Integrand size = 20, antiderivative size = 224 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\frac {11 b d^4 x}{8 c^3}+\frac {24 b d^4 x^2}{35 c^2}+\frac {11 b d^4 x^3}{24 c}+\frac {12}{35} b d^4 x^4+\frac {9}{40} b c d^4 x^5+\frac {2}{21} b c^2 d^4 x^6+\frac {1}{56} b c^3 d^4 x^7+\frac {1}{4} d^4 x^4 (a+b \text {arctanh}(c x))+\frac {4}{5} c d^4 x^5 (a+b \text {arctanh}(c x))+c^2 d^4 x^6 (a+b \text {arctanh}(c x))+\frac {4}{7} c^3 d^4 x^7 (a+b \text {arctanh}(c x))+\frac {1}{8} c^4 d^4 x^8 (a+b \text {arctanh}(c x))+\frac {769 b d^4 \log (1-c x)}{560 c^4}-\frac {b d^4 \log (1+c x)}{560 c^4} \] Output:
11/8*b*d^4*x/c^3+24/35*b*d^4*x^2/c^2+11/24*b*d^4*x^3/c+12/35*b*d^4*x^4+9/4 0*b*c*d^4*x^5+2/21*b*c^2*d^4*x^6+1/56*b*c^3*d^4*x^7+1/4*d^4*x^4*(a+b*arcta nh(c*x))+4/5*c*d^4*x^5*(a+b*arctanh(c*x))+c^2*d^4*x^6*(a+b*arctanh(c*x))+4 /7*c^3*d^4*x^7*(a+b*arctanh(c*x))+1/8*c^4*d^4*x^8*(a+b*arctanh(c*x))+769/5 60*b*d^4*ln(-c*x+1)/c^4-1/560*b*d^4*ln(c*x+1)/c^4
Time = 0.07 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.79 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\frac {d^4 \left (2310 b c x+1152 b c^2 x^2+770 b c^3 x^3+420 a c^4 x^4+576 b c^4 x^4+1344 a c^5 x^5+378 b c^5 x^5+1680 a c^6 x^6+160 b c^6 x^6+960 a c^7 x^7+30 b c^7 x^7+210 a c^8 x^8+6 b c^4 x^4 \left (70+224 c x+280 c^2 x^2+160 c^3 x^3+35 c^4 x^4\right ) \text {arctanh}(c x)+2307 b \log (1-c x)-3 b \log (1+c x)\right )}{1680 c^4} \] Input:
Integrate[x^3*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]
Output:
(d^4*(2310*b*c*x + 1152*b*c^2*x^2 + 770*b*c^3*x^3 + 420*a*c^4*x^4 + 576*b* c^4*x^4 + 1344*a*c^5*x^5 + 378*b*c^5*x^5 + 1680*a*c^6*x^6 + 160*b*c^6*x^6 + 960*a*c^7*x^7 + 30*b*c^7*x^7 + 210*a*c^8*x^8 + 6*b*c^4*x^4*(70 + 224*c*x + 280*c^2*x^2 + 160*c^3*x^3 + 35*c^4*x^4)*ArcTanh[c*x] + 2307*b*Log[1 - c *x] - 3*b*Log[1 + c*x]))/(1680*c^4)
Time = 0.51 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6498, 27, 2333, 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.
