Integrand size = 12, antiderivative size = 65 \[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=\frac {2^{1-\frac {d}{2}} e^c (1+a+b x)^{\frac {2+d}{2}} \operatorname {Hypergeometric2F1}\left (\frac {d}{2},\frac {2+d}{2},\frac {4+d}{2},\frac {1}{2} (1+a+b x)\right )}{b (2+d)} \] Output:
2^(1-1/2*d)*exp(c)*(b*x+a+1)^(1+1/2*d)*hypergeom([1/2*d, 1+1/2*d],[2+1/2*d ],1/2*b*x+1/2*a+1/2)/b/(2+d)
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.80 \[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=\frac {4 e^{c+(2+d) \text {arctanh}(a+b x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {d}{2},2+\frac {d}{2},-e^{2 \text {arctanh}(a+b x)}\right )}{b (2+d)} \] Input:
Integrate[E^(c + d*ArcTanh[a + b*x]),x]
Output:
(4*E^(c + (2 + d)*ArcTanh[a + b*x])*Hypergeometric2F1[2, 1 + d/2, 2 + d/2, -E^(2*ArcTanh[a + b*x])])/(b*(2 + d))
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.
\(\displaystyle \int e^{d \text {arctanh}(a+b x)+c} \, dx\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {\int e^{c+d \text {arctanh}(a+b x)}d(a+b x)}{b}\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {\int e^{c+d \text {arctanh}(a+b x)}d(a+b x)}{b}\) |
Input:
Int[E^(c + d*ArcTanh[a + b*x]),x]
Output:
$Aborted
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
\[\int {\mathrm e}^{c +d \,\operatorname {arctanh}\left (b x +a \right )}d x\]
Input:
int(exp(c+d*arctanh(b*x+a)),x)
Output:
int(exp(c+d*arctanh(b*x+a)),x)
\[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=\int { e^{\left (d \operatorname {artanh}\left (b x + a\right ) + c\right )} \,d x } \] Input:
integrate(exp(c+d*arctanh(b*x+a)),x, algorithm="fricas")
Output:
integral(e^(d*arctanh(b*x + a) + c), x)
\[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=e^{c} \int e^{d \operatorname {atanh}{\left (a + b x \right )}}\, dx \] Input:
integrate(exp(c+d*atanh(b*x+a)),x)
Output:
exp(c)*Integral(exp(d*atanh(a + b*x)), x)
\[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=\int { e^{\left (d \operatorname {artanh}\left (b x + a\right ) + c\right )} \,d x } \] Input:
integrate(exp(c+d*arctanh(b*x+a)),x, algorithm="maxima")
Output:
integrate(e^(d*arctanh(b*x + a) + c), x)
\[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=\int { e^{\left (d \operatorname {artanh}\left (b x + a\right ) + c\right )} \,d x } \] Input:
integrate(exp(c+d*arctanh(b*x+a)),x, algorithm="giac")
Output:
integrate(e^(d*arctanh(b*x + a) + c), x)
Timed out. \[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=\int \frac {{\mathrm {e}}^c\,{\left (a+b\,x+1\right )}^{d/2}}{{\left (1-b\,x-a\right )}^{d/2}} \,d x \] Input:
int(exp(c + d*atanh(a + b*x)),x)
Output:
int((exp(c)*(a + b*x + 1)^(d/2))/(1 - b*x - a)^(d/2), x)
\[ \int e^{c+d \text {arctanh}(a+b x)} \, dx=e^{c} \left (\int e^{\mathit {atanh} \left (b x +a \right ) d}d x \right ) \] Input:
int(exp(c+d*atanh(b*x+a)),x)
Output:
e**c*int(e**(atanh(a + b*x)*d),x)