3.267 \(\int \frac {1}{(a+b \cosh ^{-1}(-1+d x^2))^{7/2}} \, dx\)

Optimal. Leaf size=302 \[ \frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{15 b^{7/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{15 b^{7/2} d x}-\frac {\sqrt {d x^2} \sqrt {d x^2-2}}{15 b^3 d x \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}-\frac {x}{15 b^2 \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{3/2}}+\frac {2 x^2-d x^4}{5 b x \sqrt {d x^2} \sqrt {d x^2-2} \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2}} \]

[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)

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Rubi [A]  time = 0.06, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5889, 5886} \[ \frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{15 b^{7/2} d x}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )}{15 b^{7/2} d x}-\frac {x}{15 b^2 \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{3/2}}-\frac {\sqrt {d x^2} \sqrt {d x^2-2}}{15 b^3 d x \sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}+\frac {2 x^2-d x^4}{5 b x \sqrt {d x^2} \sqrt {d x^2-2} \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2}} \]

Antiderivative was successfully verified.

[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 5886

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 5889

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*} \int \frac {1}{\left (a+b \cosh ^{-1}\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 \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^{3/2}}+\frac {\int \frac {1}{\left (a+b \cosh ^{-1}\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 \cosh ^{-1}\left (-1+d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+b \cosh ^{-1}\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 \cosh ^{-1}\left (-1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \cosh \left (\frac {1}{2} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\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} \cosh ^{-1}\left (-1+d x^2\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\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*}

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Mathematica [A]  time = 1.08, size = 260, normalized size = 0.86 \[ \frac {\cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (4 \sqrt {b} \left (-\sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (\left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2+3 b^2\right )-b \cosh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2-1\right )\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )\right )+\sqrt {2 \pi } \left (\sinh \left (\frac {a}{2 b}\right )+\cosh \left (\frac {a}{2 b}\right )\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )+\sqrt {2 \pi } \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (d x^2-1\right )}}{\sqrt {2} \sqrt {b}}\right )\right )}{30 b^{7/2} d x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[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))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]index.cc index_m i_lex_is_greater Error: Bad Argument ValueEvaluation time: 1.05index.cc index_m o
perator + Error: Bad Argument Value

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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