Integrand size = 18, antiderivative size = 81 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {8}{3 c \sqrt {1-\frac {1}{a^2 x^2}} \left (a-\frac {1}{x}\right )}+\frac {4}{3 a^2 c \sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c} \] Output:
8/3/c/(1-1/a^2/x^2)^(1/2)/(a-1/x)+4/3/a^2/c/(1-1/a^2/x^2)^(1/2)/x-arctanh( (1-1/a^2/x^2)^(1/2))/a/c
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\frac {4 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+2 a x)}{(-1+a x)^2}-\frac {3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a}}{3 c} \] Input:
Integrate[E^(3*ArcCoth[a*x])/(c - a*c*x),x]
Output:
((4*Sqrt[1 - 1/(a^2*x^2)]*x*(-1 + 2*a*x))/(-1 + a*x)^2 - (3*Log[a*(1 + Sqr t[1 - 1/(a^2*x^2)])*x])/a)/(3*c)
Time = 0.74 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6724, 27, 570, 532, 25, 2336, 27, 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.
\(\displaystyle \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx\) |
\(\Big \downarrow \) 6724 |
\(\displaystyle a^3 c^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}{c^4 \left (a-\frac {1}{x}\right )^4}d\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^3 \int \frac {\left (1-\frac {1}{a^2 x^2}\right )^{3/2} x}{\left (a-\frac {1}{x}\right )^4}d\frac {1}{x}}{c}\) |
\(\Big \downarrow \) 570 |
\(\displaystyle \frac {\int \frac {\left (a+\frac {1}{x}\right )^4 x}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}}d\frac {1}{x}}{a^5 c}\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {\frac {8 a^3 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {1}{3} \int -\frac {\left (3 a^4+\frac {4 a^3}{x}-\frac {3 a^2}{x^2}\right ) x}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}}{a^5 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{3} \int \frac {\left (3 a^4+\frac {4 a^3}{x}-\frac {3 a^2}{x^2}\right ) x}{\left (1-\frac {1}{a^2 x^2}\right )^{3/2}}d\frac {1}{x}+\frac {8 a^3 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^5 c}\) |
\(\Big \downarrow \) 2336 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {4 a^3}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\int -\frac {3 a^4 x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}\right )+\frac {8 a^3 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^5 c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{3} \left (3 a^4 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x}+\frac {4 a^3}{x \sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {8 a^3 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^5 c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {3}{2} a^4 \int \frac {x}{\sqrt {1-\frac {1}{a^2 x^2}}}d\frac {1}{x^2}+\frac {4 a^3}{x \sqrt {1-\frac {1}{a^2 x^2}}}\right )+\frac {8 a^3 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^5 c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{3} \left (\frac {4 a^3}{x \sqrt {1-\frac {1}{a^2 x^2}}}-3 a^6 \int \frac {1}{a^2-a^2 \sqrt {1-\frac {1}{a^2 x^2}}}d\sqrt {1-\frac {1}{a^2 x^2}}\right )+\frac {8 a^3 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}}{a^5 c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {8 a^3 \left (a+\frac {1}{x}\right )}{3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {1}{3} \left (\frac {4 a^3}{x \sqrt {1-\frac {1}{a^2 x^2}}}-3 a^4 \text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\right )}{a^5 c}\) |
Input:
Int[E^(3*ArcCoth[a*x])/(c - a*c*x),x]
Output:
((8*a^3*(a + x^(-1)))/(3*(1 - 1/(a^2*x^2))^(3/2)) + ((4*a^3)/(Sqrt[1 - 1/( a^2*x^2)]*x) - 3*a^4*ArcTanh[Sqrt[1 - 1/(a^2*x^2)]])/3)/(a^5*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[(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[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[-d^n Subst[Int[(d + c*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(p + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d}, x] && EqQ[a*c + d, 0] && IntegerQ[p] && In tegerQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(344\) vs. \(2(71)=142\).