| \(\displaystyle \int x^3 (c d x+d)^4 (a+b \text {arctanh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6498 |
| \(\displaystyle -b c \int \frac {d^4 x^4 \left (35 c^4 x^4+160 c^3 x^3+280 c^2 x^2+224 c x+70\right )}{280 \left (1-c^2 x^2\right )}dx+\frac {1}{8} c^4 d^4 x^8 (a+b \text {arctanh}(c x))+\frac {4}{7} c^3 d^4 x^7 (a+b \text {arctanh}(c x))+c^2 d^4 x^6 (a+b \text {arctanh}(c x))+\frac {4}{5} c d^4 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^4 x^4 (a+b \text {arctanh}(c x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{280} b c d^4 \int \frac {x^4 \left (35 c^4 x^4+160 c^3 x^3+280 c^2 x^2+224 c x+70\right )}{1-c^2 x^2}dx+\frac {1}{8} c^4 d^4 x^8 (a+b \text {arctanh}(c x))+\frac {4}{7} c^3 d^4 x^7 (a+b \text {arctanh}(c x))+c^2 d^4 x^6 (a+b \text {arctanh}(c x))+\frac {4}{5} c d^4 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^4 x^4 (a+b \text {arctanh}(c x))\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle -\frac {1}{280} b c d^4 \int \left (-35 c^2 x^6-160 c x^5-315 x^4-\frac {384 x^3}{c}-\frac {385 x^2}{c^2}-\frac {384 x}{c^3}+\frac {384 c x+385}{c^4 \left (1-c^2 x^2\right )}-\frac {385}{c^4}\right )dx+\frac {1}{8} c^4 d^4 x^8 (a+b \text {arctanh}(c x))+\frac {4}{7} c^3 d^4 x^7 (a+b \text {arctanh}(c x))+c^2 d^4 x^6 (a+b \text {arctanh}(c x))+\frac {4}{5} c d^4 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^4 x^4 (a+b \text {arctanh}(c x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} c^4 d^4 x^8 (a+b \text {arctanh}(c x))+\frac {4}{7} c^3 d^4 x^7 (a+b \text {arctanh}(c x))+c^2 d^4 x^6 (a+b \text {arctanh}(c x))+\frac {4}{5} c d^4 x^5 (a+b \text {arctanh}(c x))+\frac {1}{4} d^4 x^4 (a+b \text {arctanh}(c x))-\frac {1}{280} b c d^4 \left (\frac {385 \text {arctanh}(c x)}{c^5}-\frac {385 x}{c^4}-\frac {192 x^2}{c^3}-5 c^2 x^7-\frac {385 x^3}{3 c^2}-\frac {192 \log \left (1-c^2 x^2\right )}{c^5}-\frac {80 c x^6}{3}-\frac {96 x^4}{c}-63 x^5\right )\) |
Input:
Int[x^3*(d + c*d*x)^4*(a + b*ArcTanh[c*x]),x]
Output:
(d^4*x^4*(a + b*ArcTanh[c*x]))/4 + (4*c*d^4*x^5*(a + b*ArcTanh[c*x]))/5 + c^2*d^4*x^6*(a + b*ArcTanh[c*x]) + (4*c^3*d^4*x^7*(a + b*ArcTanh[c*x]))/7 + (c^4*d^4*x^8*(a + b*ArcTanh[c*x]))/8 - (b*c*d^4*((-385*x)/c^4 - (192*x^2 )/c^3 - (385*x^3)/(3*c^2) - (96*x^4)/c - 63*x^5 - (80*c*x^6)/3 - 5*c^2*x^7 + (385*ArcTanh[c*x])/c^5 - (192*Log[1 - c^2*x^2])/c^5))/280
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
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 = 0.84 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.79
| method | result | size |
| parts | \(d^{4} a \left (\frac {1}{8} c^{4} x^{8}+\frac {4}{7} c^{3} x^{7}+c^{2} x^{6}+\frac {4}{5} c \,x^{5}+\frac {1}{4} x^{4}\right )+\frac {d^{4} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {4 \,\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}+\frac {4 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{7} x^{7}}{56}+\frac {2 c^{6} x^{6}}{21}+\frac {9 c^{5} x^{5}}{40}+\frac {12 c^{4} x^{4}}{35}+\frac {11 x^{3} c^{3}}{24}+\frac {24 c^{2} x^{2}}{35}+\frac {11 c x}{8}+\frac {769 \ln \left (c x -1\right )}{560}-\frac {\ln \left (c x +1\right )}{560}\right )}{c^{4}}\) | \(178\) |
