\(\int (d x)^m (a+b \text {arctanh}(c x^n)) \, dx\) [240]

Optimal result

Integrand size = 16, antiderivative size = 84 \[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\frac {x (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right )}{1+m}-\frac {b c n x^{1+n} (d x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+n}{2 n},\frac {1+m+3 n}{2 n},c^2 x^{2 n}\right )}{(1+m) (1+m+n)} \] Output:

x*(d*x)^m*(a+b*arctanh(c*x^n))/(1+m)-b*c*n*x^(1+n)*(d*x)^m*hypergeom([1, 1 
/2*(1+m+n)/n],[1/2*(1+m+3*n)/n],c^2*x^(2*n))/(1+m)/(1+m+n)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92 \[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\frac {x (d x)^m \left ((1+m+n) \left (a+b \text {arctanh}\left (c x^n\right )\right )-b c n x^n \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+n}{2 n},\frac {1+m+3 n}{2 n},c^2 x^{2 n}\right )\right )}{(1+m) (1+m+n)} \] Input:

Integrate[(d*x)^m*(a + b*ArcTanh[c*x^n]),x]
 

Output:

(x*(d*x)^m*((1 + m + n)*(a + b*ArcTanh[c*x^n]) - b*c*n*x^n*Hypergeometric2 
F1[1, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), c^2*x^(2*n)]))/((1 + m)*(1 + 
 m + n))
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6466, 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.

\(\displaystyle \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 6466

\(\displaystyle x^{-m} (d x)^m \int x^m \left (a+b \text {arctanh}\left (c x^n\right )\right )dx\)

\(\Big \downarrow \) 6452

\(\displaystyle x^{-m} (d x)^m \left (\frac {x^{m+1} \left (a+b \text {arctanh}\left (c x^n\right )\right )}{m+1}-\frac {b c n \int \frac {x^{m+n}}{1-c^2 x^{2 n}}dx}{m+1}\right )\)

\(\Big \downarrow \) 888

\(\displaystyle x^{-m} (d x)^m \left (\frac {x^{m+1} \left (a+b \text {arctanh}\left (c x^n\right )\right )}{m+1}-\frac {b c n x^{m+n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+n+1}{2 n},\frac {m+3 n+1}{2 n},c^2 x^{2 n}\right )}{(m+1) (m+n+1)}\right )\)

Input:

Int[(d*x)^m*(a + b*ArcTanh[c*x^n]),x]
 

Output:

((d*x)^m*((x^(1 + m)*(a + b*ArcTanh[c*x^n]))/(1 + m) - (b*c*n*x^(1 + m + n 
)*Hypergeometric2F1[1, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), c^2*x^(2*n) 
])/((1 + m)*(1 + m + n))))/x^m
 

Defintions of rubi rules used

rule 888

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])
 

rule 6452

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]
 

rule 6466

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_)*(x_))^(m_), x_Sym 
bol] :> Simp[d^IntPart[m]*((d*x)^FracPart[m]/x^FracPart[m])   Int[x^m*(a + 
b*ArcTanh[c*x^n])^p, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IGtQ[p, 0] & 
& (EqQ[p, 1] || RationalQ[m, n])
 

Maple [F]

\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arctanh}\left (c \,x^{n}\right )\right )d x\]

Input:

int((d*x)^m*(a+b*arctanh(c*x^n)),x)
 

Output:

int((d*x)^m*(a+b*arctanh(c*x^n)),x)
 

Fricas [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arctanh(c*x^n)),x, algorithm="fricas")
 

Output:

integral((d*x)^m*b*arctanh(c*x^n) + (d*x)^m*a, x)
 

Sympy [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (c x^{n} \right )}\right )\, dx \] Input:

integrate((d*x)**m*(a+b*atanh(c*x**n)),x)
 

Output:

Integral((d*x)**m*(a + b*atanh(c*x**n)), x)
 

Maxima [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arctanh(c*x^n)),x, algorithm="maxima")
 

Output:

1/2*(d^m*n*integrate(x^m/(c*(m + 1)*x^n + m + 1), x) + d^m*n*integrate(x^m 
/(c*(m + 1)*x^n - m - 1), x) + (d^m*x*x^m*log(c*x^n + 1) - d^m*x*x^m*log(- 
c*x^n + 1))/(m + 1))*b + (d*x)^(m + 1)*a/(d*(m + 1))
 

Giac [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arctanh(c*x^n)),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^n) + a)*(d*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right ) \,d x \] Input:

int((d*x)^m*(a + b*atanh(c*x^n)),x)
 

Output:

int((d*x)^m*(a + b*atanh(c*x^n)), x)
 

Reduce [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^n\right )\right ) \, dx=\frac {d^{m} \left (x^{m} a x +\left (\int x^{m} \mathit {atanh} \left (x^{n} c \right )d x \right ) b m +\left (\int x^{m} \mathit {atanh} \left (x^{n} c \right )d x \right ) b \right )}{m +1} \] Input:

int((d*x)^m*(a+b*atanh(c*x^n)),x)
 

Output:

(d**m*(x**m*a*x + int(x**m*atanh(x**n*c),x)*b*m + int(x**m*atanh(x**n*c),x 
)*b))/(m + 1)