\(\int \frac {1}{(a+b \text {arccosh}(-1+d x^2))^{7/2}} \, dx\) [267]

Optimal result

Integrand size = 16, antiderivative size = 302 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\frac {2 x^2-d x^4}{5 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}-\frac {\sqrt {d x^2} \sqrt {-2+d x^2}}{15 b^3 d x \sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )}{15 b^{7/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{15 b^{7/2} d x} \]

[Out]

-1/15*x/b^2/(a+b*arccosh(d*x^2-1))^(3/2)+1/30*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^(7/2)/d/x+1/30*cosh(1/2*arccosh(d*x^2-1))*e
rf(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^(7/2)/d/
x+1/5*(-d*x^4+2*x^2)/b/x/(a+b*arccosh(d*x^2-1))^(5/2)/(d*x^2)^(1/2)/(d*x^2-2)^(1/2)-1/15*(d*x^2)^(1/2)*(d*x^2-
2)^(1/2)/b^3/d/x/(a+b*arccosh(d*x^2-1))^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6010, 6007} \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\frac {\sqrt {\frac {\pi }{2}} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{15 b^{7/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (d x^2-1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{15 b^{7/2} d x}-\frac {\sqrt {d x^2} \sqrt {d x^2-2}}{15 b^3 d x \sqrt {a+b \text {arccosh}\left (d x^2-1\right )}}-\frac {x}{15 b^2 \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{3/2}}+\frac {2 x^2-d x^4}{5 b x \sqrt {d x^2} \sqrt {d x^2-2} \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^{5/2}} \]

[In]

Int[(a + b*ArcCosh[-1 + d*x^2])^(-7/2),x]

[Out]

(2*x^2 - d*x^4)/(5*b*x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]*(a + b*ArcCosh[-1 + d*x^2])^(5/2)) - x/(15*b^2*(a + b*ArcC
osh[-1 + d*x^2])^(3/2)) - (Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(15*b^3*d*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]]) + (Sqrt[
Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sin
h[a/(2*b)]))/(15*b^(7/2)*d*x) + (Sqrt[Pi/2]*Cosh[ArcCosh[-1 + d*x^2]/2]*Erf[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/(S
qrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(15*b^(7/2)*d*x)

Rule 6007

Int[((a_.) + ArcCosh[-1 + (d_.)*(x_)^2]*(b_.))^(-3/2), x_Symbol] :> Simp[(-Sqrt[d*x^2])*(Sqrt[-2 + d*x^2]/(b*d
*x*Sqrt[a + b*ArcCosh[-1 + d*x^2]])), 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]]/(b^(3/2)*d*x)), x] + Simp[Sqrt[Pi/2]*(Cosh[a/(2*b)]
- Sinh[a/(2*b)])*Cosh[ArcCosh[-1 + d*x^2]/2]*(Erfi[Sqrt[a + b*ArcCosh[-1 + d*x^2]]/Sqrt[2*b]]/(b^(3/2)*d*x)),
x]) /; FreeQ[{a, b, d}, x]

Rule 6010

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] + (Dist[1/(4*b^2*(n + 1)*(n + 2)), Int[(a + b*ArcCosh[c + d*x^2])^(n + 2), x]
, 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]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && LtQ[n, -1] && NeQ[n, -2]

Rubi steps \begin{align*} \text {integral}& = \frac {2 x^2-d x^4}{5 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}+\frac {\int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}} \, dx}{15 b^2} \\ & = \frac {2 x^2-d x^4}{5 b x \sqrt {d x^2} \sqrt {-2+d x^2} \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{3/2}}-\frac {\sqrt {d x^2} \sqrt {-2+d x^2}}{15 b^3 d x \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 )}{15 b^{7/2} 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 )}{15 b^{7/2} d x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\frac {\cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right ) \left (\sqrt {2 \pi } \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {2 \pi } \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (-1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+4 \sqrt {b} \left (-b \left (a+b \text {arccosh}\left (-1+d x^2\right )\right ) \cosh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )-\left (3 b^2+\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (-1+d x^2\right )\right )\right )\right )}{30 b^{7/2} d x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{5/2}} \]

[In]

Integrate[(a + b*ArcCosh[-1 + d*x^2])^(-7/2),x]

[Out]

(Cosh[ArcCosh[-1 + d*x^2]/2]*(Sqrt[2*Pi]*(a + b*ArcCosh[-1 + d*x^2])^(5/2)*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])^(5/2)*Erf[Sqrt[a
+ b*ArcCosh[-1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]) + 4*Sqrt[b]*(-(b*(a + b*ArcCosh[-
1 + d*x^2])*Cosh[ArcCosh[-1 + d*x^2]/2]) - (3*b^2 + (a + b*ArcCosh[-1 + d*x^2])^2)*Sinh[ArcCosh[-1 + d*x^2]/2]
)))/(30*b^(7/2)*d*x*(a + b*ArcCosh[-1 + d*x^2])^(5/2))

Maple [F]

\[\int \frac {1}{{\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{\frac {7}{2}}}d x\]

[In]

int(1/(a+b*arccosh(d*x^2-1))^(7/2),x)

[Out]

int(1/(a+b*arccosh(d*x^2-1))^(7/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a+b*arccosh(d*x^2-1))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*acosh(d*x**2-1))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*arccosh(d*x^2-1))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(d*x^2 - 1) + a)^(-7/2), x)

Giac [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*arccosh(d*x^2-1))^(7/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x^2 - 1) + a)^(-7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^{7/2}} \,d x \]

[In]

int(1/(a + b*acosh(d*x^2 - 1))^(7/2),x)

[Out]

int(1/(a + b*acosh(d*x^2 - 1))^(7/2), x)