Integrand size = 20, antiderivative size = 58 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c (1+a x) \sqrt {1-a^2 x^2}}{a^2 x}+\frac {c \arcsin (a x)}{a}-\frac {c \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{a} \] Output:
c*(a*x+1)*(-a^2*x^2+1)^(1/2)/a^2/x+c*arcsin(a*x)/a-c*arctanh((-a^2*x^2+1)^ (1/2))/a
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left ((1+a x) \sqrt {1-a^2 x^2}+a x \arcsin (a x)-a x \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )\right )}{a^2 x} \] Input:
Integrate[(c - c/(a^2*x^2))/E^ArcTanh[a*x],x]
Output:
(c*((1 + a*x)*Sqrt[1 - a^2*x^2] + a*x*ArcSin[a*x] - a*x*ArcTanh[Sqrt[1 - a ^2*x^2]]))/(a^2*x)
Time = 0.36 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6707, 6699, 536, 538, 223, 243, 73, 221}
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.
\(\displaystyle \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx\) |
\(\Big \downarrow \) 6707 |
\(\displaystyle -\frac {c \int \frac {e^{-\text {arctanh}(a x)} \left (1-a^2 x^2\right )}{x^2}dx}{a^2}\) |
\(\Big \downarrow \) 6699 |
\(\displaystyle -\frac {c \int \frac {(1-a x) \sqrt {1-a^2 x^2}}{x^2}dx}{a^2}\) |
\(\Big \downarrow \) 536 |
\(\displaystyle -\frac {c \left (\int \frac {-x a^2-a}{x \sqrt {1-a^2 x^2}}dx-\frac {(a x+1) \sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
\(\Big \downarrow \) 538 |
\(\displaystyle -\frac {c \left (a^2 \left (-\int \frac {1}{\sqrt {1-a^2 x^2}}dx\right )-a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {(a x+1) \sqrt {1-a^2 x^2}}{x}\right )}{a^2}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {c \left (-a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )}{a^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {c \left (-\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )}{a^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {c \left (\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )}{a^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {c \left (a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sqrt {1-a^2 x^2} (a x+1)}{x}-a \arcsin (a x)\right )}{a^2}\) |
Input:
Int[(c - c/(a^2*x^2))/E^ArcTanh[a*x],x]
Output:
-((c*(-(((1 + a*x)*Sqrt[1 - a^2*x^2])/x) - a*ArcSin[a*x] + a*ArcTanh[Sqrt[ 1 - a^2*x^2]]))/a^2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[c^p Int[x^m*((1 - a^2*x^2)^(p + n/2)/(1 - a*x)^n), x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c , 0]) && ILtQ[(n - 1)/2, 0] && !IntegerQ[p - n/2]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[(u/x^(2*p))*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x ] /; FreeQ[{a, c, d, n}, x] && EqQ[c + a^2*d, 0] && IntegerQ[p]
Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67
method | result | size |
risch | \(-\frac {\left (a^{2} x^{2}-1\right ) c}{x \sqrt {-a^{2} x^{2}+1}\, a^{2}}+\frac {\left (a \sqrt {-a^{2} x^{2}+1}-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right ) c}{a^{2}}\) | \(97\) |
default | \(\frac {c \left (a \left (\sqrt {-a^{2} x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )+\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}+2 a^{2} \left (\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\right )}{a^{2}}\) | \(101\) |
Input:
int((c-c/a^2/x^2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
-(a^2*x^2-1)/x/(-a^2*x^2+1)^(1/2)*c/a^2+(a*(-a^2*x^2+1)^(1/2)-a*arctanh(1/ (-a^2*x^2+1)^(1/2))+a^2/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2 )))*c/a^2
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.47 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {2 \, a c x \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - a c x \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - a c x - \sqrt {-a^{2} x^{2} + 1} {\left (a c x + c\right )}}{a^{2} x} \] Input:
integrate((c-c/a^2/x^2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas")
Output:
-(2*a*c*x*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - a*c*x*log((sqrt(-a^2*x^ 2 + 1) - 1)/x) - a*c*x - sqrt(-a^2*x^2 + 1)*(a*c*x + c))/(a^2*x)
Result contains complex when optimal does not.
