Integrand size = 25, antiderivative size = 187 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2}}{3 x^3 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{2 x^2 \sqrt {c-a^2 c x^2}}-\frac {a^2 \sqrt {1-a^2 x^2}}{x \sqrt {c-a^2 c x^2}}+\frac {a^3 \sqrt {1-a^2 x^2} \log (x)}{\sqrt {c-a^2 c x^2}}-\frac {a^3 \sqrt {1-a^2 x^2} \log (1-a x)}{\sqrt {c-a^2 c x^2}} \] Output:
-1/3*(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^(1/2)-1/2*a*(-a^2*x^2+1)^(1/2)/ x^2/(-a^2*c*x^2+c)^(1/2)-a^2*(-a^2*x^2+1)^(1/2)/x/(-a^2*c*x^2+c)^(1/2)+a^3 *(-a^2*x^2+1)^(1/2)*ln(x)/(-a^2*c*x^2+c)^(1/2)-a^3*(-a^2*x^2+1)^(1/2)*ln(- a*x+1)/(-a^2*c*x^2+c)^(1/2)
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-\frac {1}{3 x^3}-\frac {a}{2 x^2}-\frac {a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)\right )}{\sqrt {c-a^2 c x^2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x^4*Sqrt[c - a^2*c*x^2]),x]
Output:
(Sqrt[1 - a^2*x^2]*(-1/3*1/x^3 - a/(2*x^2) - a^2/x + a^3*Log[x] - a^3*Log[ 1 - a*x]))/Sqrt[c - a^2*c*x^2]
Time = 0.47 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.39, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6703, 6700, 54, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {1-a^2 x^2}}dx}{\sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {1}{x^4 (1-a x)}dx}{\sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \left (-\frac {a^4}{a x-1}+\frac {a^3}{x}+\frac {a^2}{x^2}+\frac {a}{x^3}+\frac {1}{x^4}\right )dx}{\sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (a^3 \log (x)-a^3 \log (1-a x)-\frac {a^2}{x}-\frac {a}{2 x^2}-\frac {1}{3 x^3}\right )}{\sqrt {c-a^2 c x^2}}\) |
Input:
Int[E^ArcTanh[a*x]/(x^4*Sqrt[c - a^2*c*x^2]),x]
Output:
(Sqrt[1 - a^2*x^2]*(-1/3*1/x^3 - a/(2*x^2) - a^2/x + a^3*Log[x] - a^3*Log[ 1 - a*x]))/Sqrt[c - a^2*c*x^2]
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[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.39
method | result | size |
default | \(-\frac {\left (6 a^{3} \ln \left (a x -1\right ) x^{3}-6 \ln \left (x \right ) x^{3} a^{3}+6 a^{2} x^{2}+3 a x +2\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{6 \sqrt {-a^{2} x^{2}+1}\, c \,x^{3}}\) | \(73\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^(1/2),x,method=_RETURNVE RBOSE)
Output:
-1/6*(6*a^3*ln(a*x-1)*x^3-6*ln(x)*x^3*a^3+6*a^2*x^2+3*a*x+2)/(-a^2*x^2+1)^ (1/2)*(-c*(a^2*x^2-1))^(1/2)/c/x^3
Time = 0.12 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.58 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=\left [\frac {3 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt {c} \log \left (-\frac {4 \, a^{5} c x^{5} - {\left (2 \, a^{6} - 4 \, a^{5} + 6 \, a^{4} - 4 \, a^{3} + a^{2}\right )} c x^{6} - {\left (4 \, a^{4} + 4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} c x^{4} + 5 \, a^{2} c x^{2} - 4 \, a c x + {\left (4 \, a^{3} x^{3} - {\left (4 \, a^{3} - 6 \, a^{2} + 4 \, a - 1\right )} x^{4} - 6 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} + c}{a^{4} x^{6} - 2 \, a^{3} x^{5} + 2 \, a x^{3} - x^{2}}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (6 \, a^{2} x^{2} - {\left (6 \, a^{2} + 3 \, a + 2\right )} x^{3} + 3 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} c x^{5} - c