\(\int (a+b \text {arccosh}(1+d x^2))^2 \, dx\) [242]

Optimal result

Integrand size = 14, antiderivative size = 72 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2 \, dx=8 b^2 x-\frac {4 b \left (2 x^2+d x^4\right ) \left (a+b \text {arccosh}\left (1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {2+d x^2}}+x \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2 \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6001, 8} \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2 \, dx=x \left (a+b \text {arccosh}\left (d x^2+1\right )\right )^2-\frac {4 b \left (d x^4+2 x^2\right ) \left (a+b \text {arccosh}\left (d x^2+1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2+2}}+8 b^2 x \]

[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 8

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

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]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.44 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2 \, dx=\left (a^2+8 b^2\right ) x-\frac {4 a b \sqrt {d x^2} \sqrt {2+d x^2}}{d x}+\frac {2 b \left (a d x^2-2 b \sqrt {d x^2} \sqrt {2+d x^2}\right ) \text {arccosh}\left (1+d x^2\right )}{d x}+b^2 x \text {arccosh}\left (1+d x^2\right )^2 \]

[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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.82 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2 \, dx=\frac {b^{2} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x^{2} - 4 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} a b + 2 \, {\left (a b d x^{2} - 2 \, \sqrt {d^{2} x^{4} + 2 \, d x^{2}} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} + 2 \, d x^{2}} + 1\right )}{d x} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.78 \[ \int \left (a+b \text {arccosh}\left (1+d x^2\right )\right )^2 \, dx=b^{2} x \operatorname {arcosh}\left (d x^{2} + 1\right )^{2} + 4 \, b^{2} d {\left (\frac {2 \, x}{d} - \frac {{\left (d^{\frac {3}{2}} x^{2} + 2 \, \sqrt {d}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2} \sqrt {d x^{2}} + 1\right )}{\sqrt {d x^{2} + 2} d^{2}}\right )} + 2 \, {\left (x \operatorname {arcosh}\left (d x^{2} + 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2} d}\right )} a b + a^{2} x \]

[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

Giac [F(-2)]

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

[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:index.cc index_m
operator + Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int((a + b*acosh(d*x^2 + 1))^2,x)

[Out]

int((a + b*acosh(d*x^2 + 1))^2, x)