Time = 2.78 (sec) , antiderivative size = 177, normalized size of antiderivative = 3.05 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (\begin {cases} i \sqrt {a^{2} x^{2} - 1} - \log {\left (a x \right )} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} + i \operatorname {asin}{\left (\frac {1}{a x} \right )} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\sqrt {- a^{2} x^{2} + 1} + \frac {\log {\left (a^{2} x^{2} \right )}}{2} - \log {\left (\sqrt {- a^{2} x^{2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )}{a} - \frac {c \left (\begin {cases} - \frac {i a^{2} x}{\sqrt {a^{2} x^{2} - 1}} + i a \operatorname {acosh}{\left (a x \right )} + \frac {i}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} x}{\sqrt {- a^{2} x^{2} + 1}} - a \operatorname {asin}{\left (a x \right )} - \frac {1}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right )}{a^{2}} \] Input:
integrate((c-c/a**2/x**2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)
Output:
c*Piecewise((I*sqrt(a**2*x**2 - 1) - log(a*x) + log(a**2*x**2)/2 + I*asin( 1/(a*x)), Abs(a**2*x**2) > 1), (sqrt(-a**2*x**2 + 1) + log(a**2*x**2)/2 - log(sqrt(-a**2*x**2 + 1) + 1), True))/a - c*Piecewise((-I*a**2*x/sqrt(a**2 *x**2 - 1) + I*a*acosh(a*x) + I/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*x/sqrt(-a**2*x**2 + 1) - a*asin(a*x) - 1/(x*sqrt(-a**2*x**2 + 1) ), True))/a**2
\[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a^{2} x^{2}}\right )}}{a x + 1} \,d x } \] Input:
integrate((c-c/a^2/x^2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")
Output:
c*(arcsin(a*x)/a + sqrt(-a^2*x^2 + 1)/a) - c*integrate(sqrt(a*x + 1)*sqrt( -a*x + 1)/(a^3*x^3 + a^2*x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (54) = 108\).
Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.21 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=-\frac {a^{2} c x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} + \frac {c \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{{\left | a \right |}} - \frac {c \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{{\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{2 \, a^{2} x {\left | a \right |}} \] Input:
integrate((c-c/a^2/x^2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
-1/2*a^2*c*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) + c*arcsin(a*x)*sgn( a)/abs(a) - c*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)) )/abs(a) + sqrt(-a^2*x^2 + 1)*c/a + 1/2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*c/ (a^2*x*abs(a))
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.31 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c\,\sqrt {1-a^2\,x^2}}{a}-\frac {c\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{a}+\frac {c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{\sqrt {-a^2}}+\frac {c\,\sqrt {1-a^2\,x^2}}{a^2\,x} \] Input:
int(((c - c/(a^2*x^2))*(1 - a^2*x^2)^(1/2))/(a*x + 1),x)
Output:
(c*(1 - a^2*x^2)^(1/2))/a - (c*atanh((1 - a^2*x^2)^(1/2)))/a + (c*asinh(x* (-a^2)^(1/2)))/(-a^2)^(1/2) + (c*(1 - a^2*x^2)^(1/2))/(a^2*x)
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.40 \[ \int e^{-\text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right ) \, dx=\frac {c \left (2 \mathit {asin} \left (a x \right ) a x +2 \sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}+\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a x -\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a x \right )}{2 a^{2} x} \] Input:
int((c-c/a^2/x^2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x)
Output:
(c*(2*asin(a*x)*a*x + 2*sqrt( - a**2*x**2 + 1)*a*x + 2*sqrt( - a**2*x**2 + 1) + log(sqrt( - a**2*x**2 + 1) - 1)*a*x - log(sqrt( - a**2*x**2 + 1) + 1 )*a*x))/(2*a**2*x)