3.241 \(\int (a+b \cosh ^{-1}(1+d x^2))^3 \, dx\)

Optimal. Leaf size=125 \[ 24 a b^2 x-\frac{6 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3-\frac{48 b^3 \sqrt{\frac{d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+24 b^3 x \cosh ^{-1}\left (d x^2+1\right ) \]

[Out]

24*a*b^2*x - (48*b^3*Sqrt[(d*x^2)/(2 + d*x^2)]*(2 + d*x^2))/(d*x) + 24*b^3*x*ArcCosh[1 + d*x^2] - (6*b*(2*x^2
+ d*x^4)*(a + b*ArcCosh[1 + d*x^2])^2)/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]) + x*(a + b*ArcCosh[1 + d*x^2])^3

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Rubi [A]  time = 0.0625657, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5880, 5901, 12, 6719, 261} \[ 24 a b^2 x-\frac{6 b \left (d x^4+2 x^2\right ) \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^2}{x \sqrt{d x^2} \sqrt{d x^2+2}}+x \left (a+b \cosh ^{-1}\left (d x^2+1\right )\right )^3-\frac{48 b^3 \sqrt{\frac{d x^2}{d x^2+2}} \left (d x^2+2\right )}{d x}+24 b^3 x \cosh ^{-1}\left (d x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[1 + d*x^2])^3,x]

[Out]

24*a*b^2*x - (48*b^3*Sqrt[(d*x^2)/(2 + d*x^2)]*(2 + d*x^2))/(d*x) + 24*b^3*x*ArcCosh[1 + d*x^2] - (6*b*(2*x^2
+ d*x^4)*(a + b*ArcCosh[1 + d*x^2])^2)/(x*Sqrt[d*x^2]*Sqrt[2 + d*x^2]) + x*(a + b*ArcCosh[1 + d*x^2])^3

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 5901

Int[ArcCosh[u_], x_Symbol] :> Simp[x*ArcCosh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(Sqrt[-1 + u]*Sqrt[1 +
u]), x], x] /; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3 \, dx &=-\frac{6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3+\left (24 b^2\right ) \int \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right ) \, dx\\ &=24 a b^2 x-\frac{6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3+\left (24 b^3\right ) \int \cosh ^{-1}\left (1+d x^2\right ) \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac{6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3-\left (24 b^3\right ) \int 2 \sqrt{\frac{d x^2}{2+d x^2}} \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac{6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3-\left (48 b^3\right ) \int \sqrt{\frac{d x^2}{2+d x^2}} \, dx\\ &=24 a b^2 x+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac{6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3-\frac{\left (48 b^3 \sqrt{\frac{d x^2}{2+d x^2}} \sqrt{2+d x^2}\right ) \int \frac{x}{\sqrt{2+d x^2}} \, dx}{x}\\ &=24 a b^2 x-\frac{48 b^3 \sqrt{\frac{d x^2}{2+d x^2}} \left (2+d x^2\right )}{d x}+24 b^3 x \cosh ^{-1}\left (1+d x^2\right )-\frac{6 b \left (2 x^2+d x^4\right ) \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^2}{x \sqrt{d x^2} \sqrt{2+d x^2}}+x \left (a+b \cosh ^{-1}\left (1+d x^2\right )\right )^3\\ \end{align*}

Mathematica [A]  time = 0.118374, size = 171, normalized size = 1.37 \[ \frac{a d x^2 \left (a^2+24 b^2\right )-6 b \left (a^2+8 b^2\right ) \sqrt{d x^2} \sqrt{d x^2+2}+3 b \cosh ^{-1}\left (d x^2+1\right ) \left (a^2 d x^2-4 a b \sqrt{d x^2} \sqrt{d x^2+2}+8 b^2 d x^2\right )+3 b^2 \cosh ^{-1}\left (d x^2+1\right )^2 \left (a d x^2-2 b \sqrt{d x^2} \sqrt{d x^2+2}\right )+b^3 d x^2 \cosh ^{-1}\left (d x^2+1\right )^3}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a^2 + 24*b^2)*d*x^2 - 6*b*(a^2 + 8*b^2)*Sqrt[d*x^2]*Sqrt[2 + d*x^2] + 3*b*(a^2*d*x^2 + 8*b^2*d*x^2 - 4*a*b
*Sqrt[d*x^2]*Sqrt[2 + d*x^2])*ArcCosh[1 + d*x^2] + 3*b^2*(a*d*x^2 - 2*b*Sqrt[d*x^2]*Sqrt[2 + d*x^2])*ArcCosh[1
+ d*x^2]^2 + b^3*d*x^2*ArcCosh[1 + d*x^2]^3)/(d*x)

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccosh} \left (d{x}^{2}+1\right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.08461, size = 441, normalized size = 3.53 \begin{align*} \frac{b^{3} d x^{2} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{3} +{\left (a^{3} + 24 \, a b^{2}\right )} d x^{2} + 3 \,{\left (a b^{2} d x^{2} - 2 \, \sqrt{d^{2} x^{4} + 2 \, d x^{2}} b^{3}\right )} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{2} + 3 \,{\left ({\left (a^{2} b + 8 \, b^{3}\right )} d x^{2} - 4 \, \sqrt{d^{2} x^{4} + 2 \, d x^{2}} a b^{2}\right )} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 \, d x^{2}} + 1\right ) - 6 \, \sqrt{d^{2} x^{4} + 2 \, d x^{2}}{\left (a^{2} b + 8 \, b^{3}\right )}}{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*d*x^2*log(d*x^2 + sqrt(d^2*x^4 + 2*d*x^2) + 1)^3 + (a^3 + 24*a*b^2)*d*x^2 + 3*(a*b^2*d*x^2 - 2*sqrt(d^2*x
^4 + 2*d*x^2)*b^3)*log(d*x^2 + sqrt(d^2*x^4 + 2*d*x^2) + 1)^2 + 3*((a^2*b + 8*b^3)*d*x^2 - 4*sqrt(d^2*x^4 + 2*
d*x^2)*a*b^2)*log(d*x^2 + sqrt(d^2*x^4 + 2*d*x^2) + 1) - 6*sqrt(d^2*x^4 + 2*d*x^2)*(a^2*b + 8*b^3))/(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 )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acosh(d*x**2 + 1))**3, 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))^3,x, algorithm="giac")

[Out]

Exception raised: RuntimeError