\(\int (a+b \text {arccosh}(-1+d x^2))^4 \, dx\) [247]

Optimal result

Integrand size = 14, antiderivative size = 147 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=384 b^4 x+\frac {192 b^3 \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+48 b^2 x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \]

[Out]

384*b^4*x+48*b^2*x*(a+b*arccosh(d*x^2-1))^2+x*(a+b*arccosh(d*x^2-1))^4+192*b^3*(-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)+8*b*(-d*x^4+2*x^2)*(a+b*arccosh(d*x^2-1))^3/x/(d*x^2)^(1/2)/(d*x^2-2)^(
1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6001, 8} \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\frac {192 b^3 \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+48 b^2 x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^2+x \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^4+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (d x^2-1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2-2}}+384 b^4 x \]

[In]

Int[(a + b*ArcCosh[-1 + d*x^2])^4,x]

[Out]

384*b^4*x + (192*b^3*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2]))/(x*Sqrt[d*x^2]*Sqrt[-2 + d*x^2]) + 48*b^2*x*
(a + b*ArcCosh[-1 + d*x^2])^2 + (8*b*(2*x^2 - d*x^4)*(a + b*ArcCosh[-1 + d*x^2])^3)/(x*Sqrt[d*x^2]*Sqrt[-2 + d
*x^2]) + x*(a + b*ArcCosh[-1 + d*x^2])^4

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 {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4+\left (48 b^2\right ) \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2 \, dx \\ & = \frac {192 b^3 \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}}+48 b^2 x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx \\ & = 384 b^4 x+\frac {192 b^3 \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}}+48 b^2 x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^2+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^3}{x \sqrt {d x^2} \sqrt {-2+d x^2}}+x \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.80 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\frac {\left (a^4+48 a^2 b^2+384 b^4\right ) d x^2-8 a b \left (a^2+24 b^2\right ) \sqrt {d x^2} \sqrt {-2+d x^2}+4 b \left (a^3 d x^2+24 a b^2 d x^2-6 a^2 b \sqrt {d x^2} \sqrt {-2+d x^2}-48 b^3 \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )+6 b^2 \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )^2+4 b^3 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {-2+d x^2}\right ) \text {arccosh}\left (-1+d x^2\right )^3+b^4 d x^2 \text {arccosh}\left (-1+d x^2\right )^4}{d x} \]

[In]

Integrate[(a + b*ArcCosh[-1 + d*x^2])^4,x]

[Out]

((a^4 + 48*a^2*b^2 + 384*b^4)*d*x^2 - 8*a*b*(a^2 + 24*b^2)*Sqrt[d*x^2]*Sqrt[-2 + d*x^2] + 4*b*(a^3*d*x^2 + 24*
a*b^2*d*x^2 - 6*a^2*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2] - 48*b^3*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])*ArcCosh[-1 + d*x^2]
+ 6*b^2*(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]^2 + 4*b^3*(a*d*x^2
- 2*b*Sqrt[d*x^2]*Sqrt[-2 + d*x^2])*ArcCosh[-1 + d*x^2]^3 + b^4*d*x^2*ArcCosh[-1 + d*x^2]^4)/(d*x)

Maple [F]

\[\int {\left (a +b \,\operatorname {arccosh}\left (d \,x^{2}-1\right )\right )}^{4}d x\]

[In]

int((a+b*arccosh(d*x^2-1))^4,x)

[Out]

int((a+b*arccosh(d*x^2-1))^4,x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (137) = 274\).

Time = 0.26 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.03 \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\frac {b^{4} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{4} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 4 \, {\left (a b^{3} d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{3} - 6 \, {\left (4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x^{2} - 6 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{2} b^{2} + 8 \, b^{4}\right )}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) - 8 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{3} b + 24 \, a b^{3}\right )}}{d x} \]

[In]

integrate((a+b*arccosh(d*x^2-1))^4,x, algorithm="fricas")

[Out]

(b^4*d*x^2*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^4 + (a^4 + 48*a^2*b^2 + 384*b^4)*d*x^2 + 4*(a*b^3*d*x^2 -
2*sqrt(d^2*x^4 - 2*d*x^2)*b^4)*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^3 - 6*(4*sqrt(d^2*x^4 - 2*d*x^2)*a*b^3
- (a^2*b^2 + 8*b^4)*d*x^2)*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1)^2 + 4*((a^3*b + 24*a*b^3)*d*x^2 - 6*sqrt(
d^2*x^4 - 2*d*x^2)*(a^2*b^2 + 8*b^4))*log(d*x^2 + sqrt(d^2*x^4 - 2*d*x^2) - 1) - 8*sqrt(d^2*x^4 - 2*d*x^2)*(a^
3*b + 24*a*b^3))/(d*x)

Sympy [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \]

[In]

integrate((a+b*acosh(d*x**2-1))**4,x)

[Out]

Integral((a + b*acosh(d*x**2 - 1))**4, x)

Maxima [F]

\[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\int { {\left (b \operatorname {arcosh}\left (d x^{2} - 1\right ) + a\right )}^{4} \,d x } \]

[In]

integrate((a+b*arccosh(d*x^2-1))^4,x, algorithm="maxima")

[Out]

b^4*x*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1)^4 + 6*a^2*b^2*x*arccosh(d*x^2 - 1)^2 + 24*a^2*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)) + 4*(x*arccosh(d
*x^2 - 1) - 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(d*x^2 - 2)*d))*a^3*b + a^4*x + integrate(4*((a*b^3*d^2 - 2*b^4*d
^2)*x^4 + 2*a*b^3 - (3*a*b^3*d - 4*b^4*d)*x^2 + ((a*b^3*d - 2*b^4*d)*sqrt(d)*x^3 - 2*(a*b^3 - b^4)*sqrt(d)*x)*
sqrt(d*x^2 - 2))*log(d*x^2 + sqrt(d*x^2 - 2)*sqrt(d)*x - 1)^3/(d^2*x^4 - 3*d*x^2 + (d^(3/2)*x^3 - 2*sqrt(d)*x)
*sqrt(d*x^2 - 2) + 2), x)

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a+b*arccosh(d*x^2-1))^4,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. \[ \int \left (a+b \text {arccosh}\left (-1+d x^2\right )\right )^4 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^4 \,d x \]

[In]

int((a + b*acosh(d*x^2 - 1))^4,x)

[Out]

int((a + b*acosh(d*x^2 - 1))^4, x)