Integrand size = 16, antiderivative size = 253 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\frac {2 x^2-d x^4}{3 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} d x}-\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{3 b^{5/2} d x} \]
1/6*cosh(1/2*arccosh(d*x^2-1))*erfi(1/2*(a+b*arccosh(d*x^2-1))^(1/2)*2^(1/ 2)/b^(1/2))*(cosh(1/2*a/b)-sinh(1/2*a/b))*2^(1/2)*Pi^(1/2)/b^(5/2)/d/x-1/6 *cosh(1/2*arccosh(d*x^2-1))*erf(1/2*(a+b*arccosh(d*x^2-1))^(1/2)*2^(1/2)/b ^(1/2))*(cosh(1/2*a/b)+sinh(1/2*a/b))*2^(1/2)*Pi^(1/2)/b^(5/2)/d/x+1/3*(-d *x^4+2*x^2)/b/x/(a+b*arccosh(d*x^2-1))^(3/2)/(d*x^2)^(1/2)/(d*x^2-2)^(1/2) -1/3*x/b^2/(a+b*arccosh(d*x^2-1))^(1/2)
Time = 0.65 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=-\frac {\cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (\sqrt {2 \pi } \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (-\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {2 \pi } \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {b} \left (\left (a+b \text {arccosh}\left (-1+d x^2\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )+b \sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )\right )\right )}{6 b^{5/2} d x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}} \]
-1/6*(Cosh[ArcCosh[-1 + d*x^2]/2]*(Sqrt[2*Pi]*(a + b*ArcCosh[-1 + d*x^2])^ (3/2)*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(-Cosh[a/(2* b)] + Sinh[a/(2*b)]) + Sqrt[2*Pi]*(a + b*ArcCosh[-1 + d*x^2])^(3/2)*Erf[Sq rt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/( 2*b)]) + 4*Sqrt[b]*((a + b*ArcCosh[-1 + d*x^2])*Cosh[ArcCosh[-1 + d*x^2]/2 ] + b*Sinh[ArcCosh[-1 + d*x^2]/2])))/(b^(5/2)*d*x*(a + b*ArcCosh[-1 + d*x^ 2])^(3/2))
Time = 0.37 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6425, 6420}
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.
\(\displaystyle \int \frac {1}{\left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6425 |
\(\displaystyle \frac {\int \frac {1}{\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}dx}{3 b^2}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}+\frac {2 x^2-d x^4}{3 b x \sqrt {d x^2} \sqrt {d x^2-2} \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{3/2}}\) |
\(\Big \downarrow \) 6420 |
\(\displaystyle \frac {\frac {\sqrt {\frac {\pi }{2}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x}-\frac {\sqrt {\frac {\pi }{2}} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b} d x}}{3 b^2}-\frac {x}{3 b^2 \sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}+\frac {2 x^2-d x^4}{3 b x \sqrt {d x^2} \sqrt {d x^2-2} \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{3/2}}\) |
(2*x^2 - d*x^4)/(3*b*x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]*(a + b*ArcCosh[-1 + d* x^2])^(3/2)) - x/(3*b^2*Sqrt[a + b*ArcCosh[-1 + d*x^2]]) + ((Sqrt[Pi/2]*Co sh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sq rt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)]))/(Sqrt[b]*d*x) - (Sqrt[Pi/2]*Cosh[ ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b ])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(Sqrt[b]*d*x))/(3*b^2)
3.3.66.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.)], x_Symbol] :> Simp[Sqr t[Pi/2]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*(Erfi[S qrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x)), x] - Simp[Sqrt[Pi /2]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*(Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(Sqrt[b]*d*x)), x] /; FreeQ[{a, b, d}, x]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[(- x)*((a + b*ArcCosh[c + d*x^2])^(n + 2)/(4*b^2*(n + 1)*(n + 2))), x] + (Simp [(2*c*x^2 + d*x^4)*((a + b*ArcCosh[c + d*x^2])^(n + 1)/(2*b*(n + 1)*x*Sqrt[ -1 + c + d*x^2]*Sqrt[1 + c + d*x^2])), x] + Simp[1/(4*b^2*(n + 1)*(n + 2)) Int[(a + b*ArcCosh[c + d*x^2])^(n + 2), x], x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]
\[\int \frac {1}{{\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^{5/2}} \,d x \]