Integrand size = 15, antiderivative size = 83 \[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=-\frac {1}{9 a \left (a-a x^2\right )^{3/2}}-\frac {2}{3 a^2 \sqrt {a-a x^2}}+\frac {x \text {arctanh}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac {2 x \text {arctanh}(x)}{3 a^2 \sqrt {a-a x^2}} \] Output:
-1/9/a/(-a*x^2+a)^(3/2)-2/3/a^2/(-a*x^2+a)^(1/2)+1/3*x*arctanh(x)/a/(-a*x^ 2+a)^(3/2)+2/3*x*arctanh(x)/a^2/(-a*x^2+a)^(1/2)
Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.54 \[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=-\frac {\sqrt {a-a x^2} \left (7-6 x^2+\left (-9 x+6 x^3\right ) \text {arctanh}(x)\right )}{9 a^3 \left (-1+x^2\right )^2} \] Input:
Integrate[ArcTanh[x]/(a - a*x^2)^(5/2),x]
Output:
-1/9*(Sqrt[a - a*x^2]*(7 - 6*x^2 + (-9*x + 6*x^3)*ArcTanh[x]))/(a^3*(-1 + x^2)^2)
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6522, 6520}
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.
\(\displaystyle \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6522 |
\(\displaystyle \frac {2 \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{3/2}}dx}{3 a}+\frac {x \text {arctanh}(x)}{3 a \left (a-a x^2\right )^{3/2}}-\frac {1}{9 a \left (a-a x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6520 |
\(\displaystyle \frac {x \text {arctanh}(x)}{3 a \left (a-a x^2\right )^{3/2}}+\frac {2 \left (\frac {x \text {arctanh}(x)}{a \sqrt {a-a x^2}}-\frac {1}{a \sqrt {a-a x^2}}\right )}{3 a}-\frac {1}{9 a \left (a-a x^2\right )^{3/2}}\) |
Input:
Int[ArcTanh[x]/(a - a*x^2)^(5/2),x]
Output:
-1/9*1/(a*(a - a*x^2)^(3/2)) + (x*ArcTanh[x])/(3*a*(a - a*x^2)^(3/2)) + (2 *(-(1/(a*Sqrt[a - a*x^2])) + (x*ArcTanh[x])/(a*Sqrt[a - a*x^2])))/(3*a)
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symb ol] :> Simp[-b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTanh[c*x])/(d* Sqrt[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTanh[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/( 2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x]) /; Fre eQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]
Time = 0.98 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.02
method | result | size |
risch | \(\frac {x \left (2 x^{2}-3\right ) \ln \left (1+x \right )}{6 a^{2} \left (x^{2}-1\right ) \sqrt {-a \left (x^{2}-1\right )}}-\frac {6 x^{3} \ln \left (1-x \right )+12 x^{2}-9 x \ln \left (1-x \right )-14}{18 a^{2} \left (x^{2}-1\right ) \sqrt {-a \left (x^{2}-1\right )}}\) | \(85\) |
orering | \(\frac {\left (4 x^{5}-\frac {80}{9} x^{3}+\frac {44}{9} x \right ) \operatorname {arctanh}\left (x \right )}{\left (-a \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {\left (6 x^{2}-7\right ) \left (-1+x \right )^{2} \left (1+x \right )^{2} \left (\frac {1}{\left (-x^{2}+1\right ) \left (-a \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {5 \,\operatorname {arctanh}\left (x \right ) a x}{\left (-a \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{9}\) | \(85\) |
