Integrand size = 12, antiderivative size = 57 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\frac {d x^2}{6 a}+c x \text {arctanh}(a x)+\frac {1}{3} d x^3 \text {arctanh}(a x)+\frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{6 a^3} \] Output:
1/6*d*x^2/a+c*x*arctanh(a*x)+1/3*d*x^3*arctanh(a*x)+1/6*(3*a^2*c+d)*ln(-a^ 2*x^2+1)/a^3
Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\frac {d x^2}{6 a}+c x \text {arctanh}(a x)+\frac {1}{3} d x^3 \text {arctanh}(a x)+\frac {c \log \left (1-a^2 x^2\right )}{2 a}+\frac {d \log \left (1-a^2 x^2\right )}{6 a^3} \] Input:
Integrate[(c + d*x^2)*ArcTanh[a*x],x]
Output:
(d*x^2)/(6*a) + c*x*ArcTanh[a*x] + (d*x^3*ArcTanh[a*x])/3 + (c*Log[1 - a^2 *x^2])/(2*a) + (d*Log[1 - a^2*x^2])/(6*a^3)
Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6538, 27, 353, 49, 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.
\(\displaystyle \int \text {arctanh}(a x) \left (c+d x^2\right ) \, dx\) |
\(\Big \downarrow \) 6538 |
\(\displaystyle -a \int \frac {x \left (d x^2+3 c\right )}{3 \left (1-a^2 x^2\right )}dx+c x \text {arctanh}(a x)+\frac {1}{3} d x^3 \text {arctanh}(a x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} a \int \frac {x \left (d x^2+3 c\right )}{1-a^2 x^2}dx+c x \text {arctanh}(a x)+\frac {1}{3} d x^3 \text {arctanh}(a x)\) |
\(\Big \downarrow \) 353 |
\(\displaystyle -\frac {1}{6} a \int \frac {d x^2+3 c}{1-a^2 x^2}dx^2+c x \text {arctanh}(a x)+\frac {1}{3} d x^3 \text {arctanh}(a x)\) |
\(\Big \downarrow \) 49 |
\(\displaystyle -\frac {1}{6} a \int \left (\frac {-3 c a^2-d}{a^2 \left (a^2 x^2-1\right )}-\frac {d}{a^2}\right )dx^2+c x \text {arctanh}(a x)+\frac {1}{3} d x^3 \text {arctanh}(a x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{6} a \left (-\frac {d x^2}{a^2}-\frac {\left (3 a^2 c+d\right ) \log \left (1-a^2 x^2\right )}{a^4}\right )+c x \text {arctanh}(a x)+\frac {1}{3} d x^3 \text {arctanh}(a x)\) |
Input:
Int[(c + d*x^2)*ArcTanh[a*x],x]
Output:
c*x*ArcTanh[a*x] + (d*x^3*ArcTanh[a*x])/3 - (a*(-((d*x^2)/a^2) - ((3*a^2*c + d)*Log[1 - a^2*x^2])/a^4))/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Sym bol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcTanh[c*x]) u , x] - Simp[b*c Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; Fre eQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00
method | result | size |
parts | \(\frac {d \,x^{3} \operatorname {arctanh}\left (a x \right )}{3}+c x \,\operatorname {arctanh}\left (a x \right )-\frac {a \left (-\frac {d \,x^{2}}{2 a^{2}}+\frac {\left (-3 a^{2} c -d \right ) \ln \left (a^{2} x^{2}-1\right )}{2 a^{4}}\right )}{3}\) | \(57\) |
derivativedivides | \(\frac {\operatorname {arctanh}\left (a x \right ) a c x +\frac {a \,\operatorname {arctanh}\left (a x \right ) d \,x^{3}}{3}-\frac {-\frac {d \,x^{2} a^{2}}{2}-\frac {\left (3 a^{2} c +d \right ) \ln \left (a x -1\right )}{2}+\frac {\left (-3 a^{2} c -d \right ) \ln \left (a x +1\right )}{2}}{3 a^{2}}}{a}\) | \(74\) |
default | \(\frac {\operatorname {arctanh}\left (a x \right ) a c x +\frac {a \,\operatorname {arctanh}\left (a x \right ) d \,x^{3}}{3}-\frac {-\frac {d \,x^{2} a^{2}}{2}-\frac {\left (3 a^{2} c +d \right ) \ln \left (a x -1\right )}{2}+\frac {\left (-3 a^{2} c -d \right ) \ln \left (a x +1\right )}{2}}{3 a^{2}}}{a}\) | \(74\) |
