Integrand size = 16, antiderivative size = 180 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {a \left (1-a^2 x^2\right )}{3 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {2 \sqrt {-1+a^2 x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-1+a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{3 c^2 \sqrt {d} \sqrt {-1+a x} \sqrt {1+a x}} \]
1/3*x*arccosh(a*x)/c/(d*x^2+c)^(3/2)-2/3*arctanh(d^(1/2)*(a^2*x^2-1)^(1/2) /a/(d*x^2+c)^(1/2))*(a^2*x^2-1)^(1/2)/c^2/d^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1 /2)+2/3*x*arccosh(a*x)/c^2/(d*x^2+c)^(1/2)+1/3*a*(-a^2*x^2+1)/c/(a^2*c+d)/ (a*x-1)^(1/2)/(a*x+1)^(1/2)/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.56 (sec) , antiderivative size = 609, normalized size of antiderivative = 3.38 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\frac {-\frac {a c \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )}{a^2 c+d}+x \left (3 c+2 d x^2\right ) \text {arccosh}(a x)+\frac {4 (-1+a x)^{3/2} \sqrt {\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) (1+a x)}{\left (a \sqrt {c}+i \sqrt {d}\right ) (-1+a x)}} \left (c+d x^2\right ) \left (\frac {a \left (-i a \sqrt {c}+\sqrt {d}\right ) \left (i \sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {1+\frac {i a \sqrt {c}}{\sqrt {d}}-a x+\frac {i \sqrt {d} x}{\sqrt {c}}}{1-a x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{2-2 a x}}\right ),\frac {4 i a \sqrt {c} \sqrt {d}}{\left (a \sqrt {c}+i \sqrt {d}\right )^2}\right )}{-1+a x}+a \sqrt {c} \left (-a \sqrt {c}+i \sqrt {d}\right ) \sqrt {\frac {\left (a^2 c+d\right ) \left (c+d x^2\right )}{c d (-1+a x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{1-a x}} \operatorname {EllipticPi}\left (\frac {2 a \sqrt {c}}{a \sqrt {c}+i \sqrt {d}},\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{2-2 a x}}\right ),\frac {4 i a \sqrt {c} \sqrt {d}}{\left (a \sqrt {c}+i \sqrt {d}\right )^2}\right )\right )}{a \left (a^2 c+d\right ) \sqrt {1+a x} \sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{1-a x}}}}{3 c^2 \left (c+d x^2\right )^{3/2}} \]
(-((a*c*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c + d*x^2))/(a^2*c + d)) + x*(3*c + 2*d*x^2)*ArcCosh[a*x] + (4*(-1 + a*x)^(3/2)*Sqrt[((a*Sqrt[c] - I*Sqrt[d])* (1 + a*x))/((a*Sqrt[c] + I*Sqrt[d])*(-1 + a*x))]*(c + d*x^2)*((a*((-I)*a*S qrt[c] + Sqrt[d])*(I*Sqrt[c] + Sqrt[d]*x)*Sqrt[(1 + (I*a*Sqrt[c])/Sqrt[d] - a*x + (I*Sqrt[d]*x)/Sqrt[c])/(1 - a*x)]*EllipticF[ArcSin[Sqrt[-((-1 + (I *Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(2 - 2*a*x))]], ((4*I)* a*Sqrt[c]*Sqrt[d])/(a*Sqrt[c] + I*Sqrt[d])^2])/(-1 + a*x) + a*Sqrt[c]*(-(a *Sqrt[c]) + I*Sqrt[d])*Sqrt[((a^2*c + d)*(c + d*x^2))/(c*d*(-1 + a*x)^2)]* Sqrt[-((-1 + (I*Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(1 - a*x ))]*EllipticPi[(2*a*Sqrt[c])/(a*Sqrt[c] + I*Sqrt[d]), ArcSin[Sqrt[-((-1 + (I*Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(2 - 2*a*x))]], ((4*I )*a*Sqrt[c]*Sqrt[d])/(a*Sqrt[c] + I*Sqrt[d])^2]))/(a*(a^2*c + d)*Sqrt[1 + a*x]*Sqrt[-((-1 + (I*Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(1 - a*x))]))/(3*c^2*(c + d*x^2)^(3/2))
Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6323, 27, 1076, 435, 87, 66, 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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6323 |
| \(\displaystyle -a \int \frac {x \left (2 d x^2+3 c\right )}{3 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (d x^2+c\right )^{3/2}}dx+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a \int \frac {x \left (2 d x^2+3 c\right )}{\sqrt {a x-1} \sqrt {a x+1} \left (d x^2+c\right )^{3/2}}dx}{3 c^2}+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1076 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {x \left (2 d x^2+3 c\right )}{\sqrt {a^2 x^2-1} \left (d x^2+c\right )^{3/2}}dx}{3 c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \int \frac {2 d x^2+3 c}{\sqrt {a^2 x^2-1} \left (d x^2+c\right )^{3/2}}dx^2}{6 c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \left (2 \int \frac {1}{\sqrt {a^2 x^2-1} \sqrt {d x^2+c}}dx^2+\frac {2 c \sqrt {a^2 x^2-1}}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \left (4 \int \frac {1}{a^2-d x^4}d\frac {\sqrt {a^2 x^2-1}}{\sqrt {d x^2+c}}+\frac {2 c \sqrt {a^2 x^2-1}}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {a \sqrt {a^2 x^2-1} \left (\frac {4 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{a \sqrt {d}}+\frac {2 c \sqrt {a^2 x^2-1}}{\left (a^2 c+d\right ) \sqrt {c+d x^2}}\right )}{6 c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 x \text {arccosh}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{3 c \left (c+d x^2\right )^{3/2}}\) |
(x*ArcCosh[a*x])/(3*c*(c + d*x^2)^(3/2)) + (2*x*ArcCosh[a*x])/(3*c^2*Sqrt[ c + d*x^2]) - (a*Sqrt[-1 + a^2*x^2]*((2*c*Sqrt[-1 + a^2*x^2])/((a^2*c + d) *Sqrt[c + d*x^2]) + (4*ArcTanh[(Sqrt[d]*Sqrt[-1 + a^2*x^2])/(a*Sqrt[c + d* x^2])])/(a*Sqrt[d])))/(6*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
3.1.50.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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[((g_.)*(x_))^(m_.)*((e1_) + (f1_.)*(x_)^(n2_.))^(r_.)*((e2_) + (f2_.)*( x_)^(n2_.))^(r_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^( q_.), x_Symbol] :> Simp[(e1 + f1*x^(n/2))^FracPart[r]*((e2 + f2*x^(n/2))^Fr acPart[r]/(e1*e2 + f1*f2*x^n)^FracPart[r]) Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e1*e2 + f1*f2*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e1, f1, e2, f 2, g, m, n, p, q, r}, x] && EqQ[n2, n/2] && EqQ[e2*f1 + e1*f2, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x] , x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])
\[\int \frac {\operatorname {arccosh}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (147) = 294\).
