\(\int (a+b \text {arccosh}(-1+d x^2))^3 \, dx\) [248]

Optimal result

Integrand size = 14, antiderivative size = 110 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=24 a b^2 x-48 b^3 \sqrt {1-\frac {2}{d x^2}} x+24 b^3 x \text {arccosh}\left (-1+d x^2\right )+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \]

[Out]

24*a*b^2*x+24*b^3*x*arccosh(d*x^2-1)+x*(a+b*arccosh(d*x^2-1))^3-48*b^3*x*(1-2/d/x^2)^(1/2)+6*b*(-d*x^4+2*x^2)*
(a+b*arccosh(d*x^2-1))^2/x/(d*x^2)^(1/2)/(d*x^2-2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6001, 6016, 12, 197} \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3+\frac {6 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2}{x \sqrt {d x^2} \sqrt {d x^2-2}}+24 a b^2 x+24 b^3 x \text {arccosh}\left (d x^2-1\right )-48 b^3 x \sqrt {1-\frac {2}{d x^2}} \]

[In]

Int[(a + b*ArcCosh[-1 + d*x^2])^3,x]

[Out]

24*a*b^2*x - 48*b^3*Sqrt[1 - 2/(d*x^2)]*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 12

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 6001

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 6016

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]

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.55 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\frac {a \left (a^2+24 b^2\right ) d x^2-6 b \left (a^2+8 b^2\right ) \sqrt {d x^2} \sqrt {-2+d x^2}+3 b \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )+3 b^2 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )^2+b^3 d x^2 \text {arccosh}\left (-1+d x^2\right )^3}{d x} \]

[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])*ArcCo
sh[-1 + d*x^2]^2 + b^3*d*x^2*ArcCosh[-1 + d*x^2]^3)/(d*x)

Maple [F]

\[\int {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{3}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (103) = 206\).

Time = 0.26 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.91 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\frac {b^{3} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{3} + {\left (a^{3} + 24 \, a b^{2}\right )} d x^{2} + 3 \, {\left (a b^{2} d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{3}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + 3 \, {\left ({\left (a^{2} b + 8 \, b^{3}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) - 6 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{2} b + 8 \, b^{3}\right )}}{d x} \]

[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)

Sympy [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{3}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\int { {\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{3} \,d x } \]

[In]

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

[Out]

3*a*b^2*x*arccosh(d*x^2 - 1)^2 + 12*a*b^2*d*(2*x/d - (d^(3/2)*x^2 - 2*sqrt(d))*log(d*x^2 + sqrt(d*x^2 - 2)*sqr
t(d*x^2) - 1)/(sqrt(d*x^2 - 2)*d^2)) + 3*(x*arccosh(d*x^2 - 1) - 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(d*x^2 - 2)*
d))*a^2*b + (x*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1)^3 - integrate(6*(d^2*x^4 - 2*d*x^2 + (d^(3/2)*x^3 -
sqrt(d)*x)*sqrt(d*x^2 - 2))*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1)^2/(d^2*x^4 - 3*d*x^2 + (d^(3/2)*x^3 - 2
*sqrt(d)*x)*sqrt(d*x^2 - 2) + 2), x))*b^3 + a^3*x

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:index.cc index_m
operator + Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^3 \,d x \]

[In]

int((a + b*acosh(d*x^2 - 1))^3,x)

[Out]

int((a + b*acosh(d*x^2 - 1))^3, x)