parallelrisch | \(-\frac {-2 x^{3} \operatorname {arctanh}\left (a x \right ) a^{3} d -6 c \,\operatorname {arctanh}\left (a x \right ) x \,a^{3}-d \,x^{2} a^{2}-6 \ln \left (a x -1\right ) a^{2} c -6 \,\operatorname {arctanh}\left (a x \right ) a^{2} c -2 \ln \left (a x -1\right ) d -2 \,\operatorname {arctanh}\left (a x \right ) d}{6 a^{3}}\) | \(78\) |
risch | \(\left (\frac {1}{6} d \,x^{3}+\frac {1}{2} c x \right ) \ln \left (a x +1\right )-\frac {d \,x^{3} \ln \left (-a x +1\right )}{6}-\frac {c x \ln \left (-a x +1\right )}{2}+\frac {d \,x^{2}}{6 a}+\frac {\ln \left (a^{2} x^{2}-1\right ) c}{2 a}+\frac {\ln \left (a^{2} x^{2}-1\right ) d}{6 a^{3}}\) | \(85\) |
meijerg | \(\frac {d \left (\frac {2 a^{2} x^{2}}{3}-\frac {2 a^{4} x^{4} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (-a^{2} x^{2}+1\right )}{3}\right )}{4 a^{3}}-\frac {c \left (\frac {2 a^{2} x^{2} \left (\ln \left (1-\sqrt {a^{2} x^{2}}\right )-\ln \left (1+\sqrt {a^{2} x^{2}}\right )\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (-a^{2} x^{2}+1\right )\right )}{4 a}\) | \(142\) |
Input:
int((d*x^2+c)*arctanh(a*x),x,method=_RETURNVERBOSE)
Output:
1/3*d*x^3*arctanh(a*x)+c*x*arctanh(a*x)-1/3*a*(-1/2*d/a^2*x^2+1/2*(-3*a^2* c-d)/a^4*ln(a^2*x^2-1))
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\frac {a^{2} d x^{2} + {\left (3 \, a^{2} c + d\right )} \log \left (a^{2} x^{2} - 1\right ) + {\left (a^{3} d x^{3} + 3 \, a^{3} c x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, a^{3}} \] Input:
integrate((d*x^2+c)*arctanh(a*x),x, algorithm="fricas")
Output:
1/6*(a^2*d*x^2 + (3*a^2*c + d)*log(a^2*x^2 - 1) + (a^3*d*x^3 + 3*a^3*c*x)* log(-(a*x + 1)/(a*x - 1)))/a^3
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.28 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\begin {cases} c x \operatorname {atanh}{\left (a x \right )} + \frac {d x^{3} \operatorname {atanh}{\left (a x \right )}}{3} + \frac {c \log {\left (x - \frac {1}{a} \right )}}{a} + \frac {c \operatorname {atanh}{\left (a x \right )}}{a} + \frac {d x^{2}}{6 a} + \frac {d \log {\left (x - \frac {1}{a} \right )}}{3 a^{3}} + \frac {d \operatorname {atanh}{\left (a x \right )}}{3 a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate((d*x**2+c)*atanh(a*x),x)
Output:
Piecewise((c*x*atanh(a*x) + d*x**3*atanh(a*x)/3 + c*log(x - 1/a)/a + c*ata nh(a*x)/a + d*x**2/(6*a) + d*log(x - 1/a)/(3*a**3) + d*atanh(a*x)/(3*a**3) , Ne(a, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.14 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\frac {1}{6} \, a {\left (\frac {d x^{2}}{a^{2}} + \frac {{\left (3 \, a^{2} c + d\right )} \log \left (a x + 1\right )}{a^{4}} + \frac {{\left (3 \, a^{2} c + d\right )} \log \left (a x - 1\right )}{a^{4}}\right )} + \frac {1}{3} \, {\left (d x^{3} + 3 \, c x\right )} \operatorname {artanh}\left (a x\right ) \] Input:
integrate((d*x^2+c)*arctanh(a*x),x, algorithm="maxima")
Output:
1/6*a*(d*x^2/a^2 + (3*a^2*c + d)*log(a*x + 1)/a^4 + (3*a^2*c + d)*log(a*x - 1)/a^4) + 1/3*(d*x^3 + 3*c*x)*arctanh(a*x)
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (51) = 102\).