| derivativedivides | \(\frac {d^{4} a \left (\frac {1}{8} c^{8} x^{8}+\frac {4}{7} c^{7} x^{7}+c^{6} x^{6}+\frac {4}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{4} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {4 \,\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}+\frac {4 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{7} x^{7}}{56}+\frac {2 c^{6} x^{6}}{21}+\frac {9 c^{5} x^{5}}{40}+\frac {12 c^{4} x^{4}}{35}+\frac {11 x^{3} c^{3}}{24}+\frac {24 c^{2} x^{2}}{35}+\frac {11 c x}{8}+\frac {769 \ln \left (c x -1\right )}{560}-\frac {\ln \left (c x +1\right )}{560}\right )}{c^{4}}\) | \(184\) |
| default | \(\frac {d^{4} a \left (\frac {1}{8} c^{8} x^{8}+\frac {4}{7} c^{7} x^{7}+c^{6} x^{6}+\frac {4}{5} c^{5} x^{5}+\frac {1}{4} c^{4} x^{4}\right )+d^{4} b \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {4 \,\operatorname {arctanh}\left (c x \right ) c^{7} x^{7}}{7}+\operatorname {arctanh}\left (c x \right ) c^{6} x^{6}+\frac {4 \,\operatorname {arctanh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arctanh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {c^{7} x^{7}}{56}+\frac {2 c^{6} x^{6}}{21}+\frac {9 c^{5} x^{5}}{40}+\frac {12 c^{4} x^{4}}{35}+\frac {11 x^{3} c^{3}}{24}+\frac {24 c^{2} x^{2}}{35}+\frac {11 c x}{8}+\frac {769 \ln \left (c x -1\right )}{560}-\frac {\ln \left (c x +1\right )}{560}\right )}{c^{4}}\) | \(184\) |
| parallelrisch | \(\frac {105 b \,c^{8} d^{4} \operatorname {arctanh}\left (c x \right ) x^{8}+105 c^{8} d^{4} x^{8} a +480 b \,c^{7} d^{4} \operatorname {arctanh}\left (c x \right ) x^{7}+480 a \,c^{7} d^{4} x^{7}+15 c^{7} d^{4} x^{7} b +840 b \,c^{6} d^{4} \operatorname {arctanh}\left (c x \right ) x^{6}+840 a \,c^{6} d^{4} x^{6}+80 b \,c^{6} d^{4} x^{6}+672 b \,c^{5} d^{4} \operatorname {arctanh}\left (c x \right ) x^{5}+672 a \,c^{5} d^{4} x^{5}+189 b \,c^{5} d^{4} x^{5}+210 d^{4} b \,\operatorname {arctanh}\left (c x \right ) x^{4} c^{4}+210 a \,c^{4} d^{4} x^{4}+288 b \,c^{4} d^{4} x^{4}+385 b \,c^{3} d^{4} x^{3}+576 b \,c^{2} d^{4} x^{2}+1155 b c \,d^{4} x +1152 \ln \left (c x -1\right ) b \,d^{4}-3 b \,d^{4} \operatorname {arctanh}\left (c x \right )+576 d^{4} b}{840 c^{4}}\) | \(255\) |
| risch | \(\frac {d^{4} b \,x^{4} \left (35 c^{4} x^{4}+160 x^{3} c^{3}+280 c^{2} x^{2}+224 c x +70\right ) \ln \left (c x +1\right )}{560}-\frac {d^{4} c^{4} b \,x^{8} \ln \left (-c x +1\right )}{16}+\frac {d^{4} c^{4} a \,x^{8}}{8}-\frac {2 d^{4} c^{3} b \,x^{7} \ln \left (-c x +1\right )}{7}+\frac {4 d^{4} c^{3} a \,x^{7}}{7}+\frac {b \,c^{3} d^{4} x^{7}}{56}-\frac {d^{4} c^{2} b \,x^{6} \ln \left (-c x +1\right )}{2}+d^{4} c^{2} a \,x^{6}+\frac {2 b \,c^{2} d^{4} x^{6}}{21}-\frac {2 d^{4} c b \,x^{5} \ln \left (-c x +1\right )}{5}+\frac {4 d^{4} c a \,x^{5}}{5}+\frac {9 b c \,d^{4} x^{5}}{40}-\frac {d^{4} b \,x^{4} \ln \left (-c x +1\right )}{8}+\frac {d^{4} a \,x^{4}}{4}+\frac {12 b \,d^{4} x^{4}}{35}+\frac {11 b \,d^{4} x^{3}}{24 c}+\frac {24 b \,d^{4} x^{2}}{35 c^{2}}+\frac {11 b \,d^{4} x}{8 c^{3}}-\frac {b \,d^{4} \ln \left (c x +1\right )}{560 c^{4}}+\frac {769 b \,d^{4} \ln \left (-c x +1\right )}{560 c^{4}}\) | \(299\) |
