3.247 \(\int (a+b \cosh ^{-1}(-1+d x^2))^4 \, dx\)

Optimal. Leaf size=147 \[ \frac {192 b^3 \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^4+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2-2}}+384 b^4 x \]

[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)

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Rubi [A]  time = 0.03, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5880, 8} \[ \frac {192 b^3 \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )}{x \sqrt {d x^2} \sqrt {d x^2-2}}+48 b^2 x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^2+\frac {8 b \left (2 x^2-d x^4\right ) \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^3}{x \sqrt {d x^2} \sqrt {d x^2-2}}+x \left (a+b \cosh ^{-1}\left (d x^2-1\right )\right )^4+384 b^4 x \]

Antiderivative was successfully verified.

[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 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]

Rubi steps

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

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Mathematica [A]  time = 0.24, size = 264, normalized size = 1.80 \[ \frac {-8 a b \left (a^2+24 b^2\right ) \sqrt {d x^2} \sqrt {d x^2-2}+6 b^2 \cosh ^{-1}\left (d x^2-1\right )^2 \left (a^2 d x^2-4 a b \sqrt {d x^2} \sqrt {d x^2-2}+8 b^2 d x^2\right )+d x^2 \left (a^4+48 a^2 b^2+384 b^4\right )+4 b \cosh ^{-1}\left (d x^2-1\right ) \left (a^3 d x^2-6 a^2 b \sqrt {d x^2} \sqrt {d x^2-2}+24 a b^2 d x^2-48 b^3 \sqrt {d x^2} \sqrt {d x^2-2}\right )+4 b^3 \cosh ^{-1}\left (d x^2-1\right )^3 \left (a d x^2-2 b \sqrt {d x^2} \sqrt {d x^2-2}\right )+b^4 d x^2 \cosh ^{-1}\left (d x^2-1\right )^4}{d x} \]

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.61, size = 298, normalized size = 2.03 \[ \frac {b^{4} d x^{2} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{4} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} + 4 \, {\left (a b^{3} d x^{2} - 2 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{3} - 6 \, {\left (4 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x^{2} - 6 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{2} b^{2} + 8 \, b^{4}\right )}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{4} - 2 \, d x^{2}} - 1\right ) - 8 \, \sqrt {d^{2} x^{4} - 2 \, d x^{2}} {\left (a^{3} b + 24 \, a b^{3}\right )}}{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [sign(x)]index.c
c index_m i_lex_is_greater Error: Bad Argument Valueindex.cc index_m operator + Error: Bad Argument Value

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d \,x^{2}-1\right )\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b^{4} x \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d} x - 1\right )^{4} + 6 \, a^{2} b^{2} x \operatorname {arcosh}\left (d x^{2} - 1\right )^{2} + 24 \, a^{2} b^{2} d {\left (\frac {2 \, x}{d} - \frac {{\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d x^{2}} - 1\right )}{\sqrt {d x^{2} - 2} d^{2}}\right )} + 4 \, {\left (x \operatorname {arcosh}\left (d x^{2} - 1\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2} d}\right )} a^{3} b + a^{4} x + \int \frac {4 \, {\left ({\left (a b^{3} d^{2} - 2 \, b^{4} d^{2}\right )} x^{4} + 2 \, a b^{3} - {\left (3 \, a b^{3} d - 4 \, b^{4} d\right )} x^{2} + {\left ({\left (a b^{3} d - 2 \, b^{4} d\right )} \sqrt {d} x^{3} - 2 \, {\left (a b^{3} - b^{4}\right )} \sqrt {d} x\right )} \sqrt {d x^{2} - 2}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2} \sqrt {d} x - 1\right )^{3}}{d^{2} x^{4} - 3 \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} - 2 \, \sqrt {d} x\right )} \sqrt {d x^{2} - 2} + 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acosh}\left (d\,x^2-1\right )\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(d*x^2 - 1))^4,x)

[Out]

int((a + b*acosh(d*x^2 - 1))^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acosh}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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