Time = 0.12 (sec) , antiderivative size = 266, normalized size of antiderivative = 4.67 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\frac {1}{3} \, a {\left (\frac {{\left (3 \, a^{2} c + d\right )} \log \left (\frac {{\left | -a x - 1 \right |}}{{\left | a x - 1 \right |}}\right )}{a^{4}} - \frac {{\left (3 \, a^{2} c + d\right )} \log \left ({\left | -\frac {a x + 1}{a x - 1} + 1 \right |}\right )}{a^{4}} + \frac {2 \, {\left (a x + 1\right )} d}{{\left (a x - 1\right )} a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{2}} + \frac {{\left (\frac {3 \, {\left (a x + 1\right )}^{2} a^{2} c}{{\left (a x - 1\right )}^{2}} - \frac {6 \, {\left (a x + 1\right )} a^{2} c}{a x - 1} + 3 \, a^{2} c + \frac {3 \, {\left (a x + 1\right )}^{2} d}{{\left (a x - 1\right )}^{2}} + d\right )} \log \left (-\frac {\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} + 1}{\frac {a {\left (\frac {a x + 1}{a x - 1} + 1\right )}}{\frac {{\left (a x + 1\right )} a}{a x - 1} - a} - 1}\right )}{a^{4} {\left (\frac {a x + 1}{a x - 1} - 1\right )}^{3}}\right )} \] Input:
integrate((d*x^2+c)*arctanh(a*x),x, algorithm="giac")
Output:
1/3*a*((3*a^2*c + d)*log(abs(-a*x - 1)/abs(a*x - 1))/a^4 - (3*a^2*c + d)*l og(abs(-(a*x + 1)/(a*x - 1) + 1))/a^4 + 2*(a*x + 1)*d/((a*x - 1)*a^4*((a*x + 1)/(a*x - 1) - 1)^2) + (3*(a*x + 1)^2*a^2*c/(a*x - 1)^2 - 6*(a*x + 1)*a ^2*c/(a*x - 1) + 3*a^2*c + 3*(a*x + 1)^2*d/(a*x - 1)^2 + d)*log(-(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) + 1)/(a*((a*x + 1)/(a*x - 1) + 1)/((a*x + 1)*a/(a*x - 1) - a) - 1))/(a^4*((a*x + 1)/(a*x - 1) - 1)^3 ))
Time = 3.54 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\frac {\frac {d\,\ln \left (a^2\,x^2-1\right )}{6}+a^2\,\left (\frac {c\,\ln \left (a^2\,x^2-1\right )}{2}+\frac {d\,x^2}{6}\right )}{a^3}+\frac {d\,x^3\,\mathrm {atanh}\left (a\,x\right )}{3}+c\,x\,\mathrm {atanh}\left (a\,x\right ) \] Input:
int(atanh(a*x)*(c + d*x^2),x)
Output:
((d*log(a^2*x^2 - 1))/6 + a^2*((c*log(a^2*x^2 - 1))/2 + (d*x^2)/6))/a^3 + (d*x^3*atanh(a*x))/3 + c*x*atanh(a*x)
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47 \[ \int \left (c+d x^2\right ) \text {arctanh}(a x) \, dx=\frac {6 \mathit {atanh} \left (a x \right ) a^{3} c x +2 \mathit {atanh} \left (a x \right ) a^{3} d \,x^{3}+6 \mathit {atanh} \left (a x \right ) a^{2} c +2 \mathit {atanh} \left (a x \right ) d +6 \,\mathrm {log}\left (a^{2} x -a \right ) a^{2} c +2 \,\mathrm {log}\left (a^{2} x -a \right ) d +a^{2} d \,x^{2}}{6 a^{3}} \] Input:
int((d*x^2+c)*atanh(a*x),x)
Output:
(6*atanh(a*x)*a**3*c*x + 2*atanh(a*x)*a**3*d*x**3 + 6*atanh(a*x)*a**2*c + 2*atanh(a*x)*d + 6*log(a**2*x - a)*a**2*c + 2*log(a**2*x - a)*d + a**2*d*x **2)/(6*a**3)