default | \(\frac {\left (1+x \right ) \left (-1+3 \,\operatorname {arctanh}\left (x \right )\right ) \sqrt {-\left (-1+x \right ) a \left (1+x \right )}}{72 \left (-1+x \right )^{2} a^{3}}-\frac {3 \left (\operatorname {arctanh}\left (x \right )-1\right ) \sqrt {-\left (-1+x \right ) a \left (1+x \right )}}{8 a^{3} \left (-1+x \right )}-\frac {3 \left (1+\operatorname {arctanh}\left (x \right )\right ) \sqrt {-\left (-1+x \right ) a \left (1+x \right )}}{8 \left (1+x \right ) a^{3}}+\frac {\left (-1+x \right ) \left (1+3 \,\operatorname {arctanh}\left (x \right )\right ) \sqrt {-\left (-1+x \right ) a \left (1+x \right )}}{72 \left (1+x \right )^{2} a^{3}}\) | \(112\) |
Input:
int(arctanh(x)/(-a*x^2+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/6/a^2*x*(2*x^2-3)/(x^2-1)/(-a*(x^2-1))^(1/2)*ln(1+x)-1/18/a^2*(6*x^3*ln( 1-x)+12*x^2-9*x*ln(1-x)-14)/(x^2-1)/(-a*(x^2-1))^(1/2)
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75 \[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\frac {\sqrt {-a x^{2} + a} {\left (12 \, x^{2} - 3 \, {\left (2 \, x^{3} - 3 \, x\right )} \log \left (-\frac {x + 1}{x - 1}\right ) - 14\right )}}{18 \, {\left (a^{3} x^{4} - 2 \, a^{3} x^{2} + a^{3}\right )}} \] Input:
integrate(arctanh(x)/(-a*x^2+a)^(5/2),x, algorithm="fricas")
Output:
1/18*sqrt(-a*x^2 + a)*(12*x^2 - 3*(2*x^3 - 3*x)*log(-(x + 1)/(x - 1)) - 14 )/(a^3*x^4 - 2*a^3*x^2 + a^3)
\[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atanh}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(atanh(x)/(-a*x**2+a)**(5/2),x)
Output:
Integral(atanh(x)/(-a*(x - 1)*(x + 1))**(5/2), x)
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a x^{2} + a} a^{2}} + \frac {x}{{\left (-a x^{2} + a\right )}^{\frac {3}{2}} a}\right )} \operatorname {artanh}\left (x\right ) - \frac {2}{3 \, \sqrt {-a x^{2} + a} a^{2}} - \frac {1}{9 \, {\left (-a x^{2} + a\right )}^{\frac {3}{2}} a} \] Input:
integrate(arctanh(x)/(-a*x^2+a)^(5/2),x, algorithm="maxima")
Output:
1/3*(2*x/(sqrt(-a*x^2 + a)*a^2) + x/((-a*x^2 + a)^(3/2)*a))*arctanh(x) - 2 /3/(sqrt(-a*x^2 + a)*a^2) - 1/9/((-a*x^2 + a)^(3/2)*a)
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=-\frac {\sqrt {-a x^{2} + a} x {\left (\frac {2 \, x^{2}}{a} - \frac {3}{a}\right )} \log \left (-\frac {x + 1}{x - 1}\right )}{6 \, {\left (a x^{2} - a\right )}^{2}} - \frac {6 \, a x^{2} - 7 \, a}{9 \, {\left (a x^{2} - a\right )} \sqrt {-a x^{2} + a} a^{2}} \] Input:
integrate(arctanh(x)/(-a*x^2+a)^(5/2),x, algorithm="giac")
Output:
-1/6*sqrt(-a*x^2 + a)*x*(2*x^2/a - 3/a)*log(-(x + 1)/(x - 1))/(a*x^2 - a)^ 2 - 1/9*(6*a*x^2 - 7*a)/((a*x^2 - a)*sqrt(-a*x^2 + a)*a^2)
Timed out. \[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {atanh}\left (x\right )}{{\left (a-a\,x^2\right )}^{5/2}} \,d x \] Input:
int(atanh(x)/(a - a*x^2)^(5/2),x)
Output:
int(atanh(x)/(a - a*x^2)^(5/2), x)
\[ \int \frac {\text {arctanh}(x)}{\left (a-a x^2\right )^{5/2}} \, dx=\frac {\int \frac {\mathit {atanh} \left (x \right )}{\sqrt {-x^{2}+1}\, x^{4}-2 \sqrt {-x^{2}+1}\, x^{2}+\sqrt {-x^{2}+1}}d x}{\sqrt {a}\, a^{2}} \] Input:
int(atanh(x)/(-a*x^2+a)^(5/2),x)
Output:
int(atanh(x)/(sqrt( - x**2 + 1)*x**4 - 2*sqrt( - x**2 + 1)*x**2 + sqrt( - x**2 + 1)),x)/(sqrt(a)*a**2)