Integrand size = 20, antiderivative size = 227 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=-\frac {3 a x}{c^4 d^3}+\frac {b x}{2 c^4 d^3}-\frac {b}{8 c^5 d^3 (1+c x)^2}+\frac {15 b}{8 c^5 d^3 (1+c x)}-\frac {19 b \text {arctanh}(c x)}{8 c^5 d^3}-\frac {3 b x \text {arctanh}(c x)}{c^4 d^3}+\frac {x^2 (a+b \text {arctanh}(c x))}{2 c^3 d^3}-\frac {a+b \text {arctanh}(c x)}{2 c^5 d^3 (1+c x)^2}+\frac {4 (a+b \text {arctanh}(c x))}{c^5 d^3 (1+c x)}-\frac {6 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{c^5 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{c^5 d^3} \]
-3*a*x/c^4/d^3+1/2*b*x/c^4/d^3-1/8*b/c^5/d^3/(c*x+1)^2+15/8*b/c^5/d^3/(c*x +1)-19/8*b*arctanh(c*x)/c^5/d^3-3*b*x*arctanh(c*x)/c^4/d^3+1/2*x^2*(a+b*ar ctanh(c*x))/c^3/d^3+1/2*(-a-b*arctanh(c*x))/c^5/d^3/(c*x+1)^2+4*(a+b*arcta nh(c*x))/c^5/d^3/(c*x+1)-6*(a+b*arctanh(c*x))*ln(2/(c*x+1))/c^5/d^3-3/2*b* ln(-c^2*x^2+1)/c^5/d^3+3*b*polylog(2,1-2/(c*x+1))/c^5/d^3
Time = 0.62 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\frac {-96 a c x+16 a c^2 x^2-\frac {16 a}{(1+c x)^2}+\frac {128 a}{1+c x}+192 a \log (1+c x)+b \left (16 c x+28 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-48 \log \left (1-c^2 x^2\right )+96 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )-28 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))+4 \text {arctanh}(c x) \left (-4-24 c x+4 c^2 x^2+14 \cosh (2 \text {arctanh}(c x))-\cosh (4 \text {arctanh}(c x))-48 \log \left (1+e^{-2 \text {arctanh}(c x)}\right )-14 \sinh (2 \text {arctanh}(c x))+\sinh (4 \text {arctanh}(c x))\right )\right )}{32 c^5 d^3} \]
(-96*a*c*x + 16*a*c^2*x^2 - (16*a)/(1 + c*x)^2 + (128*a)/(1 + c*x) + 192*a *Log[1 + c*x] + b*(16*c*x + 28*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 - c^2*x^2] + 96*PolyLog[2, -E^(-2*ArcTanh[c*x])] - 28*Sinh[2*A rcTanh[c*x]] + Sinh[4*ArcTanh[c*x]] + 4*ArcTanh[c*x]*(-4 - 24*c*x + 4*c^2* x^2 + 14*Cosh[2*ArcTanh[c*x]] - Cosh[4*ArcTanh[c*x]] - 48*Log[1 + E^(-2*Ar cTanh[c*x])] - 14*Sinh[2*ArcTanh[c*x]] + Sinh[4*ArcTanh[c*x]])))/(32*c^5*d ^3)
Time = 0.54 (sec) , antiderivative size = 227, 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.
\(\displaystyle \int \frac {x^4 (a+b \text {arctanh}(c x))}{(c d x+d)^3} \, dx\) |
\(\Big \downarrow \) 6502 |
\(\displaystyle \int \left (\frac {6 (a+b \text {arctanh}(c x))}{c^4 d^3 (c x+1)}-\frac {4 (a+b \text {arctanh}(c x))}{c^4 d^3 (c x+1)^2}-\frac {3 (a+b \text {arctanh}(c x))}{c^4 d^3}+\frac {a+b \text {arctanh}(c x)}{c^4 d^3 (c x+1)^3}+\frac {x (a+b \text {arctanh}(c x))}{c^3 d^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 (a+b \text {arctanh}(c x))}{c^5 d^3 (c x+1)}-\frac {a+b \text {arctanh}(c x)}{2 c^5 d^3 (c x+1)^2}-\frac {6 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{c^5 d^3}+\frac {x^2 (a+b \text {arctanh}(c x))}{2 c^3 d^3}-\frac {3 a x}{c^4 d^3}-\frac {19 b \text {arctanh}(c x)}{8 c^5 d^3}-\frac {3 b x \text {arctanh}(c x)}{c^4 d^3}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{c^5 d^3}+\frac {15 b}{8 c^5 d^3 (c x+1)}-\frac {b}{8 c^5 d^3 (c x+1)^2}+\frac {b x}{2 c^4 d^3}-\frac {3 b \log \left (1-c^2 x^2\right )}{2 c^5 d^3}\) |
(-3*a*x)/(c^4*d^3) + (b*x)/(2*c^4*d^3) - b/(8*c^5*d^3*(1 + c*x)^2) + (15*b )/(8*c^5*d^3*(1 + c*x)) - (19*b*ArcTanh[c*x])/(8*c^5*d^3) - (3*b*x*ArcTanh [c*x])/(c^4*d^3) + (x^2*(a + b*ArcTanh[c*x]))/(2*c^3*d^3) - (a + b*ArcTanh [c*x])/(2*c^5*d^3*(1 + c*x)^2) + (4*(a + b*ArcTanh[c*x]))/(c^5*d^3*(1 + c* x)) - (6*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/(c^5*d^3) - (3*b*Log[1 - c ^2*x^2])/(2*c^5*d^3) + (3*b*PolyLog[2, 1 - 2/(1 + c*x)])/(c^5*d^3)