Input:
int(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x,method=_RETURNVERBOSE)
Output:
d^4*a*(1/8*c^4*x^8+4/7*c^3*x^7+c^2*x^6+4/5*c*x^5+1/4*x^4)+d^4*b/c^4*(1/8*a rctanh(c*x)*c^8*x^8+4/7*arctanh(c*x)*c^7*x^7+arctanh(c*x)*c^6*x^6+4/5*arct anh(c*x)*c^5*x^5+1/4*arctanh(c*x)*c^4*x^4+1/56*c^7*x^7+2/21*c^6*x^6+9/40*c ^5*x^5+12/35*c^4*x^4+11/24*x^3*c^3+24/35*c^2*x^2+11/8*c*x+769/560*ln(c*x-1 )-1/560*ln(c*x+1))
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.99 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\frac {210 \, a c^{8} d^{4} x^{8} + 30 \, {\left (32 \, a + b\right )} c^{7} d^{4} x^{7} + 80 \, {\left (21 \, a + 2 \, b\right )} c^{6} d^{4} x^{6} + 42 \, {\left (32 \, a + 9 \, b\right )} c^{5} d^{4} x^{5} + 12 \, {\left (35 \, a + 48 \, b\right )} c^{4} d^{4} x^{4} + 770 \, b c^{3} d^{4} x^{3} + 1152 \, b c^{2} d^{4} x^{2} + 2310 \, b c d^{4} x - 3 \, b d^{4} \log \left (c x + 1\right ) + 2307 \, b d^{4} \log \left (c x - 1\right ) + 3 \, {\left (35 \, b c^{8} d^{4} x^{8} + 160 \, b c^{7} d^{4} x^{7} + 280 \, b c^{6} d^{4} x^{6} + 224 \, b c^{5} d^{4} x^{5} + 70 \, b c^{4} d^{4} x^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{1680 \, c^{4}} \] Input:
integrate(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x, algorithm="fricas")
Output:
1/1680*(210*a*c^8*d^4*x^8 + 30*(32*a + b)*c^7*d^4*x^7 + 80*(21*a + 2*b)*c^ 6*d^4*x^6 + 42*(32*a + 9*b)*c^5*d^4*x^5 + 12*(35*a + 48*b)*c^4*d^4*x^4 + 7 70*b*c^3*d^4*x^3 + 1152*b*c^2*d^4*x^2 + 2310*b*c*d^4*x - 3*b*d^4*log(c*x + 1) + 2307*b*d^4*log(c*x - 1) + 3*(35*b*c^8*d^4*x^8 + 160*b*c^7*d^4*x^7 + 280*b*c^6*d^4*x^6 + 224*b*c^5*d^4*x^5 + 70*b*c^4*d^4*x^4)*log(-(c*x + 1)/( c*x - 1)))/c^4
Time = 0.72 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.31 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{4} x^{8}}{8} + \frac {4 a c^{3} d^{4} x^{7}}{7} + a c^{2} d^{4} x^{6} + \frac {4 a c d^{4} x^{5}}{5} + \frac {a d^{4} x^{4}}{4} + \frac {b c^{4} d^{4} x^{8} \operatorname {atanh}{\left (c x \right )}}{8} + \frac {4 b c^{3} d^{4} x^{7} \operatorname {atanh}{\left (c x \right )}}{7} + \frac {b c^{3} d^{4} x^{7}}{56} + b c^{2} d^{4} x^{6} \operatorname {atanh}{\left (c x \right )} + \frac {2 b c^{2} d^{4} x^{6}}{21} + \frac {4 b c d^{4} x^{5} \operatorname {atanh}{\left (c x \right )}}{5} + \frac {9 b c d^{4} x^{5}}{40} + \frac {b d^{4} x^{4} \operatorname {atanh}{\left (c x \right )}}{4} + \frac {12 b d^{4} x^{4}}{35} + \frac {11 b d^{4} x^{3}}{24 c} + \frac {24 b d^{4} x^{2}}{35 c^{2}} + \frac {11 b d^{4} x}{8 c^{3}} + \frac {48 b d^{4} \log {\left (x - \frac {1}{c} \right )}}{35 c^{4}} - \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{280 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{4} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*(c*d*x+d)**4*(a+b*atanh(c*x)),x)
