Integrand size = 14, antiderivative size = 45 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{6} b c x^2+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c^3 \log \left (c^2-x^2\right ) \] Output:
1/6*b*c*x^2+1/3*x^3*(a+b*arctanh(c/x))+1/6*b*c^3*ln(c^2-x^2)
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.11 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{6} b c x^2+\frac {a x^3}{3}+\frac {1}{3} b x^3 \text {arctanh}\left (\frac {c}{x}\right )+\frac {1}{6} b c^3 \log \left (-c^2+x^2\right ) \] Input:
Integrate[x^2*(a + b*ArcTanh[c/x]),x]
Output:
(b*c*x^2)/6 + (a*x^3)/3 + (b*x^3*ArcTanh[c/x])/3 + (b*c^3*Log[-c^2 + x^2]) /6
Time = 0.34 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6452, 795, 243, 25, 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.
\(\displaystyle \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{3} b c \int \frac {x}{1-\frac {c^2}{x^2}}dx+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \frac {1}{3} b c \int \frac {x^3}{x^2-c^2}dx+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{6} b c \int -\frac {x^2}{c^2-x^2}dx^2+\frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )-\frac {1}{6} b c \int \frac {x^2}{c^2-x^2}dx^2\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )-\frac {1}{6} b c \int \left (\frac {c^2}{c^2-x^2}-1\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right )+\frac {1}{6} b c \left (c^2 \log \left (c^2-x^2\right )+x^2\right )\) |
Input:
Int[x^2*(a + b*ArcTanh[c/x]),x]
Output:
(x^3*(a + b*ArcTanh[c/x]))/3 + (b*c*(x^2 + c^2*Log[c^2 - x^2]))/6
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_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Time = 0.43 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27
method | result | size |
parallelrisch | \(\frac {\ln \left (x -c \right ) b \,c^{3}}{3}+\frac {b \,x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3}+\frac {\operatorname {arctanh}\left (\frac {c}{x}\right ) b \,c^{3}}{3}+\frac {a \,x^{3}}{3}+\frac {b c \,x^{2}}{6}+\frac {b \,c^{3}}{6}\) | \(57\) |
parts | \(\frac {a \,x^{3}}{3}-b \,c^{3} \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{3}}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {\ln \left (1+\frac {c}{x}\right )}{6}-\frac {x^{2}}{6 c^{2}}+\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\) | \(65\) |
derivativedivides | \(-c^{3} \left (-\frac {a \,x^{3}}{3 c^{3}}+b \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{3}}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {\ln \left (1+\frac {c}{x}\right )}{6}-\frac {x^{2}}{6 c^{2}}+\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\right )\) | \(69\) |
default | \(-c^{3} \left (-\frac {a \,x^{3}}{3 c^{3}}+b \left (-\frac {x^{3} \operatorname {arctanh}\left (\frac {c}{x}\right )}{3 c^{3}}-\frac {\ln \left (\frac {c}{x}-1\right )}{6}-\frac {\ln \left (1+\frac {c}{x}\right )}{6}-\frac {x^{2}}{6 c^{2}}+\frac {\ln \left (\frac {c}{x}\right )}{3}\right )\right )\) | \(69\) |
risch | \(\frac {b \,x^{3} \ln \left (x +c \right )}{6}-\frac {b \,x^{3} \ln \left (c -x \right )}{6}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{3}}{12}+\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (c -x \right )\right ) \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )}{12}-\frac {i \pi b \,x^{3}}{6}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}}{12}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{3}}{12}+\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}}{12}+\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}}{6}+\frac {i \pi b \,x^{3} \operatorname {csgn}\left (i \left (x +c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )^{2}}{12}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x +c \right )\right ) \operatorname {csgn}\left (\frac {i \left (x +c \right )}{x}\right )}{12}-\frac {i \pi b \,x^{3} \operatorname {csgn}\left (i \left (c -x \right )\right ) \operatorname {csgn}\left (\frac {i \left (c -x \right )}{x}\right )^{2}}{12}+\frac {a \,x^{3}}{3}+\frac {b c \,x^{2}}{6}+\frac {b \,c^{3} \ln \left (-c^{2}+x^{2}\right )}{6}\) | \(307\) |
Input:
int(x^2*(a+b*arctanh(c/x)),x,method=_RETURNVERBOSE)
Output:
1/3*ln(x-c)*b*c^3+1/3*b*x^3*arctanh(c/x)+1/3*arctanh(c/x)*b*c^3+1/3*a*x^3+ 1/6*b*c*x^2+1/6*b*c^3
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{6} \, b c^{3} \log \left (-c^{2} + x^{2}\right ) + \frac {1}{6} \, b x^{3} \log \left (-\frac {c + x}{c - x}\right ) + \frac {1}{6} \, b c x^{2} + \frac {1}{3} \, a x^{3} \] Input:
integrate(x^2*(a+b*arctanh(c/x)),x, algorithm="fricas")
Output:
1/6*b*c^3*log(-c^2 + x^2) + 1/6*b*x^3*log(-(c + x)/(c - x)) + 1/6*b*c*x^2 + 1/3*a*x^3
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.09 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=\frac {a x^{3}}{3} + \frac {b c^{3} \log {\left (- c + x \right )}}{3} + \frac {b c^{3} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{3} + \frac {b c x^{2}}{6} + \frac {b x^{3} \operatorname {atanh}{\left (\frac {c}{x} \right )}}{3} \] Input:
integrate(x**2*(a+b*atanh(c/x)),x)
Output:
a*x**3/3 + b*c**3*log(-c + x)/3 + b*c**3*atanh(c/x)/3 + b*c*x**2/6 + b*x** 3*atanh(c/x)/3
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=\frac {1}{3} \, a x^{3} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (\frac {c}{x}\right ) + {\left (c^{2} \log \left (-c^{2} + x^{2}\right ) + x^{2}\right )} c\right )} b \] Input:
integrate(x^2*(a+b*arctanh(c/x)),x, algorithm="maxima")
Output:
1/3*a*x^3 + 1/6*(2*x^3*arctanh(c/x) + (c^2*log(-c^2 + x^2) + x^2)*c)*b
Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (39) = 78\).
