |
| default | \(\text {Expression too large to display}\) | \(4602\) |
| parts | \(\text {Expression too large to display}\) | \(4602\) |
Input:
int(x^4*(a+b*arctanh(c*x))*(d+e*ln(-c^2*x^2+1)),x,method=_RETURNVERBOSE)
Output:
1/600*(-154*b*e+60*d*b+60*b*c^2*d*x^2+60*b*d*ln(-c^2*x^2+1)+120*c^5*d*x^5* a+30*d*x^4*c^4*b+120*b*c^5*d*arctanh(c*x)*x^5-240*a*c*e*x-80*a*c^3*e*x^3-2 7*b*c^4*e*x^4-154*b*c^2*e*x^2-48*a*c^5*e*x^5+120*b*e*ln(-c^2*x^2+1)*arctan h(c*x)*x^5*c^5-274*ln(-c^2*x^2+1)*b*e+240*arctanh(c*x)*a*e+120*e*b*arctanh (c*x)^2+30*e*b*ln(-c^2*x^2+1)^2-80*e*b*arctanh(c*x)*x^3*c^3-240*e*b*arctan h(c*x)*x*c+30*b*e*ln(-c^2*x^2+1)*x^4*c^4+120*a*e*ln(-c^2*x^2+1)*x^5*c^5-48 *x^5*arctanh(c*x)*b*c^5*e+60*x^2*ln(-c^2*x^2+1)*b*c^2*e)/c^5
Time = 0.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.95 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {80 \, a c^{3} e x^{3} - 24 \, {\left (5 \, a c^{5} d - 2 \, a c^{5} e\right )} x^{5} - 3 \, {\left (10 \, b c^{4} d - 9 \, b c^{4} e\right )} x^{4} + 240 \, a c e x - 30 \, b e \log \left (-c^{2} x^{2} + 1\right )^{2} - 30 \, b e \log \left (-\frac {c x + 1}{c x - 1}\right )^{2} - 2 \, {\left (30 \, b c^{2} d - 77 \, b c^{2} e\right )} x^{2} - 2 \, {\left (60 \, a c^{5} e x^{5} + 15 \, b c^{4} e x^{4} + 30 \, b c^{2} e x^{2} + 30 \, b d - 137 \, b e\right )} \log \left (-c^{2} x^{2} + 1\right ) - 4 \, {\left (15 \, b c^{5} e x^{5} \log \left (-c^{2} x^{2} + 1\right ) - 10 \, b c^{3} e x^{3} + 3 \, {\left (5 \, b c^{5} d - 2 \, b c^{5} e\right )} x^{5} - 30 \, b c e x + 30 \, a e\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{600 \, c^{5}} \] Input:
integrate(x^4*(a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="frica s")
Output:
-1/600*(80*a*c^3*e*x^3 - 24*(5*a*c^5*d - 2*a*c^5*e)*x^5 - 3*(10*b*c^4*d - 9*b*c^4*e)*x^4 + 240*a*c*e*x - 30*b*e*log(-c^2*x^2 + 1)^2 - 30*b*e*log(-(c *x + 1)/(c*x - 1))^2 - 2*(30*b*c^2*d - 77*b*c^2*e)*x^2 - 2*(60*a*c^5*e*x^5 + 15*b*c^4*e*x^4 + 30*b*c^2*e*x^2 + 30*b*d - 137*b*e)*log(-c^2*x^2 + 1) - 4*(15*b*c^5*e*x^5*log(-c^2*x^2 + 1) - 10*b*c^3*e*x^3 + 3*(5*b*c^5*d - 2*b *c^5*e)*x^5 - 30*b*c*e*x + 30*a*e)*log(-(c*x + 1)/(c*x - 1)))/c^5
Time = 1.86 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.29 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\begin {cases} \frac {a d x^{5}}{5} + \frac {a e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )}}{5} - \frac {2 a e x^{5}}{25} - \frac {2 a e x^{3}}{15 c^{2}} - \frac {2 a e x}{5 c^{4}} + \frac {2 a e \operatorname {atanh}{\left (c x \right )}}{5 c^{5}} + \frac {b d x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {b e x^{5} \log {\left (- c^{2} x^{2} + 1 \right )} \operatorname {atanh}{\left (c x \right )}}{5} - \frac {2 b e x^{5} \operatorname {atanh}{\left (c x \right )}}{25} + \frac {b d x^{4}}{20 c} + \frac {b e x^{4} \log {\left (- c^{2} x^{2} + 1 \right )}}{20 c} - \frac {9 b e x^{4}}{200 c} - \frac {2 b e x^{3} \operatorname {atanh}{\left (c x \right )}}{15 c^{2}} + \frac {b d x^{2}}{10 c^{3}} + \frac {b e x^{2} \log {\left (- c^{2} x^{2} + 1 \right )}}{10 c^{3}} - \frac {77 b e x^{2}}{300 c^{3}} - \frac {2 b e x \operatorname {atanh}{\left (c x \right )}}{5 c^{4}} + \frac {b d \log {\left (- c^{2} x^{2} + 1 \right )}}{10 c^{5}} + \frac {b e \log {\left (- c^{2} x^{2} + 1 \right )}^{2}}{20 c^{5}} - \frac {137 b e \log {\left (- c^{2} x^{2} + 1 \right )}}{300 c^{5}} + \frac {b e \operatorname {atanh}^{2}{\left (c x \right )}}{5 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(x**4*(a+b*atanh(c*x))*(d+e*ln(-c**2*x**2+1)),x)
Output:
Piecewise((a*d*x**5/5 + a*e*x**5*log(-c**2*x**2 + 1)/5 - 2*a*e*x**5/25 - 2 *a*e*x**3/(15*c**2) - 2*a*e*x/(5*c**4) + 2*a*e*atanh(c*x)/(5*c**5) + b*d*x **5*atanh(c*x)/5 + b*e*x**5*log(-c**2*x**2 + 1)*atanh(c*x)/5 - 2*b*e*x**5* atanh(c*x)/25 + b*d*x**4/(20*c) + b*e*x**4*log(-c**2*x**2 + 1)/(20*c) - 9* b*e*x**4/(200*c) - 2*b*e*x**3*atanh(c*x)/(15*c**2) + b*d*x**2/(10*c**3) + b*e*x**2*log(-c**2*x**2 + 1)/(10*c**3) - 77*b*e*x**2/(300*c**3) - 2*b*e*x* atanh(c*x)/(5*c**4) + b*d*log(-c**2*x**2 + 1)/(10*c**5) + b*e*log(-c**2*x* *2 + 1)**2/(20*c**5) - 137*b*e*log(-c**2*x**2 + 1)/(300*c**5) + b*e*atanh( c*x)**2/(5*c**5), Ne(c, 0)), (a*d*x**5/5, True))
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.21 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {1}{5} \, a d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b e \operatorname {artanh}\left (c x\right ) + \frac {1}{20} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \log \left (-c^{2} x^{2} + 1\right ) - c^{2} {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} a e - \frac {{\left (3 \, {\left (-10 i \, \pi c^{4} + 9 \, c^{4}\right )} x^{4} + 2 \, {\left (-30 i \, \pi c^{2} + 77 \, c^{2}\right )} x^{2} + 2 \, {\left (-30 i \, \pi - 15 \, c^{4} x^{4} - 30 \, c^{2} x^{2} - 60 \, \log \left (c x - 1\right ) + 137\right )} \log \left (c x + 1\right ) + 2 \, {\left (-30 i \, \pi - 15 \, c^{4} x^{4} - 30 \, c^{2} x^{2} + 137\right )} \log \left (c x - 1\right )\right )} b e}{600 \, c^{5}} \] Input:
integrate(x^4*(a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="maxim a")
Output:
