3.3.60 \(\int \frac {1}{(a+b \cosh ^{-1}(1+d x^2))^{7/2}} \, dx\) [260]

Optimal. Leaf size=301 \[ -\frac {2 x^2+d x^4}{5 b x \sqrt {d x^2} \sqrt {2+d x^2} \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2}}-\frac {x}{15 b^2 \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{3/2}}-\frac {\sqrt {d x^2} \sqrt {2+d x^2}}{15 b^3 d x \sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )}}+\frac {\sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{15 b^{7/2} d x}-\frac {\sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}\left (1+d x^2\right )}}{\sqrt {2} \sqrt {b}}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{15 b^{7/2} d x} \]

[Out]

-1/15*x/b^2/(a+b*arccosh(d*x^2+1))^(3/2)+1/30*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^(7/2)/d/x-1/30*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))*sinh(1/2*arccosh(d*x^2+1))*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 = 0.07, antiderivative size = 301, 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 = {6010, 6006} \begin {gather*} -\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} \cosh ^{-1}\left (d x^2+1\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}} \left (\cosh \left (\frac {a}{2 b}\right )-\sinh \left (\frac {a}{2 b}\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (d x^2+1\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 {d x^4+2 x^2}{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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*(2*x^2 + d*x^4)/(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*Arc
Cosh[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]*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])/(15*b^(7/2)*d*x) - (Sqrt[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])/(15*b^(7/2)*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]

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*} \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}} \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 ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{15 b^{7/2} d x}-\frac {\sqrt {\frac {\pi }{2}} \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 ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )}{15 b^{7/2} d x}\\ \end {align*}

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Mathematica [A]
time = 0.87, size = 291, normalized size = 0.97 \begin {gather*} -\frac {x \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right ) \left (\sqrt {2 \pi } \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2} \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 )+\sqrt {2 \pi } \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2} \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 )+4 \sqrt {b} \left (\left (3 b^2+\left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2\right ) \cosh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )+b \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right ) \sinh \left (\frac {1}{2} \cosh ^{-1}\left (1+d x^2\right )\right )\right )\right )}{30 b^{7/2} \sqrt {d x^2} \sqrt {\frac {d x^2}{2+d x^2}} \sqrt {2+d x^2} \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (d\,x^2+1\right )\right )}^{7/2}} \,d x \end {gather*}

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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