x^{3}\right )}}, -\frac {6 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} \sqrt {-c} \arctan \left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a^{2} - 2 \, a + 1\right )} x^{2} - 2 \, a x + 1\right )} \sqrt {-c}}{2 \, a^{3} c x^{3} - {\left (2 \, a^{3} - a^{2}\right )} c x^{4} - {\left (a^{2} - 2 \, a + 1\right )} c x^{2} - 2 \, a c x + c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (6 \, a^{2} x^{2} - {\left (6 \, a^{2} + 3 \, a + 2\right )} x^{3} + 3 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{2} c x^{5} - c x^{3}\right )}}\right ] \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^(1/2),x, algorithm ="fricas")
Output:
[1/6*(3*(a^5*x^5 - a^3*x^3)*sqrt(c)*log(-(4*a^5*c*x^5 - (2*a^6 - 4*a^5 + 6 *a^4 - 4*a^3 + a^2)*c*x^6 - (4*a^4 + 4*a^3 - 6*a^2 + 4*a - 1)*c*x^4 + 5*a^ 2*c*x^2 - 4*a*c*x + (4*a^3*x^3 - (4*a^3 - 6*a^2 + 4*a - 1)*x^4 - 6*a^2*x^2 + 4*a*x - 1)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) + c)/(a^4*x^ 6 - 2*a^3*x^5 + 2*a*x^3 - x^2)) + sqrt(-a^2*c*x^2 + c)*(6*a^2*x^2 - (6*a^2 + 3*a + 2)*x^3 + 3*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2*c*x^5 - c*x^3), -1/6 *(6*(a^5*x^5 - a^3*x^3)*sqrt(-c)*arctan(-sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^ 2 + 1)*((2*a^2 - 2*a + 1)*x^2 - 2*a*x + 1)*sqrt(-c)/(2*a^3*c*x^3 - (2*a^3 - a^2)*c*x^4 - (a^2 - 2*a + 1)*c*x^2 - 2*a*c*x + c)) - sqrt(-a^2*c*x^2 + c )*(6*a^2*x^2 - (6*a^2 + 3*a + 2)*x^3 + 3*a*x + 2)*sqrt(-a^2*x^2 + 1))/(a^2 *c*x^5 - c*x^3)]
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=\int \frac {a x + 1}{x^{4} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**4/(-a**2*c*x**2+c)**(1/2),x)
Output:
Integral((a*x + 1)/(x**4*sqrt(-(a*x - 1)*(a*x + 1))*sqrt(-c*(a*x - 1)*(a*x + 1))), x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{4}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^(1/2),x, algorithm ="maxima")
Output:
integrate((a*x + 1)/(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^4), x)
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.26 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=-\frac {a^{3} \log \left ({\left | -a x + 1 \right |}\right )}{\sqrt {c}} + \frac {a^{3} \log \left ({\left | x \right |}\right )}{\sqrt {c}} - \frac {6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, \sqrt {c} x^{3}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^(1/2),x, algorithm ="giac")
Output:
-a^3*log(abs(-a*x + 1))/sqrt(c) + a^3*log(abs(x))/sqrt(c) - 1/6*(6*a^2*x^2 + 3*a*x + 2)/(sqrt(c)*x^3)
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=\int \frac {a\,x+1}{x^4\,\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((a*x + 1)/(x^4*(c - a^2*c*x^2)^(1/2)*(1 - a^2*x^2)^(1/2)),x)
Output:
int((a*x + 1)/(x^4*(c - a^2*c*x^2)^(1/2)*(1 - a^2*x^2)^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.26 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {c}\, \left (-6 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+6 \,\mathrm {log}\left (x \right ) a^{3} x^{3}-6 a^{2} x^{2}-3 a x -2\right )}{6 c \,x^{3}} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^4/(-a^2*c*x^2+c)^(1/2),x)
Output:
(sqrt(c)*( - 6*log(a*x - 1)*a**3*x**3 + 6*log(x)*a**3*x**3 - 6*a**2*x**2 - 3*a*x - 2))/(6*c*x**3)