3.1.58.3.1 Defintions of rubi rules used
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 = 1.21 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {c^{2} x^{2}}{2}-3 c x +6 \ln \left (c x +1\right )+\frac {4}{c x +1}-\frac {1}{2 \left (c x +1\right )^{2}}\right )}{d^{3}}+\frac {b \left (\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}-3 c x \,\operatorname {arctanh}\left (c x \right )+6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {c x}{2}+\frac {1}{2}-\frac {5 \ln \left (c x -1\right )}{16}-\frac {1}{8 \left (c x +1\right )^{2}}+\frac {15}{8 \left (c x +1\right )}-\frac {43 \ln \left (c x +1\right )}{16}+3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-\frac {3 \ln \left (c x +1\right )^{2}}{2}\right )}{d^{3}}}{c^{5}}\) | \(197\) |
default | \(\frac {\frac {a \left (\frac {c^{2} x^{2}}{2}-3 c x +6 \ln \left (c x +1\right )+\frac {4}{c x +1}-\frac {1}{2 \left (c x +1\right )^{2}}\right )}{d^{3}}+\frac {b \left (\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}-3 c x \,\operatorname {arctanh}\left (c x \right )+6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {c x}{2}+\frac {1}{2}-\frac {5 \ln \left (c x -1\right )}{16}-\frac {1}{8 \left (c x +1\right )^{2}}+\frac {15}{8 \left (c x +1\right )}-\frac {43 \ln \left (c x +1\right )}{16}+3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-\frac {3 \ln \left (c x +1\right )^{2}}{2}\right )}{d^{3}}}{c^{5}}\) | \(197\) |
parts | \(\frac {a \left (\frac {\frac {1}{2} c \,x^{2}-3 x}{c^{4}}+\frac {6 \ln \left (c x +1\right )}{c^{5}}-\frac {1}{2 c^{5} \left (c x +1\right )^{2}}+\frac {4}{c^{5} \left (c x +1\right )}\right )}{d^{3}}+\frac {b \left (\frac {c^{2} x^{2} \operatorname {arctanh}\left (c x \right )}{2}-3 c x \,\operatorname {arctanh}\left (c x \right )+6 \,\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )+\frac {4 \,\operatorname {arctanh}\left (c x \right )}{c x +1}-\frac {\operatorname {arctanh}\left (c x \right )}{2 \left (c x +1\right )^{2}}+\frac {c x}{2}+\frac {1}{2}-\frac {5 \ln \left (c x -1\right )}{16}-\frac {1}{8 \left (c x +1\right )^{2}}+\frac {15}{8 \left (c x +1\right )}-\frac {43 \ln \left (c x +1\right )}{16}+3 \left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )-3 \operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )-\frac {3 \ln \left (c x +1\right )^{2}}{2}\right )}{d^{3} c^{5}}\) | \(207\) |
risch | \(-\frac {3 a x}{c^{4} d^{3}}+\frac {b x}{2 c^{4} d^{3}}-\frac {b}{8 c^{5} d^{3} \left (c x +1\right )^{2}}+\frac {2 b}{c^{5} d^{3} \left (c x +1\right )}-\frac {b \ln \left (-c x +1\right ) x}{c^{4} d^{3} \left (-c x -1\right )}-\frac {b \ln \left (-c x +1\right ) x^{2}}{16 c^{3} d^{3} \left (-c x -1\right )^{2}}-\frac {b \ln \left (-c x +1\right ) x}{8 c^{4} d^{3} \left (-c x -1\right )^{2}}-\frac {7 b \ln \left (c x +1\right )}{4 c^{5} d^{3}}+\frac {9 b}{8 c^{5} d^{3}}+\frac {5 a}{2 c^{5} d^{3}}-\frac {b \,x^{2} \ln \left (-c x +1\right )}{4 c^{3} d^{3}}+\frac {3 b x \ln \left (-c x +1\right )}{2 c^{4} d^{3}}-\frac {3 b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right )}{c^{5} d^{3}}+\frac {3 b \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{c^{5} d^{3}}+\frac {b \ln \left (-c x +1\right )}{c^{5} d^{3} \left (-c x -1\right )}+\frac {3 b \ln \left (-c x +1\right )}{16 c^{5} d^{3} \left (-c x -1\right )^{2}}+\left (\frac {b \left (\frac {1}{2} c \,x^{2}-3 x \right )}{2 c^{4} d^{3}}+\frac {4 b \,d^{3} x +\frac {7 d^{3} b}{2 c}}{2 c^{4} d^{6} \left (c x +1\right )^{2}}\right ) \ln \left (c x +1\right )+\frac {a \,x^{2}}{2 c^{3} d^{3}}-\frac {4 a}{c^{5} d^{3} \left (-c x -1\right )}-\frac {a}{2 c^{5} d^{3} \left (-c x -1\right )^{2}}-\frac {5 b \ln \left (-c x +1\right )}{4 c^{5} d^{3}}+\frac {3 b \operatorname {dilog}\left (-\frac {c x}{2}+\frac {1}{2}\right )}{c^{5} d^{3}}-\frac {15 b \ln \left (-c x -1\right )}{16 c^{5} d^{3}}+\frac {6 a \ln \left (-c x -1\right )}{c^{5} d^{3}}+\frac {b}{8 c^{5} d^{3} \left (-c x -1\right )}+\frac {3 b \ln \left (c x +1\right )^{2}}{2 c^{5} d^{3}}\) | \(496\) |
1/c^5*(a/d^3*(1/2*c^2*x^2-3*c*x+6*ln(c*x+1)+4/(c*x+1)-1/2/(c*x+1)^2)+b/d^3 *(1/2*c^2*x^2*arctanh(c*x)-3*c*x*arctanh(c*x)+6*arctanh(c*x)*ln(c*x+1)+4/( c*x+1)*arctanh(c*x)-1/2/(c*x+1)^2*arctanh(c*x)+1/2*c*x+1/2-5/16*ln(c*x-1)- 1/8/(c*x+1)^2+15/8/(c*x+1)-43/16*ln(c*x+1)+3*(ln(c*x+1)-ln(1/2*c*x+1/2))*l n(-1/2*c*x+1/2)-3*dilog(1/2*c*x+1/2)-3/2*ln(c*x+1)^2))
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{4}}{{\left (c d x + d\right )}^{3}} \,d x } \]
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\frac {\int \frac {a x^{4}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx + \int \frac {b x^{4} \operatorname {atanh}{\left (c x \right )}}{c^{3} x^{3} + 3 c^{2} x^{2} + 3 c x + 1}\, dx}{d^{3}} \]
(Integral(a*x**4/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x) + Integral(b*x* *4*atanh(c*x)/(c**3*x**3 + 3*c**2*x**2 + 3*c*x + 1), x))/d**3
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{4}}{{\left (c d x + d\right )}^{3}} \,d x } \]
1/32*(c^5*(2*(9*c*x + 8)/(c^12*d^3*x^2 + 2*c^11*d^3*x + c^10*d^3) + 4*(c*x ^2 - 4*x)/(c^9*d^3) + 31*log(c*x + 1)/(c^10*d^3) + log(c*x - 1)/(c^10*d^3) ) + 32*c^5*integrate(1/2*x^5*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2 *c^5*d^3*x - c^4*d^3), x) + 3*c^4*(2*(7*c*x + 6)/(c^11*d^3*x^2 + 2*c^10*d^ 3*x + c^9*d^3) - 8*x/(c^8*d^3) + 17*log(c*x + 1)/(c^9*d^3) - log(c*x - 1)/ (c^9*d^3)) - 32*c^4*integrate(1/2*x^4*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^ 3*x^3 - 2*c^5*d^3*x - c^4*d^3), x) - 15*c^3*(2*(5*c*x + 4)/(c^10*d^3*x^2 + 2*c^9*d^3*x + c^8*d^3) + 7*log(c*x + 1)/(c^8*d^3) + log(c*x - 1)/(c^8*d^3 )) + 192*c^3*integrate(1/2*x^3*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2*c^5*d^3*x - c^4*d^3), x) + 9*c^2*(2*(3*c*x + 2)/(c^9*d^3*x^2 + 2*c^8*d^ 3*x + c^7*d^3) + log(c*x + 1)/(c^7*d^3) - log(c*x - 1)/(c^7*d^3)) + 576*c^ 2*integrate(1/2*x^2*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2*c^5*d^3* x - c^4*d^3), x) + 9*c*(2*x/(c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) - log(c* x + 1)/(c^6*d^3) + log(c*x - 1)/(c^6*d^3)) + 576*c*integrate(1/2*x*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^3*x^3 - 2*c^5*d^3*x - c^4*d^3), x) - 8*(c^4*x ^4 - 4*c^3*x^3 - 11*c^2*x^2 + 2*c*x + 12*(c^2*x^2 + 2*c*x + 1)*log(c*x + 1 ) + 7)*log(-c*x + 1)/(c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) + 14*(c*x + 2)/ (c^7*d^3*x^2 + 2*c^6*d^3*x + c^5*d^3) - 7*log(c*x + 1)/(c^5*d^3) + 7*log(c *x - 1)/(c^5*d^3) + 192*integrate(1/2*log(c*x + 1)/(c^8*d^3*x^4 + 2*c^7*d^ 3*x^3 - 2*c^5*d^3*x - c^4*d^3), x))*b + 1/2*a*((8*c*x + 7)/(c^7*d^3*x^2...
\[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{4}}{{\left (c d x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arctanh}(c x))}{(d+c d x)^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{{\left (d+c\,d\,x\right )}^3} \,d x \]