\(\int (d x)^m (a+b \text {arctanh}(c x^2)) \, dx\) [96]

Optimal result

Integrand size = 16, antiderivative size = 74 \[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {(d x)^{1+m} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d (1+m)}-\frac {2 b c (d x)^{3+m} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{4},\frac {7+m}{4},c^2 x^4\right )}{d^3 (1+m) (3+m)} \] Output:

(d*x)^(1+m)*(a+b*arctanh(c*x^2))/d/(1+m)-2*b*c*(d*x)^(3+m)*hypergeom([1, 3 
/4+1/4*m],[7/4+1/4*m],c^2*x^4)/d^3/(1+m)/(3+m)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.86 \[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=-\frac {x (d x)^m \left (-\left ((3+m) \left (a+b \text {arctanh}\left (c x^2\right )\right )\right )+2 b c x^2 \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{4},\frac {7+m}{4},c^2 x^4\right )\right )}{(1+m) (3+m)} \] Input:

Integrate[(d*x)^m*(a + b*ArcTanh[c*x^2]),x]
 

Output:

-((x*(d*x)^m*(-((3 + m)*(a + b*ArcTanh[c*x^2])) + 2*b*c*x^2*Hypergeometric 
2F1[1, (3 + m)/4, (7 + m)/4, c^2*x^4]))/((1 + m)*(3 + m)))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6464, 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.

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

\(\Big \downarrow \) 6464

\(\displaystyle \frac {(d x)^{m+1} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d (m+1)}-\frac {2 b c \int \frac {(d x)^{m+2}}{1-c^2 x^4}dx}{d^2 (m+1)}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {(d x)^{m+1} \left (a+b \text {arctanh}\left (c x^2\right )\right )}{d (m+1)}-\frac {2 b c (d x)^{m+3} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{4},\frac {m+7}{4},c^2 x^4\right )}{d^3 (m+1) (m+3)}\)

Input:

Int[(d*x)^m*(a + b*ArcTanh[c*x^2]),x]
 

Output:

((d*x)^(1 + m)*(a + b*ArcTanh[c*x^2]))/(d*(1 + m)) - (2*b*c*(d*x)^(3 + m)* 
Hypergeometric2F1[1, (3 + m)/4, (7 + m)/4, c^2*x^4])/(d^3*(1 + m)*(3 + 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 6464

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]
 

Maple [F]

\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )d x\]

Input:

int((d*x)^m*(a+b*arctanh(c*x^2)),x)
 

Output:

int((d*x)^m*(a+b*arctanh(c*x^2)),x)
 

Fricas [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arctanh(c*x^2)),x, algorithm="fricas")
 

Output:

integral((b*arctanh(c*x^2) + a)*(d*x)^m, x)
 

Sympy [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )\, dx \] Input:

integrate((d*x)**m*(a+b*atanh(c*x**2)),x)
 

Output:

Integral((d*x)**m*(a + b*atanh(c*x**2)), x)
 

Maxima [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arctanh(c*x^2)),x, algorithm="maxima")
 

Output:

1/2*(4*c*d^m*integrate(x^2*x^m/(c^2*(m + 1)*x^4 - m - 1), x) + (d^m*x*x^m* 
log(c*x^2 + 1) - d^m*x*x^m*log(-c*x^2 + 1))/(m + 1))*b + (d*x)^(m + 1)*a/( 
d*(m + 1))
 

Giac [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(a+b*arctanh(c*x^2)),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^2) + a)*(d*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right ) \,d x \] Input:

int((d*x)^m*(a + b*atanh(c*x^2)),x)
 

Output:

int((d*x)^m*(a + b*atanh(c*x^2)), x)
 

Reduce [F]

\[ \int (d x)^m \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {d^{m} \left (x^{m} a x +\left (\int x^{m} \mathit {atanh} \left (c \,x^{2}\right )d x \right ) b m +\left (\int x^{m} \mathit {atanh} \left (c \,x^{2}\right )d x \right ) b \right )}{m +1} \] Input:

int((d*x)^m*(a+b*atanh(c*x^2)),x)
 

Output:

(d**m*(x**m*a*x + int(x**m*atanh(c*x**2),x)*b*m + int(x**m*atanh(c*x**2),x 
)*b))/(m + 1)