\(\int \frac {1}{(a+b \text {arccosh}(1+d x^2))^{3/2}} \, dx\) [258]

Optimal result

Integrand size = 16, antiderivative size = 213 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=-\frac {\sqrt {d x^2} \sqrt {2+d x^2}}{b d x \sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{b^{3/2} d x}-\frac {\sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{b^{3/2} d x} \]

[Out]

1/2*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))*sinh(1/2*arccosh(d*x^
2+1))*2^(1/2)*Pi^(1/2)/b^(3/2)/d/x-1/2*erf(1/2*(a+b*arccosh(d*x^2+1))^(1/2)*2^(1/2)/b^(1/2))*(cosh(1/2*a/b)+si
nh(1/2*a/b))*sinh(1/2*arccosh(d*x^2+1))*2^(1/2)*Pi^(1/2)/b^(3/2)/d/x-(d*x^2)^(1/2)*(d*x^2+2)^(1/2)/b/d/x/(a+b*
arccosh(d*x^2+1))^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6006} \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (d x^2+1\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{b^{3/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (d x^2+1\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (d x^2+1\right )}}{\sqrt {2} \sqrt {b}}\right )}{b^{3/2} d x}-\frac {\sqrt {d x^2} \sqrt {d x^2+2}}{b d x \sqrt {a+b \text {arccosh}\left (d x^2+1\right )}} \]

[In]

Int[(a + b*ArcCosh[1 + d*x^2])^(-3/2),x]

[Out]

-((Sqrt[d*x^2]*Sqrt[2 + d*x^2])/(b*d*x*Sqrt[a + b*ArcCosh[1 + d*x^2]])) + (Sqrt[Pi/2]*Erfi[Sqrt[a + b*ArcCosh[
1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] - Sinh[a/(2*b)])*Sinh[ArcCosh[1 + d*x^2]/2])/(b^(3/2)*d*x) - (Sq
rt[Pi/2]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)])*Sinh[ArcCosh[1
+ d*x^2]/2])/(b^(3/2)*d*x)

Rule 6006

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)])*Sinh[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)] - Si
nh[a/(2*b)])*Sinh[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d x^2} \sqrt {2+d x^2}}{b d x \sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{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 ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{b^{3/2} d x}-\frac {\sqrt {\frac {\pi }{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 ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{b^{3/2} d x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=-\frac {x \left (4 \sqrt {b} \cosh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )+\sqrt {2 \pi } \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (-\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {2 \pi } \sqrt {a+b \text {arccosh}\left (1+d x^2\right )} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )\right ) \sinh \left (\frac {1}{2} \text {arccosh}\left (1+d x^2\right )\right )}{2 b^{3/2} \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2} \sqrt {a+b \text {arccosh}\left (1+d x^2\right )}} \]

[In]

Integrate[(a + b*ArcCosh[1 + d*x^2])^(-3/2),x]

[Out]

-1/2*(x*(4*Sqrt[b]*Cosh[ArcCosh[1 + d*x^2]/2] + Sqrt[2*Pi]*Sqrt[a + b*ArcCosh[1 + d*x^2]]*Erfi[Sqrt[a + b*ArcC
osh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(-Cosh[a/(2*b)] + Sinh[a/(2*b)]) + Sqrt[2*Pi]*Sqrt[a + b*ArcCosh[1 + d*x^2]
]*Erf[Sqrt[a + b*ArcCosh[1 + d*x^2]]/(Sqrt[2]*Sqrt[b])]*(Cosh[a/(2*b)] + Sinh[a/(2*b)]))*Sinh[ArcCosh[1 + d*x^
2]/2])/(b^(3/2)*Sqrt[d*x^2]*Sqrt[(d*x^2)/(2 + d*x^2)]*Sqrt[2 + d*x^2]*Sqrt[a + b*ArcCosh[1 + d*x^2]])

Maple [F]

\[\int \frac {1}{{\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}+1\right )\right )}^{\frac {3}{2}}}d x\]

[In]

int(1/(a+b*arccosh(d*x^2+1))^(3/2),x)

[Out]

int(1/(a+b*arccosh(d*x^2+1))^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a+b*arccosh(d*x^2+1))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {acosh}{\left (d x^{2} + 1 \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(a+b*acosh(d*x**2+1))**(3/2),x)

[Out]

Integral((a + b*acosh(d*x**2 + 1))**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(a+b*arccosh(d*x^2+1))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(d*x^2 + 1) + a)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} + 1\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(a+b*arccosh(d*x^2+1))^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(d*x^2 + 1) + a)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \text {arccosh}\left (1+d x^2\right )\right )^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^{3/2}} \,d x \]

[In]

int(1/(a + b*acosh(d*x^2 + 1))^(3/2),x)

[Out]

int(1/(a + b*acosh(d*x^2 + 1))^(3/2), x)