1/5*a*d*x^5 + 1/75*(15*x^5*log(-c^2*x^2 + 1) - c^2*(2*(3*c^4*x^5 + 5*c^2*x ^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*b*e*arctanh(c *x) + 1/20*(4*x^5*arctanh(c*x) + c*((c^2*x^4 + 2*x^2)/c^4 + 2*log(c^2*x^2 - 1)/c^6))*b*d + 1/75*(15*x^5*log(-c^2*x^2 + 1) - c^2*(2*(3*c^4*x^5 + 5*c^ 2*x^3 + 15*x)/c^6 - 15*log(c*x + 1)/c^7 + 15*log(c*x - 1)/c^7))*a*e - 1/60 0*(3*(-10*I*pi*c^4 + 9*c^4)*x^4 + 2*(-30*I*pi*c^2 + 77*c^2)*x^2 + 2*(-30*I *pi - 15*c^4*x^4 - 30*c^2*x^2 - 60*log(c*x - 1) + 137)*log(c*x + 1) + 2*(- 30*I*pi - 15*c^4*x^4 - 30*c^2*x^2 + 137)*log(c*x - 1))*b*e/c^5
Time = 0.26 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.19 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=-\frac {1}{10} \, b e x^{5} \log \left (-c x + 1\right )^{2} + \frac {1}{25} \, {\left (5 \, a d - 2 \, a e\right )} x^{5} + \frac {{\left (10 \, b d - 9 \, b e\right )} x^{4}}{200 \, c} - \frac {2 \, a e x^{3}}{15 \, c^{2}} + \frac {1}{10} \, {\left (b e x^{5} + \frac {b e}{c^{5}}\right )} \log \left (c x + 1\right )^{2} + \frac {1}{300} \, {\left (6 \, {\left (5 \, b d + 10 \, a e - 2 \, b e\right )} x^{5} + \frac {15 \, b e x^{4}}{c} - \frac {20 \, b e x^{3}}{c^{2}} + \frac {30 \, b e x^{2}}{c^{3}} - \frac {60 \, b e x}{c^{4}}\right )} \log \left (c x + 1\right ) - \frac {1}{300} \, {\left (6 \, {\left (5 \, b d - 10 \, a e - 2 \, b e\right )} x^{5} - \frac {15 \, b e x^{4}}{c} - \frac {20 \, b e x^{3}}{c^{2}} - \frac {30 \, b e x^{2}}{c^{3}} - \frac {60 \, b e x}{c^{4}} - \frac {60 \, b e \log \left (c x - 1\right )}{c^{5}}\right )} \log \left (-c x + 1\right ) + \frac {{\left (30 \, b d - 77 \, b e\right )} x^{2}}{300 \, c^{3}} - \frac {2 \, a e x}{5 \, c^{4}} - \frac {b e \log \left (c x - 1\right )^{2}}{10 \, c^{5}} + \frac {{\left (30 \, b d + 60 \, a e - 137 \, b e\right )} \log \left (c x + 1\right )}{300 \, c^{5}} + \frac {{\left (30 \, b d - 60 \, a e - 137 \, b e\right )} \log \left (c x - 1\right )}{300 \, c^{5}} \] Input:
integrate(x^4*(a+b*arctanh(c*x))*(d+e*log(-c^2*x^2+1)),x, algorithm="giac" )
Output:
-1/10*b*e*x^5*log(-c*x + 1)^2 + 1/25*(5*a*d - 2*a*e)*x^5 + 1/200*(10*b*d - 9*b*e)*x^4/c - 2/15*a*e*x^3/c^2 + 1/10*(b*e*x^5 + b*e/c^5)*log(c*x + 1)^2 + 1/300*(6*(5*b*d + 10*a*e - 2*b*e)*x^5 + 15*b*e*x^4/c - 20*b*e*x^3/c^2 + 30*b*e*x^2/c^3 - 60*b*e*x/c^4)*log(c*x + 1) - 1/300*(6*(5*b*d - 10*a*e - 2*b*e)*x^5 - 15*b*e*x^4/c - 20*b*e*x^3/c^2 - 30*b*e*x^2/c^3 - 60*b*e*x/c^4 - 60*b*e*log(c*x - 1)/c^5)*log(-c*x + 1) + 1/300*(30*b*d - 77*b*e)*x^2/c^ 3 - 2/5*a*e*x/c^4 - 1/10*b*e*log(c*x - 1)^2/c^5 + 1/300*(30*b*d + 60*a*e - 137*b*e)*log(c*x + 1)/c^5 + 1/300*(30*b*d - 60*a*e - 137*b*e)*log(c*x - 1 )/c^5