Output:
Piecewise((a*c**4*d**4*x**8/8 + 4*a*c**3*d**4*x**7/7 + a*c**2*d**4*x**6 + 4*a*c*d**4*x**5/5 + a*d**4*x**4/4 + b*c**4*d**4*x**8*atanh(c*x)/8 + 4*b*c* *3*d**4*x**7*atanh(c*x)/7 + b*c**3*d**4*x**7/56 + b*c**2*d**4*x**6*atanh(c *x) + 2*b*c**2*d**4*x**6/21 + 4*b*c*d**4*x**5*atanh(c*x)/5 + 9*b*c*d**4*x* *5/40 + b*d**4*x**4*atanh(c*x)/4 + 12*b*d**4*x**4/35 + 11*b*d**4*x**3/(24* c) + 24*b*d**4*x**2/(35*c**2) + 11*b*d**4*x/(8*c**3) + 48*b*d**4*log(x - 1 /c)/(35*c**4) - b*d**4*atanh(c*x)/(280*c**4), Ne(c, 0)), (a*d**4*x**4/4, T rue))
Time = 0.04 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.67 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\frac {1}{8} \, a c^{4} d^{4} x^{8} + \frac {4}{7} \, a c^{3} d^{4} x^{7} + a c^{2} d^{4} x^{6} + \frac {4}{5} \, a c d^{4} x^{5} + \frac {1}{1680} \, {\left (210 \, x^{8} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (15 \, c^{6} x^{7} + 21 \, c^{4} x^{5} + 35 \, c^{2} x^{3} + 105 \, x\right )}}{c^{8}} - \frac {105 \, \log \left (c x + 1\right )}{c^{9}} + \frac {105 \, \log \left (c x - 1\right )}{c^{9}}\right )}\right )} b c^{4} d^{4} + \frac {1}{21} \, {\left (12 \, x^{7} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, c^{4} x^{6} + 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} + \frac {6 \, \log \left (c^{2} x^{2} - 1\right )}{c^{8}}\right )}\right )} b c^{3} d^{4} + \frac {1}{4} \, a d^{4} x^{4} + \frac {1}{30} \, {\left (30 \, x^{6} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (3 \, c^{4} x^{5} + 5 \, c^{2} x^{3} + 15 \, x\right )}}{c^{6}} - \frac {15 \, \log \left (c x + 1\right )}{c^{7}} + \frac {15 \, \log \left (c x - 1\right )}{c^{7}}\right )}\right )} b c^{2} d^{4} + \frac {1}{5} \, {\left (4 \, x^{5} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {c^{2} x^{4} + 2 \, x^{2}}{c^{4}} + \frac {2 \, \log \left (c^{2} x^{2} - 1\right )}{c^{6}}\right )}\right )} b c d^{4} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )}\right )} b d^{4} \] Input:
integrate(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x, algorithm="maxima")
Output:
1/8*a*c^4*d^4*x^8 + 4/7*a*c^3*d^4*x^7 + a*c^2*d^4*x^6 + 4/5*a*c*d^4*x^5 + 1/1680*(210*x^8*arctanh(c*x) + c*(2*(15*c^6*x^7 + 21*c^4*x^5 + 35*c^2*x^3 + 105*x)/c^8 - 105*log(c*x + 1)/c^9 + 105*log(c*x - 1)/c^9))*b*c^4*d^4 + 1 /21*(12*x^7*arctanh(c*x) + c*((2*c^4*x^6 + 3*c^2*x^4 + 6*x^2)/c^6 + 6*log( c^2*x^2 - 1)/c^8))*b*c^3*d^4 + 1/4*a*d^4*x^4 + 1/30*(30*x^6*arctanh(c*x) + c*(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*c^2*d^4 + 1/5*(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*c*d^4 + 1/24*(6*x^4*arctanh(c*x) + c*(2*(c^ 2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5))*b*d^4
Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (198) = 396\).
