Integrand size = 14, antiderivative size = 72 \[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=\frac {(d x)^{1+m} (a+b \text {arctanh}(c x))}{d (1+m)}-\frac {b c (d x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},c^2 x^2\right )}{d^2 (1+m) (2+m)} \] Output:
(d*x)^(1+m)*(a+b*arctanh(c*x))/d/(1+m)-b*c*(d*x)^(2+m)*hypergeom([1, 1+1/2 *m],[2+1/2*m],c^2*x^2)/d^2/(1+m)/(2+m)
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82 \[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=-\frac {x (d x)^m \left (-((2+m) (a+b \text {arctanh}(c x)))+b c x \operatorname {Hypergeometric2F1}\left (1,1+\frac {m}{2},2+\frac {m}{2},c^2 x^2\right )\right )}{(1+m) (2+m)} \] Input:
Integrate[(d*x)^m*(a + b*ArcTanh[c*x]),x]
Output:
-((x*(d*x)^m*(-((2 + m)*(a + b*ArcTanh[c*x])) + b*c*x*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2]))/((1 + m)*(2 + m)))
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6464, 278}
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.
\(\displaystyle \int (d x)^m (a+b \text {arctanh}(c x)) \, dx\) |
\(\Big \downarrow \) 6464 |
\(\displaystyle \frac {(d x)^{m+1} (a+b \text {arctanh}(c x))}{d (m+1)}-\frac {b c \int \frac {(d x)^{m+1}}{1-c^2 x^2}dx}{d (m+1)}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {(d x)^{m+1} (a+b \text {arctanh}(c x))}{d (m+1)}-\frac {b c (d x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,\frac {m+2}{2},\frac {m+4}{2},c^2 x^2\right )}{d^2 (m+1) (m+2)}\) |
Input:
Int[(d*x)^m*(a + b*ArcTanh[c*x]),x]
Output:
((d*x)^(1 + m)*(a + b*ArcTanh[c*x]))/(d*(1 + m)) - (b*c*(d*x)^(2 + m)*Hype rgeometric2F1[1, (2 + m)/2, (4 + m)/2, c^2*x^2])/(d^2*(1 + m)*(2 + m))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))*((d_)*(x_))^(m_), x_Symbol] : > Simp[(d*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(d*(m + 1))), x] - Simp[b*c*(n /(d^n*(m + 1))) Int[(d*x)^(m + n)/(1 - c^2*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegerQ[n] && NeQ[m, -1]
\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arctanh}\left (c x \right )\right )d x\]
Input:
int((d*x)^m*(a+b*arctanh(c*x)),x)
Output:
int((d*x)^m*(a+b*arctanh(c*x)),x)
\[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(a+b*arctanh(c*x)),x, algorithm="fricas")
Output:
integral((b*arctanh(c*x) + a)*(d*x)^m, x)
\[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )\, dx \] Input:
integrate((d*x)**m*(a+b*atanh(c*x)),x)
Output:
Integral((d*x)**m*(a + b*atanh(c*x)), x)
\[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(a+b*arctanh(c*x)),x, algorithm="maxima")
Output:
1/2*(2*c*d^m*integrate(x*x^m/(c^2*(m + 1)*x^2 - m - 1), x) + (d^m*x*x^m*lo g(c*x + 1) - d^m*x*x^m*log(-c*x + 1))/(m + 1))*b + (d*x)^(m + 1)*a/(d*(m + 1))
\[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=\int { {\left (b \operatorname {artanh}\left (c x\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:
integrate((d*x)^m*(a+b*arctanh(c*x)),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)*(d*x)^m, x)
Timed out. \[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=\int \left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )\,{\left (d\,x\right )}^m \,d x \] Input:
int((a + b*atanh(c*x))*(d*x)^m,x)
Output:
int((a + b*atanh(c*x))*(d*x)^m, x)
\[ \int (d x)^m (a+b \text {arctanh}(c x)) \, dx=\frac {d^{m} \left (x^{m} \mathit {atanh} \left (c x \right ) b c m x +x^{m} a c m x +x^{m} b +\left (\int \frac {x^{m}}{c^{2} m \,x^{3}+c^{2} x^{3}-m x -x}d x \right ) b \,m^{2}+\left (\int \frac {x^{m}}{c^{2} m \,x^{3}+c^{2} x^{3}-m x -x}d x \right ) b m \right )}{c m \left (m +1\right )} \] Input:
int((d*x)^m*(a+b*atanh(c*x)),x)
Output:
(d**m*(x**m*atanh(c*x)*b*c*m*x + x**m*a*c*m*x + x**m*b + int(x**m/(c**2*m* x**3 + c**2*x**3 - m*x - x),x)*b*m**2 + int(x**m/(c**2*m*x**3 + c**2*x**3 - m*x - x),x)*b*m))/(c*m*(m + 1))