Time = 6.73 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.28 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {a\,d\,x^5}{5}-\frac {2\,a\,e\,x^5}{25}+\frac {b\,d\,x^5\,\ln \left (c\,x+1\right )}{10}-\frac {b\,d\,x^5\,\ln \left (1-c\,x\right )}{10}-\frac {b\,e\,x^5\,\ln \left (c\,x+1\right )}{25}+\frac {b\,e\,x^5\,\ln \left (1-c\,x\right )}{25}+\frac {b\,e\,{\ln \left (c\,x+1\right )}^2}{10\,c^5}+\frac {b\,e\,{\ln \left (1-c\,x\right )}^2}{10\,c^5}-\frac {2\,a\,e\,x}{5\,c^4}-\frac {2\,a\,e\,x^3}{15\,c^2}+\frac {b\,d\,x^4}{20\,c}+\frac {b\,d\,x^2}{10\,c^3}-\frac {9\,b\,e\,x^4}{200\,c}-\frac {77\,b\,e\,x^2}{300\,c^3}+\frac {a\,e\,x^5\,\ln \left (1-c^2\,x^2\right )}{5}-\frac {a\,e\,\ln \left (c\,x-1\right )}{5\,c^5}+\frac {a\,e\,\ln \left (c\,x+1\right )}{5\,c^5}+\frac {b\,d\,\ln \left (c\,x-1\right )}{10\,c^5}+\frac {b\,d\,\ln \left (c\,x+1\right )}{10\,c^5}-\frac {137\,b\,e\,\ln \left (c\,x-1\right )}{300\,c^5}-\frac {137\,b\,e\,\ln \left (c\,x+1\right )}{300\,c^5}-\frac {b\,e\,\ln \left (c\,x+1\right )\,\ln \left (-\frac {2\,a\,e-2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,\ln \left (c\,x+1\right )\,\ln \left (-\frac {2\,a\,e+2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,\ln \left (1-c\,x\right )\,\ln \left (-\frac {2\,a\,e-2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,\ln \left (1-c\,x\right )\,\ln \left (-\frac {2\,a\,e+2\,a\,c\,e\,x}{5\,c^4}\right )}{10\,c^5}-\frac {b\,e\,x\,\ln \left (c\,x+1\right )}{5\,c^4}+\frac {b\,e\,x\,\ln \left (1-c\,x\right )}{5\,c^4}+\frac {b\,e\,x^4\,\ln \left (1-c^2\,x^2\right )}{20\,c}+\frac {b\,e\,x^2\,\ln \left (1-c^2\,x^2\right )}{10\,c^3}+\frac {b\,e\,\ln \left (-\frac {2\,a\,e-2\,a\,c\,e\,x}{5\,c^4}\right )\,\ln \left (1-c^2\,x^2\right )}{10\,c^5}+\frac {b\,e\,\ln \left (-\frac {2\,a\,e+2\,a\,c\,e\,x}{5\,c^4}\right )\,\ln \left (1-c^2\,x^2\right )}{10\,c^5}-\frac {b\,e\,x^3\,\ln \left (c\,x+1\right )}{15\,c^2}+\frac {b\,e\,x^3\,\ln \left (1-c\,x\right )}{15\,c^2}+\frac {b\,e\,x^5\,\ln \left (c\,x+1\right )\,\ln \left (1-c^2\,x^2\right )}{10}-\frac {b\,e\,x^5\,\ln \left (1-c\,x\right )\,\ln \left (1-c^2\,x^2\right )}{10} \] Input:
int(x^4*(a + b*atanh(c*x))*(d + e*log(1 - c^2*x^2)),x)
Output:
(a*d*x^5)/5 - (2*a*e*x^5)/25 + (b*d*x^5*log(c*x + 1))/10 - (b*d*x^5*log(1 - c*x))/10 - (b*e*x^5*log(c*x + 1))/25 + (b*e*x^5*log(1 - c*x))/25 + (b*e* log(c*x + 1)^2)/(10*c^5) + (b*e*log(1 - c*x)^2)/(10*c^5) - (2*a*e*x)/(5*c^ 4) - (2*a*e*x^3)/(15*c^2) + (b*d*x^4)/(20*c) + (b*d*x^2)/(10*c^3) - (9*b*e *x^4)/(200*c) - (77*b*e*x^2)/(300*c^3) + (a*e*x^5*log(1 - c^2*x^2))/5 - (a *e*log(c*x - 1))/(5*c^5) + (a*e*log(c*x + 1))/(5*c^5) + (b*d*log(c*x - 1)) /(10*c^5) + (b*d*log(c*x + 1))/(10*c^5) - (137*b*e*log(c*x - 1))/(300*c^5) - (137*b*e*log(c*x + 1))/(300*c^5) - (b*e*log(c*x + 1)*log(-(2*a*e - 2*a* c*e*x)/(5*c^4)))/(10*c^5) - (b*e*log(c*x + 1)*log(-(2*a*e + 2*a*c*e*x)/(5* c^4)))/(10*c^5) - (b*e*log(1 - c*x)*log(-(2*a*e - 2*a*c*e*x)/(5*c^4)))/(10 *c^5) - (b*e*log(1 - c*x)*log(-(2*a*e + 2*a*c*e*x)/(5*c^4)))/(10*c^5) - (b *e*x*log(c*x + 1))/(5*c^4) + (b*e*x*log(1 - c*x))/(5*c^4) + (b*e*x^4*log(1 - c^2*x^2))/(20*c) + (b*e*x^2*log(1 - c^2*x^2))/(10*c^3) + (b*e*log(-(2*a *e - 2*a*c*e*x)/(5*c^4))*log(1 - c^2*x^2))/(10*c^5) + (b*e*log(-(2*a*e + 2 *a*c*e*x)/(5*c^4))*log(1 - c^2*x^2))/(10*c^5) - (b*e*x^3*log(c*x + 1))/(15 *c^2) + (b*e*x^3*log(1 - c*x))/(15*c^2) + (b*e*x^5*log(c*x + 1)*log(1 - c^ 2*x^2))/10 - (b*e*x^5*log(1 - c*x)*log(1 - c^2*x^2))/10
Time = 0.17 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.17 \[ \int x^4 (a+b \text {arctanh}(c x)) \left (d+e \log \left (1-c^2 x^2\right )\right ) \, dx=\frac {120 \mathit {atanh} \left (c x \right )^{2} b e +120 \mathit {atanh} \left (c x \right ) \mathrm {log}\left (-c^{2} x^{2}+1\right ) b \,c^{5} e \,x^{5}+120 \mathit {atanh} \left (c x \right ) b \,c^{5} d \,x^{5}-48 \mathit {atanh} \left (c x \right ) b \,c^{5} e \,x^{5}-80 \mathit {atanh} \left (c x \right ) b \,c^{3} e \,x^{3}-240 \mathit {atanh} \left (c x \right ) b c e x +30 \mathrm {log}\left (-c^{2} x^{2}+1\right )^{2} b e +120 \,\mathrm {log}\left (-c^{2} x^{2}+1\right ) a \,c^{5} e \,x^{5}+120 \,\mathrm {log}\left (-c^{2} x^{2}+1\right ) a e +30 \,\mathrm {log}\left (-c^{2} x^{2}+1\right ) b \,c^{4} e \,x^{4}+60 \,\mathrm {log}\left (-c^{2} x^{2}+1\right ) b \,c^{2} e \,x^{2}+60 \,\mathrm {log}\left (-c^{2} x^{2}+1\right ) b d -274 \,\mathrm {log}\left (-c^{2} x^{2}+1\right ) b e -240 \,\mathrm {log}\left (c^{2} x -c \right ) a e +120 a \,c^{5} d \,x^{5}-48 a \,c^{5} e \,x^{5}-80 a \,c^{3} e \,x^{3}-240 a c e x +30 b \,c^{4} d \,x^{4}-27 b \,c^{4} e \,x^{4}+60 b \,c^{2} d \,x^{2}-154 b \,c^{2} e \,x^{2}}{600 c^{5}} \] Input:
int(x^4*(a+b*atanh(c*x))*(d+e*log(-c^2*x^2+1)),x)
Output:
(120*atanh(c*x)**2*b*e + 120*atanh(c*x)*log( - c**2*x**2 + 1)*b*c**5*e*x** 5 + 120*atanh(c*x)*b*c**5*d*x**5 - 48*atanh(c*x)*b*c**5*e*x**5 - 80*atanh( c*x)*b*c**3*e*x**3 - 240*atanh(c*x)*b*c*e*x + 30*log( - c**2*x**2 + 1)**2* b*e + 120*log( - c**2*x**2 + 1)*a*c**5*e*x**5 + 120*log( - c**2*x**2 + 1)* a*e + 30*log( - c**2*x**2 + 1)*b*c**4*e*x**4 + 60*log( - c**2*x**2 + 1)*b* c**2*e*x**2 + 60*log( - c**2*x**2 + 1)*b*d - 274*log( - c**2*x**2 + 1)*b*e - 240*log(c**2*x - c)*a*e + 120*a*c**5*d*x**5 - 48*a*c**5*e*x**5 - 80*a*c **3*e*x**3 - 240*a*c*e*x + 30*b*c**4*d*x**4 - 27*b*c**4*e*x**4 + 60*b*c**2 *d*x**2 - 154*b*c**2*e*x**2)/(600*c**5)