Time = 0.13 (sec) , antiderivative size = 345, normalized size of antiderivative = 4.26
method | result | size |
default | \(-\frac {3 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+3 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{3} x^{3}-9 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-3 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} a x -9 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+9 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x +\left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+9 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, a x -3 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-3 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{3 a \sqrt {a^{2}}\, \left (a x -1\right ) c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(345\) |
Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x,method=_RETURNVERBOSE)
Output:
-1/3/a*(3*ln((a^2*x+((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^4* x^3+3*(a^2)^(1/2)*((a*x-1)*(a*x+1))^(1/2)*a^3*x^3-9*ln((a^2*x+((a*x-1)*(a* x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^3*x^2-3*(a^2)^(1/2)*((a*x-1)*(a*x+ 1))^(3/2)*a*x-9*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*a^2*x^2+9*ln((a^2*x+(( a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))*a^2*x+((a*x-1)*(a*x+1))^(3 /2)*(a^2)^(1/2)+9*((a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2)*a*x-3*a*ln((a^2*x+(( a*x-1)*(a*x+1))^(1/2)*(a^2)^(1/2))/(a^2)^(1/2))-3*((a*x-1)*(a*x+1))^(1/2)* (a^2)^(1/2))/(a^2)^(1/2)/(a*x-1)/c/((a*x-1)*(a*x+1))^(1/2)/(a*x+1)/((a*x-1 )/(a*x+1))^(3/2)
Time = 0.08 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.48 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - 4 \, {\left (2 \, a^{2} x^{2} + a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x, algorithm="fricas")
Output:
-1/3*(3*(a^2*x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 3*(a^2* x^2 - 2*a*x + 1)*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - 4*(2*a^2*x^2 + a*x - 1)*sqrt((a*x - 1)/(a*x + 1)))/(a^3*c*x^2 - 2*a^2*c*x + a*c)
\[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=- \frac {\int \frac {1}{\frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(3/2)/(-a*c*x+c),x)
Output:
-Integral(1/(a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - 2*a*x *sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) + sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x)/c
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.17 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=-\frac {1}{3} \, a {\left (\frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac {2 \, {\left (\frac {3 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x, algorithm="maxima")
Output:
-1/3*a*(3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/(a^2*c) - 3*log(sqrt((a*x - 1 )/(a*x + 1)) - 1)/(a^2*c) - 2*(3*(a*x - 1)/(a*x + 1) + 1)/(a^2*c*((a*x - 1 )/(a*x + 1))^(3/2)))
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.43 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{c {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x, algorithm="giac")
Output:
log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(c*abs(a)*sgn(a*x + 1))
Time = 13.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.78 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {\frac {2\,\left (a\,x-1\right )}{a\,x+1}+\frac {2}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c} \] Input:
int(1/((c - a*c*x)*((a*x - 1)/(a*x + 1))^(3/2)),x)
Output:
((2*(a*x - 1))/(a*x + 1) + 2/3)/(a*c*((a*x - 1)/(a*x + 1))^(3/2)) - (2*ata nh(((a*x - 1)/(a*x + 1))^(1/2)))/(a*c)
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.21 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx=\frac {-2 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right ) a x +2 \sqrt {a x -1}\, \mathrm {log}\left (\frac {\sqrt {a x -1}+\sqrt {a x +1}}{\sqrt {2}}\right )+\frac {8 \sqrt {a x +1}\, a x}{3}-\frac {4 \sqrt {a x +1}}{3}}{\sqrt {a x -1}\, a c \left (a x -1\right )} \] Input:
int(1/((a*x-1)/(a*x+1))^(3/2)/(-a*c*x+c),x)
Output:
(2*( - 3*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2))*a*x + 3*sqrt(a*x - 1)*log((sqrt(a*x - 1) + sqrt(a*x + 1))/sqrt(2)) + 4*sqrt(a*x + 1)*a*x - 2*sqrt(a*x + 1)))/(3*sqrt(a*x - 1)*a*c*(a*x - 1))