Integrand size = 16, antiderivative size = 128 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {a}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {\left (3 a^2 c+2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{3 c^2 \left (a^2 c+d\right )^{3/2}} \]
1/3*x*arccoth(a*x)/c/(d*x^2+c)^(3/2)-1/3*(3*a^2*c+2*d)*arctanh(a*(d*x^2+c) ^(1/2)/(a^2*c+d)^(1/2))/c^2/(a^2*c+d)^(3/2)+1/3*a/c/(a^2*c+d)/(d*x^2+c)^(1 /2)+2/3*x*arccoth(a*x)/c^2/(d*x^2+c)^(1/2)
Time = 0.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.77 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {\frac {2 a c}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {2 x \left (3 c+2 d x^2\right ) \coth ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}}+\frac {\left (3 a^2 c+2 d\right ) \log (1-a x)}{\left (a^2 c+d\right )^{3/2}}+\frac {\left (3 a^2 c+2 d\right ) \log (1+a x)}{\left (a^2 c+d\right )^{3/2}}-\frac {\left (3 a^2 c+2 d\right ) \log \left (a c-d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{\left (a^2 c+d\right )^{3/2}}-\frac {\left (3 a^2 c+2 d\right ) \log \left (a c+d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{\left (a^2 c+d\right )^{3/2}}}{6 c^2} \]
((2*a*c)/((a^2*c + d)*Sqrt[c + d*x^2]) + (2*x*(3*c + 2*d*x^2)*ArcCoth[a*x] )/(c + d*x^2)^(3/2) + ((3*a^2*c + 2*d)*Log[1 - a*x])/(a^2*c + d)^(3/2) + ( (3*a^2*c + 2*d)*Log[1 + a*x])/(a^2*c + d)^(3/2) - ((3*a^2*c + 2*d)*Log[a*c - d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(a^2*c + d)^(3/2) - ((3*a^2*c + 2*d)*Log[a*c + d*x + Sqrt[a^2*c + d]*Sqrt[c + d*x^2]])/(a^2*c + d)^(3/2)) /(6*c^2)
Time = 0.35 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6539, 27, 435, 87, 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.
| \(\displaystyle \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6539 |
| \(\displaystyle -a \int \frac {x \left (2 d x^2+3 c\right )}{3 c^2 \left (1-a^2 x^2\right ) \left (d x^2+c\right )^{3/2}}dx+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int \frac {x \left (2 d x^2+3 c\right )}{\left (1-a^2 x^2\right ) \left (d x^2+c\right )^{3/2}}dx}{3 c^2}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {a \int \frac {2 d x^2+3 c}{\left (1-a^2 x^2\right ) \left (d x^2+c\right )^{3/2}}dx^2}{6 c^2}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {a \left (\frac {\left (3 a^2 c+2 d\right ) \int \frac {1}{\left (1-a^2 x^2\right ) \sqrt {d x^2+c}}dx^2}{a^2 c+d}-\frac {2 c}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {a \left (\frac {2 \left (3 a^2 c+2 d\right ) \int \frac {1}{-\frac {a^2 x^4}{d}+\frac {a^2 c}{d}+1}d\sqrt {d x^2+c}}{d \left (a^2 c+d\right )}-\frac {2 c}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \left (\frac {2 \left (3 a^2 c+2 d\right ) \text {arctanh}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{a \left (a^2 c+d\right )^{3/2}}-\frac {2 c}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2}+\frac {2 x \coth ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
(x*ArcCoth[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCoth[a*x])/(3*c^2*Sqrt[ c + d*x^2]) - (a*((-2*c)/((a^2*c + d)*Sqrt[c + d*x^2]) + (2*(3*a^2*c + 2*d )*ArcTanh[(a*Sqrt[c + d*x^2])/Sqrt[a^2*c + d]])/(a*(a^2*c + d)^(3/2))))/(6 *c^2)
3.1.45.3.1 Defintions of rubi rules used
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Sym bol] :> With[{u = IntHide[(d + e*x^2)^q, x]}, Simp[(a + b*ArcCoth[c*x]) u , x] - Simp[b*c Int[SimplifyIntegrand[u/(1 - c^2*x^2), x], x], x]] /; Fre eQ[{a, b, c, d, e}, x] && (IntegerQ[q] || ILtQ[q + 1/2, 0])
\[\int \frac {\operatorname {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 345 vs. \(2 (108) = 216\).
