Integrand size = 20, antiderivative size = 82 \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=-\frac {2^{1+\frac {n}{2}} (1-a x)^{\frac {2-n}{2}} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},-\frac {n}{2},\frac {5-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n)} \] Output:
-2^(1+1/2*n)*(-a*x+1)^(1-1/2*n)*(-a*c*x+c)^(1/2)*hypergeom([-1/2*n, 3/2-1/ 2*n],[5/2-1/2*n],-1/2*a*x+1/2)/a/(3-n)
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.94 \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\frac {2^{1+\frac {n}{2}} (1-a x)^{1-\frac {n}{2}} \sqrt {c-a c x} \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-\frac {n}{2},-\frac {n}{2},\frac {5}{2}-\frac {n}{2},\frac {1}{2}-\frac {a x}{2}\right )}{a (-3+n)} \] Input:
Integrate[E^(n*ArcTanh[a*x])*Sqrt[c - a*c*x],x]
Output:
(2^(1 + n/2)*(1 - a*x)^(1 - n/2)*Sqrt[c - a*c*x]*Hypergeometric2F1[3/2 - n /2, -1/2*n, 5/2 - n/2, 1/2 - (a*x)/2])/(a*(-3 + n))
Time = 0.43 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6680, 37, 79}
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.
\(\displaystyle \int \sqrt {c-a c x} e^{n \text {arctanh}(a x)} \, dx\) |
\(\Big \downarrow \) 6680 |
\(\displaystyle \int \sqrt {c-a c x} (1-a x)^{-n/2} (a x+1)^{n/2}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {\sqrt {c-a c x} \int (1-a x)^{\frac {1-n}{2}} (a x+1)^{n/2}dx}{\sqrt {1-a x}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{\frac {n}{2}+1} \sqrt {c-a c x} (1-a x)^{\frac {3-n}{2}-\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (\frac {3-n}{2},-\frac {n}{2},\frac {5-n}{2},\frac {1}{2} (1-a x)\right )}{a (3-n)}\) |
Input:
Int[E^(n*ArcTanh[a*x])*Sqrt[c - a*c*x],x]
Output:
-((2^(1 + n/2)*(1 - a*x)^(-1/2 + (3 - n)/2)*Sqrt[c - a*c*x]*Hypergeometric 2F1[(3 - n)/2, -1/2*n, (5 - n)/2, (1 - a*x)/2])/(a*(3 - n)))
Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Int[u*(c + d*x)^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] /; FreeQ[{a, c , d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && !(IntegerQ[p] || GtQ[c, 0])
\[\int {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \sqrt {-a c x +c}d x\]
Input:
int(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x)
Output:
int(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x)
\[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int { \sqrt {-a c x + c} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(-a*c*x + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int \sqrt {- c \left (a x - 1\right )} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:
integrate(exp(n*atanh(a*x))*(-a*c*x+c)**(1/2),x)
Output:
Integral(sqrt(-c*(a*x - 1))*exp(n*atanh(a*x)), x)
\[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int { \sqrt {-a c x + c} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-a*c*x + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int { \sqrt {-a c x + c} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arctanh(a*x))*(-a*c*x+c)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-a*c*x + c)*(-(a*x + 1)/(a*x - 1))^(1/2*n), x)
Timed out. \[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\sqrt {c-a\,c\,x} \,d x \] Input:
int(exp(n*atanh(a*x))*(c - a*c*x)^(1/2),x)
Output:
int(exp(n*atanh(a*x))*(c - a*c*x)^(1/2), x)
\[ \int e^{n \text {arctanh}(a x)} \sqrt {c-a c x} \, dx=\sqrt {c}\, \left (\int e^{\mathit {atanh} \left (a x \right ) n} \sqrt {-a x +1}d x \right ) \] Input:
int(exp(n*atanh(a*x))*(-a*c*x+c)^(1/2),x)
Output:
sqrt(c)*int(e**(atanh(a*x)*n)*sqrt( - a*x + 1),x)