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

Optimal. Leaf size=72 \[ -\frac{4 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2+8 b^2 x \]

[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*ArcCosh[
1 + d*x^2])^2

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Rubi [A]  time = 0.0143908, antiderivative size = 72, normalized size of antiderivative = 1., 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} \[ -\frac{4 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2+8 b^2 x \]

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*ArcCosh[
1 + d*x^2])^2

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]

Rule 8

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

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*}

Mathematica [A]  time = 0.0606404, size = 104, normalized size = 1.44 \[ x \left (a^2+8 b^2\right )-\frac{4 a b \sqrt{d x^2} \sqrt{d x^2+2}}{d x}+\frac{2 b \cosh ^{-1}\left (d x^2+1\right ) \left (a d x^2-2 b \sqrt{d x^2} \sqrt{d x^2+2}\right )}{d x}+b^2 x \cosh ^{-1}\left (d x^2+1\right )^2 \]

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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Maple [F]  time = 0.125, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}+1\right ) \right ) ^{2}\, dx \end{align*}

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 = 1.29604, size = 173, normalized size = 2.4 \begin{align*} 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 \end{align*}

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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Fricas [A]  time = 2.02929, size = 277, normalized size = 3.85 \begin{align*} \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} \end{align*}

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

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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