Time = 0.30 (sec) , antiderivative size = 728, normalized size of antiderivative = 5.69 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {{\left (3 \, a^{2} c^{3} + {\left (3 \, a^{2} c d^{2} + 2 \, d^{3}\right )} x^{4} + 2 \, c^{2} d + 2 \, {\left (3 \, a^{2} c^{2} d + 2 \, c d^{2}\right )} x^{2}\right )} \sqrt {a^{2} c + d} \log \left (\frac {a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} + 8 \, a^{2} c d + 2 \, {\left (4 \, a^{4} c d + 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} d x^{2} + 2 \, a^{3} c + a d\right )} \sqrt {a^{2} c + d} \sqrt {d x^{2} + c} + d^{2}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1}\right ) + 2 \, {\left (2 \, a^{3} c^{3} + 2 \, a c^{2} d + 2 \, {\left (a^{3} c^{2} d + a c d^{2}\right )} x^{2} + {\left (2 \, {\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a^{4} c^{6} + 2 \, a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{4} c^{4} d^{2} + 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{4} c^{5} d + 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, \frac {{\left (3 \, a^{2} c^{3} + {\left (3 \, a^{2} c d^{2} + 2 \, d^{3}\right )} x^{4} + 2 \, c^{2} d + 2 \, {\left (3 \, a^{2} c^{2} d + 2 \, c d^{2}\right )} x^{2}\right )} \sqrt {-a^{2} c - d} \arctan \left (\frac {{\left (a^{2} d x^{2} + 2 \, a^{2} c + d\right )} \sqrt {-a^{2} c - d} \sqrt {d x^{2} + c}}{2 \, {\left (a^{3} c^{2} + a c d + {\left (a^{3} c d + a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, a^{3} c^{3} + 2 \, a c^{2} d + 2 \, {\left (a^{3} c^{2} d + a c d^{2}\right )} x^{2} + {\left (2 \, {\left (a^{4} c^{2} d + 2 \, a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} x\right )} \log \left (\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {d x^{2} + c}}{6 \, {\left (a^{4} c^{6} + 2 \, a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{4} c^{4} d^{2} + 2 \, a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{4} c^{5} d + 2 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}\right ] \]
[1/12*((3*a^2*c^3 + (3*a^2*c*d^2 + 2*d^3)*x^4 + 2*c^2*d + 2*(3*a^2*c^2*d + 2*c*d^2)*x^2)*sqrt(a^2*c + d)*log((a^4*d^2*x^4 + 8*a^4*c^2 + 8*a^2*c*d + 2*(4*a^4*c*d + 3*a^2*d^2)*x^2 - 4*(a^3*d*x^2 + 2*a^3*c + a*d)*sqrt(a^2*c + d)*sqrt(d*x^2 + c) + d^2)/(a^4*x^4 - 2*a^2*x^2 + 1)) + 2*(2*a^3*c^3 + 2*a *c^2*d + 2*(a^3*c^2*d + a*c*d^2)*x^2 + (2*(a^4*c^2*d + 2*a^2*c*d^2 + d^3)* x^3 + 3*(a^4*c^3 + 2*a^2*c^2*d + c*d^2)*x)*log((a*x + 1)/(a*x - 1)))*sqrt( d*x^2 + c))/(a^4*c^6 + 2*a^2*c^5*d + c^4*d^2 + (a^4*c^4*d^2 + 2*a^2*c^3*d^ 3 + c^2*d^4)*x^4 + 2*(a^4*c^5*d + 2*a^2*c^4*d^2 + c^3*d^3)*x^2), 1/6*((3*a ^2*c^3 + (3*a^2*c*d^2 + 2*d^3)*x^4 + 2*c^2*d + 2*(3*a^2*c^2*d + 2*c*d^2)*x ^2)*sqrt(-a^2*c - d)*arctan(1/2*(a^2*d*x^2 + 2*a^2*c + d)*sqrt(-a^2*c - d) *sqrt(d*x^2 + c)/(a^3*c^2 + a*c*d + (a^3*c*d + a*d^2)*x^2)) + (2*a^3*c^3 + 2*a*c^2*d + 2*(a^3*c^2*d + a*c*d^2)*x^2 + (2*(a^4*c^2*d + 2*a^2*c*d^2 + d ^3)*x^3 + 3*(a^4*c^3 + 2*a^2*c^2*d + c*d^2)*x)*log((a*x + 1)/(a*x - 1)))*s qrt(d*x^2 + c))/(a^4*c^6 + 2*a^2*c^5*d + c^4*d^2 + (a^4*c^4*d^2 + 2*a^2*c^ 3*d^3 + c^2*d^4)*x^4 + 2*(a^4*c^5*d + 2*a^2*c^4*d^2 + c^3*d^3)*x^2)]
\[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (108) = 216\).
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.74 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{6} \, a {\left (\frac {\frac {a d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{2} c^{2} + c d\right )} \sqrt {a^{2} c + d}} + \frac {2 \, d}{{\left (a^{2} c^{2} + c d\right )} \sqrt {d x^{2} + c}}}{d} + \frac {2 \, \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{\sqrt {a^{2} c + d} a c^{2}}\right )} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {d x^{2} + c} c^{2}} + \frac {x}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \operatorname {arcoth}\left (a x\right ) \]
1/6*a*((a*d*log((sqrt(d*x^2 + c)*a^2 - sqrt(a^2*c + d)*a)/(sqrt(d*x^2 + c) *a^2 + sqrt(a^2*c + d)*a))/((a^2*c^2 + c*d)*sqrt(a^2*c + d)) + 2*d/((a^2*c ^2 + c*d)*sqrt(d*x^2 + c)))/d + 2*log((sqrt(d*x^2 + c)*a^2 - sqrt(a^2*c + d)*a)/(sqrt(d*x^2 + c)*a^2 + sqrt(a^2*c + d)*a))/(sqrt(a^2*c + d)*a*c^2)) + 1/3*(2*x/(sqrt(d*x^2 + c)*c^2) + x/((d*x^2 + c)^(3/2)*c))*arccoth(a*x)
Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.12 \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {1}{3} \, a {\left (\frac {{\left (3 \, a^{2} c + 2 \, d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c - d}}\right )}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {-a^{2} c - d} a} + \frac {1}{{\left (a^{2} c^{2} + c d\right )} \sqrt {d x^{2} + c}}\right )} + \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )} \log \left (-\frac {\frac {1}{a x} + 1}{\frac {1}{a x} - 1}\right )}{6 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
1/3*a*((3*a^2*c + 2*d)*arctan(sqrt(d*x^2 + c)*a/sqrt(-a^2*c - d))/((a^2*c^ 3 + c^2*d)*sqrt(-a^2*c - d)*a) + 1/((a^2*c^2 + c*d)*sqrt(d*x^2 + c))) + 1/ 6*x*(2*d*x^2/c^2 + 3/c)*log(-(1/(a*x) + 1)/(1/(a*x) - 1))/(d*x^2 + c)^(3/2 )
Timed out. \[ \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]