Time = 0.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 5.04 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=-\frac {b c^{4} \log \left (-\frac {c + x}{c - x} - 1\right ) - b c^{4} \log \left (-\frac {c + x}{c - x}\right ) + \frac {{\left (b c^{4} + \frac {3 \, b {\left (c + x\right )}^{2} c^{4}}{{\left (c - x\right )}^{2}}\right )} \log \left (-\frac {c + x}{c - x}\right )}{\frac {{\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {3 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {3 \, {\left (c + x\right )}}{c - x} + 1} + \frac {2 \, {\left (a c^{4} + \frac {3 \, a {\left (c + x\right )}^{2} c^{4}}{{\left (c - x\right )}^{2}} + \frac {b {\left (c + x\right )}^{2} c^{4}}{{\left (c - x\right )}^{2}} + \frac {b {\left (c + x\right )} c^{4}}{c - x}\right )}}{\frac {{\left (c + x\right )}^{3}}{{\left (c - x\right )}^{3}} + \frac {3 \, {\left (c + x\right )}^{2}}{{\left (c - x\right )}^{2}} + \frac {3 \, {\left (c + x\right )}}{c - x} + 1}}{3 \, c} \] Input:
integrate(x^2*(a+b*arctanh(c/x)),x, algorithm="giac")
Output:
-1/3*(b*c^4*log(-(c + x)/(c - x) - 1) - b*c^4*log(-(c + x)/(c - x)) + (b*c ^4 + 3*b*(c + x)^2*c^4/(c - x)^2)*log(-(c + x)/(c - x))/((c + x)^3/(c - x) ^3 + 3*(c + x)^2/(c - x)^2 + 3*(c + x)/(c - x) + 1) + 2*(a*c^4 + 3*a*(c + x)^2*c^4/(c - x)^2 + b*(c + x)^2*c^4/(c - x)^2 + b*(c + x)*c^4/(c - x))/(( c + x)^3/(c - x)^3 + 3*(c + x)^2/(c - x)^2 + 3*(c + x)/(c - x) + 1))/c
Time = 3.44 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=\frac {a\,x^3}{3}+\frac {b\,c^3\,\ln \left (x^2-c^2\right )}{6}+\frac {b\,x^3\,\mathrm {atanh}\left (\frac {c}{x}\right )}{3}+\frac {b\,c\,x^2}{6} \] Input:
int(x^2*(a + b*atanh(c/x)),x)
Output:
(a*x^3)/3 + (b*c^3*log(x^2 - c^2))/6 + (b*x^3*atanh(c/x))/3 + (b*c*x^2)/6
Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int x^2 \left (a+b \text {arctanh}\left (\frac {c}{x}\right )\right ) \, dx=-\frac {\mathit {atanh} \left (\frac {c}{x}\right ) b \,c^{3}}{3}+\frac {\mathit {atanh} \left (\frac {c}{x}\right ) b \,x^{3}}{3}+\frac {\mathrm {log}\left (-c -x \right ) b \,c^{3}}{3}+\frac {a \,x^{3}}{3}+\frac {b c \,x^{2}}{6} \] Input:
int(x^2*(a+b*atanh(c/x)),x)
Output:
( - 2*atanh(c/x)*b*c**3 + 2*atanh(c/x)*b*x**3 + 2*log( - c - x)*b*c**3 + 2 *a*x**3 + b*c*x**2)/6