3.249 \(\int (a+b \cosh ^{-1}(-1+d x^2))^2 \, dx\)

Optimal. Leaf size=73 \[ x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2+\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+8 b^2 x \]

[Out]

8*b^2*x+x*(a+b*arccosh(d*x^2-1))^2+4*b*(-d*x^4+2*x^2)*(a+b*arccosh(d*x^2-1))/x/(d*x^2)^(1/2)/(d*x^2-2)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5880, 8} \[ \frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2+8 b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[-1 + d*x^2])^2,x]

[Out]

8*b^2*x + (4*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2]))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + x*(a + b*ArcCos
h[-1 + d*x^2])^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5880

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*(a + b*ArcCosh[c + d*x^2])^n, x] +
(Dist[4*b^2*n*(n - 1), Int[(a + b*ArcCosh[c + d*x^2])^(n - 2), x], x] - Simp[(2*b*n*(2*c*d*x^2 + d^2*x^4)*(a +
b*ArcCosh[c + d*x^2])^(n - 1))/(d*x*Sqrt[-1 + c + d*x^2]*Sqrt[1 + c + d*x^2]), x]) /; FreeQ[{a, b, c, d}, x]
&& EqQ[c^2, 1] && GtQ[n, 1]

Rubi steps

\begin {align*} \int \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2 \, dx &=\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2+\left (8 b^2\right ) \int 1 \, dx\\ &=8 b^2 x+\frac {4 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \cosh ^{-1}\left (-1+d x^2\right )\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 104, normalized size = 1.42 \[ x \left (a^2+8 b^2\right )-\frac {4 a b \sqrt {d x^2} \sqrt {d x^2-2}}{d x}+\frac {2 b \cosh ^{-1}\left (d x^2-1\right ) \left (a d x^2-2 b \sqrt {d x^2} \sqrt {d x^2-2}\right )}{d x}+b^2 x \cosh ^{-1}\left (d x^2-1\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2 + 8*b^2)*x - (4*a*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])/(d*x) + (2*b*(a*d*x^2 - 2*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2
])*ArcCosh[-1 + d*x^2])/(d*x) + b^2*x*ArcCosh[-1 + d*x^2]^2

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fricas [A]  time = 0.48, size = 131, normalized size = 1.79 \[ \frac {b^{2} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b + 2 \, {\left (a b d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )}{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*d*x^2*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^2 + (a^2 + 8*b^2)*d*x^2 - 4*sqrt(d^2*x^4 - 2*d*x^2)*a*b +
2*(a*b*d*x^2 - 2*sqrt(d^2*x^4 - 2*d*x^2)*b^2)*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1))/(d*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [sign(x)]index.c
c index_m i_lex_is_greater Error: Bad Argument Valueindex.cc index_m operator + Error: Bad Argument Value

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 128, normalized size = 1.75 \[ b^{2} x \operatorname {arcosh}\left (d x^{2} - 1\right )^{2} + 4 \, b^{2} d {\left (\frac {2 \, x}{d} - \frac {{\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d x^{2}} - 1\right )}{\sqrt {d x^{2} - 2} d^{2}}\right )} + 2 \, {\left (x \operatorname {arcosh}\left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2} d}\right )} a b + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*x*arccosh(d*x^2 - 1)^2 + 4*b^2*d*(2*x/d - (d^(3/2)*x^2 - 2*sqrt(d))*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d*x^2
) - 1)/(sqrt(d*x^2 - 2)*d^2)) + 2*(x*arccosh(d*x^2 - 1) - 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(d*x^2 - 2)*d))*a*b
+ a^2*x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(d*x^2 - 1))^2,x)

[Out]

int((a + b*acosh(d*x^2 - 1))^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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