Integrand size = 21, antiderivative size = 60 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=-\frac {1}{2 c x^2}-\frac {\sqrt {1+\frac {1}{c^2 x^2}}}{2 x}+\frac {1}{2} c \text {csch}^{-1}(c x)-c \log (x)+\frac {1}{2} c \log \left (1+c^2 x^2\right ) \] Output:
-1/2/c/x^2-1/2*(1+1/c^2/x^2)^(1/2)/x+1/2*c*arccsch(c*x)-c*ln(x)+1/2*c*ln(c ^2*x^2+1)
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=\frac {1}{2} \left (-\frac {1}{c x^2}-\frac {\sqrt {1+\frac {1}{c^2 x^2}}}{x}+c \text {arcsinh}\left (\frac {1}{c x}\right )-2 c \log (x)+c \log \left (1+c^2 x^2\right )\right ) \] Input:
Integrate[E^ArcCsch[c*x]/(x^2*(1 + c^2*x^2)),x]
Output:
(-(1/(c*x^2)) - Sqrt[1 + 1/(c^2*x^2)]/x + c*ArcSinh[1/(c*x)] - 2*c*Log[x] + c*Log[1 + c^2*x^2])/2
Time = 0.55 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.35, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6896, 243, 54, 858, 262, 222, 2009}
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.
\(\displaystyle \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (c^2 x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 6896 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^4}dx}{c^2}+\frac {\int \frac {1}{x^3 \left (c^2 x^2+1\right )}dx}{c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^4}dx}{c^2}+\frac {\int \frac {1}{x^4 \left (c^2 x^2+1\right )}dx^2}{2 c}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^4}dx}{c^2}+\frac {\int \left (\frac {c^4}{c^2 x^2+1}-\frac {c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{2 c}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {\int \left (\frac {c^4}{c^2 x^2+1}-\frac {c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{2 c}-\frac {\int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}d\frac {1}{x}}{c^2}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {\int \left (\frac {c^4}{c^2 x^2+1}-\frac {c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{2 c}-\frac {\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{2 x}-\frac {1}{2} c^2 \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}}{c^2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\int \left (\frac {c^4}{c^2 x^2+1}-\frac {c^2}{x^2}+\frac {1}{x^4}\right )dx^2}{2 c}-\frac {\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{2 x}-\frac {1}{2} c^3 \text {arcsinh}\left (\frac {1}{c x}\right )}{c^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}}{2 c}-\frac {\frac {c^2 \sqrt {\frac {1}{c^2 x^2}+1}}{2 x}-\frac {1}{2} c^3 \text {arcsinh}\left (\frac {1}{c x}\right )}{c^2}\) |
Input:
Int[E^ArcCsch[c*x]/(x^2*(1 + c^2*x^2)),x]
Output:
-(((c^2*Sqrt[1 + 1/(c^2*x^2)])/(2*x) - (c^3*ArcSinh[1/(c*x)])/2)/c^2) + (- x^(-2) - c^2*Log[x^2] + c^2*Log[1 + c^2*x^2])/(2*c)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[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_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[(E^ArcCsch[(c_.)*(x_)]*((d_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^2), x_Sym bol] :> Simp[d^2/(a*c^2) Int[(d*x)^(m - 2)/Sqrt[1 + 1/(c^2*x^2)], x], x] + Simp[d/c Int[(d*x)^(m - 1)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d, m} , x] && EqQ[b - a*c^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(215\) vs. \(2(50)=100\).
Time = 0.40 (sec) , antiderivative size = 216, normalized size of antiderivative = 3.60
method | result | size |
default | \(-\frac {\sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, \left (c^{2} \left (\frac {c^{2} x^{2}+1}{c^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{c^{2}}}+\sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, \sqrt {\frac {1}{c^{2}}}\, c^{2} x^{2}-2 \sqrt {\frac {1}{c^{2}}}\, \sqrt {-\frac {\left (-c^{2} x +\sqrt {-c^{2}}\right ) \left (c^{2} x +\sqrt {-c^{2}}\right )}{c^{4}}}\, c^{2} x^{2}-\ln \left (\frac {2 \sqrt {\frac {1}{c^{2}}}\, \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, c^{2}+2}{c^{2} x}\right ) x^{2}\right )}{2 x \sqrt {\frac {c^{2} x^{2}+1}{c^{2}}}\, \sqrt {\frac {1}{c^{2}}}}+\frac {\frac {c^{2} \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )}{c}\) | \(216\) |
Input:
int((1/c/x+(1+1/c^2/x^2)^(1/2))/x^2/(c^2*x^2+1),x,method=_RETURNVERBOSE)
Output:
-1/2*((c^2*x^2+1)/c^2/x^2)^(1/2)/x*(c^2*((c^2*x^2+1)/c^2)^(3/2)*(1/c^2)^(1 /2)+((c^2*x^2+1)/c^2)^(1/2)*(1/c^2)^(1/2)*c^2*x^2-2*(1/c^2)^(1/2)*(-(-c^2* x+(-c^2)^(1/2))*(c^2*x+(-c^2)^(1/2))/c^4)^(1/2)*c^2*x^2-ln(2*((1/c^2)^(1/2 )*((c^2*x^2+1)/c^2)^(1/2)*c^2+1)/c^2/x)*x^2)/((c^2*x^2+1)/c^2)^(1/2)/(1/c^ 2)^(1/2)+1/c*(1/2*c^2*ln(c^2*x^2+1)-1/2/x^2-c^2*ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (50) = 100\).
Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.17 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=\frac {c^{2} x^{2} \log \left (c^{2} x^{2} + 1\right ) + c^{2} x^{2} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - c^{2} x^{2} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 2 \, c^{2} x^{2} \log \left (x\right ) - c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - 1}{2 \, c x^{2}} \] Input:
integrate((1/c/x+(1+1/c^2/x^2)^(1/2))/x^2/(c^2*x^2+1),x, algorithm="fricas ")
Output:
1/2*(c^2*x^2*log(c^2*x^2 + 1) + c^2*x^2*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^ 2)) - c*x + 1) - c^2*x^2*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) - 2*c^2*x^2*log(x) - c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - 1)/(c*x^2)
\[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=\frac {\int \frac {c x \sqrt {1 + \frac {1}{c^{2} x^{2}}}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {1}{c^{2} x^{5} + x^{3}}\, dx}{c} \] Input:
integrate((1/c/x+(1+1/c**2/x**2)**(1/2))/x**2/(c**2*x**2+1),x)
Output:
(Integral(c*x*sqrt(1 + 1/(c**2*x**2))/(c**2*x**5 + x**3), x) + Integral(1/ (c**2*x**5 + x**3), x))/c
\[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=\int { \frac {\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}}{{\left (c^{2} x^{2} + 1\right )} x^{2}} \,d x } \] Input:
integrate((1/c/x+(1+1/c^2/x^2)^(1/2))/x^2/(c^2*x^2+1),x, algorithm="maxima ")
Output:
1/2*c*log(c^2*x^2 + 1) - c*log(x) - 1/2/(c*x^2) + integrate(sqrt(c^2*x^2 + 1)/(c^3*x^5 + c*x^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (50) = 100\).
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.90 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=\frac {1}{2} \, c \log \left (c^{2} x^{2} + 1\right ) + \frac {1}{4} \, {\left ({\left | c \right |} \mathrm {sgn}\left (x\right ) - 2 \, c\right )} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) - \frac {1}{4} \, {\left ({\left | c \right |} \mathrm {sgn}\left (x\right ) + 2 \, c\right )} \log \left (\sqrt {c^{2} x^{2} + 1} - 1\right ) - \frac {\sqrt {c^{2} x^{2} + 1} {\left | c \right |} \mathrm {sgn}\left (x\right ) + c}{2 \, {\left (\sqrt {c^{2} x^{2} + 1} + 1\right )} {\left (\sqrt {c^{2} x^{2} + 1} - 1\right )}} \] Input:
integrate((1/c/x+(1+1/c^2/x^2)^(1/2))/x^2/(c^2*x^2+1),x, algorithm="giac")
Output:
1/2*c*log(c^2*x^2 + 1) + 1/4*(abs(c)*sgn(x) - 2*c)*log(sqrt(c^2*x^2 + 1) + 1) - 1/4*(abs(c)*sgn(x) + 2*c)*log(sqrt(c^2*x^2 + 1) - 1) - 1/2*(sqrt(c^2 *x^2 + 1)*abs(c)*sgn(x) + c)/((sqrt(c^2*x^2 + 1) + 1)*(sqrt(c^2*x^2 + 1) - 1))
Time = 24.60 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=\frac {\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{c^2}}}{x}\right )}{2\,\sqrt {\frac {1}{c^2}}}+\frac {c\,\ln \left (-c^2\,x^2-1\right )}{2}-c\,\ln \left (x\right )-\frac {\sqrt {\frac {1}{c^2\,x^2}+1}}{2\,x}-\frac {1}{2\,c\,x^2} \] Input:
int(((1/(c^2*x^2) + 1)^(1/2) + 1/(c*x))/(x^2*(c^2*x^2 + 1)),x)
Output:
asinh((1/c^2)^(1/2)/x)/(2*(1/c^2)^(1/2)) + (c*log(- c^2*x^2 - 1))/2 - c*lo g(x) - (1/(c^2*x^2) + 1)^(1/2)/(2*x) - 1/(2*c*x^2)
Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.60 \[ \int \frac {e^{\text {csch}^{-1}(c x)}}{x^2 \left (1+c^2 x^2\right )} \, dx=\frac {-\sqrt {c^{2} x^{2}+1}-\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) c^{2} x^{2}+\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) c^{2} x^{2}+\mathrm {log}\left (c^{2} x^{2}+1\right ) c^{2} x^{2}-2 \,\mathrm {log}\left (x \right ) c^{2} x^{2}-1}{2 c \,x^{2}} \] Input:
int((1/c/x+(1+1/c^2/x^2)^(1/2))/x^2/(c^2*x^2+1),x)
Output:
( - sqrt(c**2*x**2 + 1) - log(sqrt(c**2*x**2 + 1) + c*x - 1)*c**2*x**2 + l og(sqrt(c**2*x**2 + 1) + c*x + 1)*c**2*x**2 + log(c**2*x**2 + 1)*c**2*x**2 - 2*log(x)*c**2*x**2 - 1)/(2*c*x**2)