Integrand size = 20, antiderivative size = 187 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c (1-a x)^{2-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{a (2-n)}-\frac {2 c (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-2+n),\frac {n}{2},\frac {1+a x}{1-a x}\right )}{a (2-n)}+\frac {2^{n/2} c (1-n) (1-a x)^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n) (4-n)} \]
c*(-a*x+1)^(2-1/2*n)*(a*x+1)^(-1+1/2*n)/a/(2-n)-2*c*(-a*x+1)^(1-1/2*n)*(a* x+1)^(-1+1/2*n)*hypergeom([1, -1+1/2*n],[1/2*n],(a*x+1)/(-a*x+1))/a/(2-n)+ 2^(1/2*n)*c*(1-n)*(-a*x+1)^(2-1/2*n)*hypergeom([2-1/2*n, 1-1/2*n],[3-1/2*n ],-1/2*a*x+1/2)/a/(n^2-6*n+8)
Time = 0.07 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.80 \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=-\frac {2 c (1-a x)^{-n/2} \left ((-2+n) (1+a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {1-a x}{1+a x}\right )+2^{n/2} \left (n (-1+a x) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-a x)\right )-(-2+n) \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-a x)\right )\right )\right )}{a (-2+n) n} \]
(-2*c*((-2 + n)*(1 + a*x)^(n/2)*Hypergeometric2F1[1, -1/2*n, 1 - n/2, (1 - a*x)/(1 + a*x)] + 2^(n/2)*(n*(-1 + a*x)*Hypergeometric2F1[1 - n/2, -1/2*n , 2 - n/2, (1 - a*x)/2] - (-2 + n)*Hypergeometric2F1[-1/2*n, -1/2*n, 1 - n /2, (1 - a*x)/2])))/(a*(-2 + n)*n*(1 - a*x)^(n/2))
Time = 0.44 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6681, 6679, 139, 88, 79, 141}
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.
\(\displaystyle \int \left (c-\frac {c}{a x}\right ) e^{n \text {arctanh}(a x)} \, dx\) |
\(\Big \downarrow \) 6681 |
\(\displaystyle -\frac {c \int \frac {e^{n \text {arctanh}(a x)} (1-a x)}{x}dx}{a}\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle -\frac {c \int \frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{n/2}}{x}dx}{a}\) |
\(\Big \downarrow \) 139 |
\(\displaystyle -\frac {c \left (\int \frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{x}dx+a \int (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}} (a x+2)dx\right )}{a}\) |
\(\Big \downarrow \) 88 |
\(\displaystyle -\frac {c \left (\int \frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{x}dx+a \left (\frac {(1-n) \int (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}dx}{2-n}-\frac {(1-a x)^{2-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}}}{a (2-n)}\right )\right )}{a}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {c \left (\int \frac {(1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-4}{2}}}{x}dx+a \left (-\frac {2^{n/2} (1-n) (1-a x)^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n) (4-n)}-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{2-\frac {n}{2}}}{a (2-n)}\right )\right )}{a}\) |
\(\Big \downarrow \) 141 |
\(\displaystyle -\frac {c \left (\frac {2 (a x+1)^{\frac {n-2}{2}} (1-a x)^{\frac {2-n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {n-2}{2},\frac {n}{2},\frac {a x+1}{1-a x}\right )}{2-n}+a \left (-\frac {2^{n/2} (1-n) (1-a x)^{2-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (\frac {2-n}{2},2-\frac {n}{2},3-\frac {n}{2},\frac {1}{2} (1-a x)\right )}{a (2-n) (4-n)}-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{2-\frac {n}{2}}}{a (2-n)}\right )\right )}{a}\) |
-((c*((2*(1 - a*x)^((2 - n)/2)*(1 + a*x)^((-2 + n)/2)*Hypergeometric2F1[1, (-2 + n)/2, n/2, (1 + a*x)/(1 - a*x)])/(2 - n) + a*(-(((1 - a*x)^(2 - n/2 )*(1 + a*x)^((-2 + n)/2))/(a*(2 - n))) - (2^(n/2)*(1 - n)*(1 - a*x)^(2 - n /2)*Hypergeometric2F1[(2 - n)/2, 2 - n/2, 3 - n/2, (1 - a*x)/2])/(a*(2 - n )*(4 - n)))))/a)
3.7.21.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[f^(p - 1)/d^p Int[(a + b*x)^m*((d*e*p - c*f*(p - 1) + d*f*x)/(c + d*x)^(m + 1)), x], x] + Simp[f^(p - 1) Int[(a + b*x)^m*((e + f*x)^p/(c + d*x)^(m + 1))*ExpandToSum[f^(-p + 1)*(c + d*x)^(-p + 1) - (d*e *p - c*f*(p - 1) + d*f*x)/(d^p*(e + f*x)^p), x], x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p, 0] && ILtQ[p, 0] && (LtQ[m, 0] || SumS implerQ[m, 1] || !(LtQ[n, 0] || SumSimplerQ[n, 1]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f ))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
\[\int {\mathrm e}^{n \,\operatorname {arctanh}\left (a x \right )} \left (c -\frac {c}{a x}\right )d x\]
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { {\left (c - \frac {c}{a x}\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\frac {c \left (\int a e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x}\right )\, dx\right )}{a} \]
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { {\left (c - \frac {c}{a x}\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
\[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int { {\left (c - \frac {c}{a x}\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]
Timed out. \[ \int e^{n \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right ) \, dx=\int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-\frac {c}{a\,x}\right ) \,d x \]