Integrand size = 12, antiderivative size = 67 \[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^n\right )\right )-\frac {b c n x^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (2+n)} \] Output:
1/2*x^2*(a+b*arctanh(c*x^n))-b*c*n*x^(2+n)*hypergeom([1, 1/2*(2+n)/n],[3/2 +1/n],c^2*x^(2*n))/(4+2*n)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\frac {a x^2}{2}+\frac {1}{2} b x^2 \text {arctanh}\left (c x^n\right )-\frac {b c n x^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2 n},1+\frac {2+n}{2 n},c^2 x^{2 n}\right )}{2 (2+n)} \] Input:
Integrate[x*(a + b*ArcTanh[c*x^n]),x]
Output:
(a*x^2)/2 + (b*x^2*ArcTanh[c*x^n])/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[ 1, (2 + n)/(2*n), 1 + (2 + n)/(2*n), c^2*x^(2*n)])/(2*(2 + n))
Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6452, 888}
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.
\(\displaystyle \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^n\right )\right )-\frac {1}{2} b c n \int \frac {x^{n+1}}{1-c^2 x^{2 n}}dx\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {1}{2} x^2 \left (a+b \text {arctanh}\left (c x^n\right )\right )-\frac {b c n x^{n+2} \operatorname {Hypergeometric2F1}\left (1,\frac {n+2}{2 n},\frac {1}{2} \left (3+\frac {2}{n}\right ),c^2 x^{2 n}\right )}{2 (n+2)}\) |
Input:
Int[x*(a + b*ArcTanh[c*x^n]),x]
Output:
(x^2*(a + b*ArcTanh[c*x^n]))/2 - (b*c*n*x^(2 + n)*Hypergeometric2F1[1, (2 + n)/(2*n), (3 + 2/n)/2, c^2*x^(2*n)])/(2*(2 + n))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
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]
\[\int x \left (a +b \,\operatorname {arctanh}\left (c \,x^{n}\right )\right )d x\]
Input:
int(x*(a+b*arctanh(c*x^n)),x)
Output:
int(x*(a+b*arctanh(c*x^n)),x)
\[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )} x \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x^n)),x, algorithm="fricas")
Output:
integral(b*x*arctanh(c*x^n) + a*x, x)
\[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int x \left (a + b \operatorname {atanh}{\left (c x^{n} \right )}\right )\, dx \] Input:
integrate(x*(a+b*atanh(c*x**n)),x)
Output:
Integral(x*(a + b*atanh(c*x**n)), x)
\[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )} x \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x^n)),x, algorithm="maxima")
Output:
1/2*a*x^2 + 1/4*(x^2*log(c*x^n + 1) - x^2*log(-c*x^n + 1) + 2*n*integrate( 1/2*x/(c*x^n + 1), x) + 2*n*integrate(1/2*x/(c*x^n - 1), x))*b
\[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )} x \,d x } \] Input:
integrate(x*(a+b*arctanh(c*x^n)),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x^n) + a)*x, x)
Timed out. \[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int x\,\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right ) \,d x \] Input:
int(x*(a + b*atanh(c*x^n)),x)
Output:
int(x*(a + b*atanh(c*x^n)), x)
\[ \int x \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\left (\int \mathit {atanh} \left (x^{n} c \right ) x d x \right ) b +\frac {a \,x^{2}}{2} \] Input:
int(x*(a+b*atanh(c*x^n)),x)
Output:
(2*int(atanh(x**n*c)*x,x)*b + a*x**2)/2