Time = 0.13 (sec) , antiderivative size = 817, normalized size of antiderivative = 3.65 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\text {Too large to display} \] Input:
integrate(x^3*(c*d*x+d)^4*(a+b*arctanh(c*x)),x, algorithm="giac")
Output:
-4/105*c*(36*b*d^4*log(-(c*x + 1)/(c*x - 1) + 1)/c^5 - 12*(35*(c*x + 1)^7* b*d^4/(c*x - 1)^7 - 70*(c*x + 1)^6*b*d^4/(c*x - 1)^6 + 175*(c*x + 1)^5*b*d ^4/(c*x - 1)^5 - 210*(c*x + 1)^4*b*d^4/(c*x - 1)^4 + 168*(c*x + 1)^3*b*d^4 /(c*x - 1)^3 - 84*(c*x + 1)^2*b*d^4/(c*x - 1)^2 + 24*(c*x + 1)*b*d^4/(c*x - 1) - 3*b*d^4)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^8*c^5/(c*x - 1)^8 - 8 *(c*x + 1)^7*c^5/(c*x - 1)^7 + 28*(c*x + 1)^6*c^5/(c*x - 1)^6 - 56*(c*x + 1)^5*c^5/(c*x - 1)^5 + 70*(c*x + 1)^4*c^5/(c*x - 1)^4 - 56*(c*x + 1)^3*c^5 /(c*x - 1)^3 + 28*(c*x + 1)^2*c^5/(c*x - 1)^2 - 8*(c*x + 1)*c^5/(c*x - 1) + c^5) - 36*b*d^4*log(-(c*x + 1)/(c*x - 1))/c^5 - (840*(c*x + 1)^7*a*d^4/( c*x - 1)^7 - 1680*(c*x + 1)^6*a*d^4/(c*x - 1)^6 + 4200*(c*x + 1)^5*a*d^4/( c*x - 1)^5 - 5040*(c*x + 1)^4*a*d^4/(c*x - 1)^4 + 4032*(c*x + 1)^3*a*d^4/( c*x - 1)^3 - 2016*(c*x + 1)^2*a*d^4/(c*x - 1)^2 + 576*(c*x + 1)*a*d^4/(c*x - 1) - 72*a*d^4 + 384*(c*x + 1)^7*b*d^4/(c*x - 1)^7 - 1830*(c*x + 1)^6*b* d^4/(c*x - 1)^6 + 4304*(c*x + 1)^5*b*d^4/(c*x - 1)^5 - 6031*(c*x + 1)^4*b* d^4/(c*x - 1)^4 + 5228*(c*x + 1)^3*b*d^4/(c*x - 1)^3 - 2782*(c*x + 1)^2*b* d^4/(c*x - 1)^2 + 836*(c*x + 1)*b*d^4/(c*x - 1) - 109*b*d^4)/((c*x + 1)^8* c^5/(c*x - 1)^8 - 8*(c*x + 1)^7*c^5/(c*x - 1)^7 + 28*(c*x + 1)^6*c^5/(c*x - 1)^6 - 56*(c*x + 1)^5*c^5/(c*x - 1)^5 + 70*(c*x + 1)^4*c^5/(c*x - 1)^4 - 56*(c*x + 1)^3*c^5/(c*x - 1)^3 + 28*(c*x + 1)^2*c^5/(c*x - 1)^2 - 8*(c*x + 1)*c^5/(c*x - 1) + c^5))
Time = 4.34 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.50 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\frac {a\,d^4\,x^4}{4}+\frac {12\,b\,d^4\,x^4}{35}+a\,c^2\,d^4\,x^6+\frac {4\,a\,c^3\,d^4\,x^7}{7}+\frac {a\,c^4\,d^4\,x^8}{8}+\frac {11\,b\,d^4\,x^3}{24\,c}+\frac {24\,b\,d^4\,x^2}{35\,c^2}+\frac {2\,b\,c^2\,d^4\,x^6}{21}+\frac {b\,c^3\,d^4\,x^7}{56}+\frac {769\,b\,d^4\,\ln \left (c\,x-1\right )}{560\,c^4}-\frac {b\,d^4\,\ln \left (c\,x+1\right )}{560\,c^4}+\frac {b\,d^4\,x^4\,\ln \left (c\,x+1\right )}{8}-\frac {b\,d^4\,x^4\,\ln \left (1-c\,x\right )}{8}+\frac {4\,a\,c\,d^4\,x^5}{5}+\frac {11\,b\,d^4\,x}{8\,c^3}+\frac {9\,b\,c\,d^4\,x^5}{40}+\frac {b\,c^2\,d^4\,x^6\,\ln \left (c\,x+1\right )}{2}-\frac {b\,c^2\,d^4\,x^6\,\ln \left (1-c\,x\right )}{2}+\frac {2\,b\,c^3\,d^4\,x^7\,\ln \left (c\,x+1\right )}{7}-\frac {2\,b\,c^3\,d^4\,x^7\,\ln \left (1-c\,x\right )}{7}+\frac {b\,c^4\,d^4\,x^8\,\ln \left (c\,x+1\right )}{16}-\frac {b\,c^4\,d^4\,x^8\,\ln \left (1-c\,x\right )}{16}+\frac {2\,b\,c\,d^4\,x^5\,\ln \left (c\,x+1\right )}{5}-\frac {2\,b\,c\,d^4\,x^5\,\ln \left (1-c\,x\right )}{5} \] Input:
int(x^3*(a + b*atanh(c*x))*(d + c*d*x)^4,x)
Output:
(a*d^4*x^4)/4 + (12*b*d^4*x^4)/35 + a*c^2*d^4*x^6 + (4*a*c^3*d^4*x^7)/7 + (a*c^4*d^4*x^8)/8 + (11*b*d^4*x^3)/(24*c) + (24*b*d^4*x^2)/(35*c^2) + (2*b *c^2*d^4*x^6)/21 + (b*c^3*d^4*x^7)/56 + (769*b*d^4*log(c*x - 1))/(560*c^4) - (b*d^4*log(c*x + 1))/(560*c^4) + (b*d^4*x^4*log(c*x + 1))/8 - (b*d^4*x^ 4*log(1 - c*x))/8 + (4*a*c*d^4*x^5)/5 + (11*b*d^4*x)/(8*c^3) + (9*b*c*d^4* x^5)/40 + (b*c^2*d^4*x^6*log(c*x + 1))/2 - (b*c^2*d^4*x^6*log(1 - c*x))/2 + (2*b*c^3*d^4*x^7*log(c*x + 1))/7 - (2*b*c^3*d^4*x^7*log(1 - c*x))/7 + (b *c^4*d^4*x^8*log(c*x + 1))/16 - (b*c^4*d^4*x^8*log(1 - c*x))/16 + (2*b*c*d ^4*x^5*log(c*x + 1))/5 - (2*b*c*d^4*x^5*log(1 - c*x))/5
Time = 0.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.88 \[ \int x^3 (d+c d x)^4 (a+b \text {arctanh}(c x)) \, dx=\frac {d^{4} \left (105 \mathit {atanh} \left (c x \right ) b \,c^{8} x^{8}+480 \mathit {atanh} \left (c x \right ) b \,c^{7} x^{7}+840 \mathit {atanh} \left (c x \right ) b \,c^{6} x^{6}+672 \mathit {atanh} \left (c x \right ) b \,c^{5} x^{5}+210 \mathit {atanh} \left (c x \right ) b \,c^{4} x^{4}-3 \mathit {atanh} \left (c x \right ) b +1152 \,\mathrm {log}\left (c^{2} x -c \right ) b +105 a \,c^{8} x^{8}+480 a \,c^{7} x^{7}+840 a \,c^{6} x^{6}+672 a \,c^{5} x^{5}+210 a \,c^{4} x^{4}+15 b \,c^{7} x^{7}+80 b \,c^{6} x^{6}+189 b \,c^{5} x^{5}+288 b \,c^{4} x^{4}+385 b \,c^{3} x^{3}+576 b \,c^{2} x^{2}+1155 b c x \right )}{840 c^{4}} \] Input:
int(x^3*(c*d*x+d)^4*(a+b*atanh(c*x)),x)
Output:
(d**4*(105*atanh(c*x)*b*c**8*x**8 + 480*atanh(c*x)*b*c**7*x**7 + 840*atanh (c*x)*b*c**6*x**6 + 672*atanh(c*x)*b*c**5*x**5 + 210*atanh(c*x)*b*c**4*x** 4 - 3*atanh(c*x)*b + 1152*log(c**2*x - c)*b + 105*a*c**8*x**8 + 480*a*c**7 *x**7 + 840*a*c**6*x**6 + 672*a*c**5*x**5 + 210*a*c**4*x**4 + 15*b*c**7*x* *7 + 80*b*c**6*x**6 + 189*b*c**5*x**5 + 288*b*c**4*x**4 + 385*b*c**3*x**3 + 576*b*c**2*x**2 + 1155*b*c*x))/(840*c**4)