Time = 0.31 (sec) , antiderivative size = 613, normalized size of antiderivative = 3.41 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\left [\frac {{\left (a^{2} c^{3} + {\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (8 \, a^{4} d^{2} x^{4} + a^{4} c^{2} - 6 \, a^{2} c d + 8 \, {\left (a^{4} c d - a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{3} d x^{2} + a^{3} c - a d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c} \sqrt {d} + d^{2}\right ) + 2 \, {\left (2 \, {\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \sqrt {d x^{2} + c} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c}}{6 \, {\left (a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, \frac {{\left (a^{2} c^{3} + {\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {{\left (2 \, a^{2} d x^{2} + a^{2} c - d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c} \sqrt {-d}}{2 \, {\left (a^{3} d^{2} x^{4} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (2 \, {\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \sqrt {d x^{2} + c} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c}}{3 \, {\left (a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}\right ] \]
[1/6*((a^2*c^3 + (a^2*c*d^2 + d^3)*x^4 + c^2*d + 2*(a^2*c^2*d + c*d^2)*x^2 )*sqrt(d)*log(8*a^4*d^2*x^4 + a^4*c^2 - 6*a^2*c*d + 8*(a^4*c*d - a^2*d^2)* x^2 - 4*(2*a^3*d*x^2 + a^3*c - a*d)*sqrt(a^2*x^2 - 1)*sqrt(d*x^2 + c)*sqrt (d) + d^2) + 2*(2*(a^2*c*d^2 + d^3)*x^3 + 3*(a^2*c^2*d + c*d^2)*x)*sqrt(d* x^2 + c)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(a*c*d^2*x^2 + a*c^2*d)*sqrt(a^2 *x^2 - 1)*sqrt(d*x^2 + c))/(a^2*c^5*d + c^4*d^2 + (a^2*c^3*d^3 + c^2*d^4)* x^4 + 2*(a^2*c^4*d^2 + c^3*d^3)*x^2), 1/3*((a^2*c^3 + (a^2*c*d^2 + d^3)*x^ 4 + c^2*d + 2*(a^2*c^2*d + c*d^2)*x^2)*sqrt(-d)*arctan(1/2*(2*a^2*d*x^2 + a^2*c - d)*sqrt(a^2*x^2 - 1)*sqrt(d*x^2 + c)*sqrt(-d)/(a^3*d^2*x^4 - a*c*d + (a^3*c*d - a*d^2)*x^2)) + (2*(a^2*c*d^2 + d^3)*x^3 + 3*(a^2*c^2*d + c*d ^2)*x)*sqrt(d*x^2 + c)*log(a*x + sqrt(a^2*x^2 - 1)) - (a*c*d^2*x^2 + a*c^2 *d)*sqrt(a^2*x^2 - 1)*sqrt(d*x^2 + c))/(a^2*c^5*d + c^4*d^2 + (a^2*c^3*d^3 + c^2*d^4)*x^4 + 2*(a^2*c^4*d^2 + c^3*d^3)*x^2)]
\[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(d-a^2*c>0)', see `assume?` for m ore detail
Time = 0.34 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=-\frac {1}{3} \, {\left (\frac {\sqrt {a^{2} x^{2} - 1} a^{2} c^{3} {\left | a \right |}}{{\left (a^{4} c^{5} + a^{2} c^{4} d\right )} \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d}} - \frac {2 \, {\left | a \right |} \log \left ({\left | -\sqrt {a^{2} x^{2} - 1} \sqrt {d} + \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d} \right |}\right )}{a^{2} c^{2} \sqrt {d}}\right )} a + \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \]
-1/3*(sqrt(a^2*x^2 - 1)*a^2*c^3*abs(a)/((a^4*c^5 + a^2*c^4*d)*sqrt(a^2*c + (a^2*x^2 - 1)*d + d)) - 2*abs(a)*log(abs(-sqrt(a^2*x^2 - 1)*sqrt(d) + sqr t(a^2*c + (a^2*x^2 - 1)*d + d)))/(a^2*c^2*sqrt(d)))*a + 1/3*x*(2*d*x^2/c^2 + 3/c)*log(a*x + sqrt(a^2*x^2 - 1))/(d*x^2 + c)^(3/2)
Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \]