3.464 \(\int x (d+e x^2) (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=122 \[ \frac {1}{2} d x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b \left (8 c^2 d+3 e\right ) \cosh ^{-1}(c x)}{32 c^4}-\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (8 c^2 d+3 e\right )}{32 c^3}-\frac {b e x^3 \sqrt {c x-1} \sqrt {c x+1}}{16 c} \]

[Out]

-1/32*b*(8*c^2*d+3*e)*arccosh(c*x)/c^4+1/2*d*x^2*(a+b*arccosh(c*x))+1/4*e*x^4*(a+b*arccosh(c*x))-1/32*b*(8*c^2
*d+3*e)*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/16*b*e*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

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Rubi [A]  time = 0.11, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {5786, 460, 90, 52} \[ \frac {1}{2} d x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b x \sqrt {c x-1} \sqrt {c x+1} \left (8 c^2 d+3 e\right )}{32 c^3}-\frac {b \left (8 c^2 d+3 e\right ) \cosh ^{-1}(c x)}{32 c^4}-\frac {b e x^3 \sqrt {c x-1} \sqrt {c x+1}}{16 c} \]

Antiderivative was successfully verified.

[In]

Int[x*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

-(b*(8*c^2*d + 3*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c^3) - (b*e*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c) -
(b*(8*c^2*d + 3*e)*ArcCosh[c*x])/(32*c^4) + (d*x^2*(a + b*ArcCosh[c*x]))/2 + (e*x^4*(a + b*ArcCosh[c*x]))/4

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 5786

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(d*(f*x)^(
m + 1)*(a + b*ArcCosh[c*x]))/(f*(m + 1)), x] + (-Dist[(b*c)/(f*(m + 1)*(m + 3)), Int[((f*x)^(m + 1)*(d*(m + 3)
+ e*(m + 1)*x^2))/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] + Simp[(e*(f*x)^(m + 3)*(a + b*ArcCosh[c*x]))/(f^3*(
m + 3)), x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && NeQ[m, -1] && NeQ[m, -3]

Rubi steps

\begin {align*} \int x \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{2} d x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{8} (b c) \int \frac {x^2 \left (4 d+2 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}+\frac {1}{2} d x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{16} \left (b c \left (8 d+\frac {3 e}{c^2}\right )\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}+\frac {1}{2} d x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \left (8 c^2 d+3 e\right )\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 c^3}\\ &=-\frac {b \left (8 c^2 d+3 e\right ) x \sqrt {-1+c x} \sqrt {1+c x}}{32 c^3}-\frac {b e x^3 \sqrt {-1+c x} \sqrt {1+c x}}{16 c}-\frac {b \left (8 c^2 d+3 e\right ) \cosh ^{-1}(c x)}{32 c^4}+\frac {1}{2} d x^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} e x^4 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 120, normalized size = 0.98 \[ \frac {c x \left (8 a c^3 x \left (2 d+e x^2\right )-b \sqrt {c x-1} \sqrt {c x+1} \left (2 c^2 \left (4 d+e x^2\right )+3 e\right )\right )+8 b c^4 x^2 \cosh ^{-1}(c x) \left (2 d+e x^2\right )-2 b \left (8 c^2 d+3 e\right ) \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )}{32 c^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

(c*x*(8*a*c^3*x*(2*d + e*x^2) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3*e + 2*c^2*(4*d + e*x^2))) + 8*b*c^4*x^2*(2*d
+ e*x^2)*ArcCosh[c*x] - 2*b*(8*c^2*d + 3*e)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(32*c^4)

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fricas [A]  time = 0.59, size = 114, normalized size = 0.93 \[ \frac {8 \, a c^{4} e x^{4} + 16 \, a c^{4} d x^{2} + {\left (8 \, b c^{4} e x^{4} + 16 \, b c^{4} d x^{2} - 8 \, b c^{2} d - 3 \, b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{3} e x^{3} + {\left (8 \, b c^{3} d + 3 \, b c e\right )} x\right )} \sqrt {c^{2} x^{2} - 1}}{32 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*a*c^4*e*x^4 + 16*a*c^4*d*x^2 + (8*b*c^4*e*x^4 + 16*b*c^4*d*x^2 - 8*b*c^2*d - 3*b*e)*log(c*x + sqrt(c^2
*x^2 - 1)) - (2*b*c^3*e*x^3 + (8*b*c^3*d + 3*b*c*e)*x)*sqrt(c^2*x^2 - 1))/c^4

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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(x*(e*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.01, size = 202, normalized size = 1.66 \[ \frac {a \,x^{4} e}{4}+\frac {a \,x^{2} d}{2}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{4} e}{4}+\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{2} d}{2}-\frac {b e \,x^{3} \sqrt {c x -1}\, \sqrt {c x +1}}{16 c}-\frac {b d x \sqrt {c x -1}\, \sqrt {c x +1}}{4 c}-\frac {b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) d}{4 c^{2} \sqrt {c^{2} x^{2}-1}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, e x}{32 c^{3}}-\frac {3 b \sqrt {c x -1}\, \sqrt {c x +1}\, e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{32 c^{4} \sqrt {c^{2} x^{2}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4*e+1/2*a*x^2*d+1/4*b*arccosh(c*x)*x^4*e+1/2*b*arccosh(c*x)*x^2*d-1/16*b*e*x^3*(c*x-1)^(1/2)*(c*x+1)^(
1/2)/c-1/4*b*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/4/c^2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*ln(c*x+
(c^2*x^2-1)^(1/2))*d-3/32/c^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e*x-3/32/c^4*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^
2-1)^(1/2)*e*ln(c*x+(c^2*x^2-1)^(1/2))

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maxima [A]  time = 0.46, size = 156, normalized size = 1.28 \[ \frac {1}{4} \, a e x^{4} + \frac {1}{2} \, a d x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b d + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} x}{c^{4}} + \frac {3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{5}}\right )} c\right )} b e \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e*x^4 + 1/2*a*d*x^2 + 1/4*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2*x^
2 - 1)*c)/c^3))*b*d + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*
log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^5)*c)*b*e

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*acosh(c*x))*(d + e*x^2),x)

[Out]

int(x*(a + b*acosh(c*x))*(d + e*x^2), x)

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sympy [A]  time = 1.19, size = 160, normalized size = 1.31 \[ \begin {cases} \frac {a d x^{2}}{2} + \frac {a e x^{4}}{4} + \frac {b d x^{2} \operatorname {acosh}{\left (c x \right )}}{2} + \frac {b e x^{4} \operatorname {acosh}{\left (c x \right )}}{4} - \frac {b d x \sqrt {c^{2} x^{2} - 1}}{4 c} - \frac {b e x^{3} \sqrt {c^{2} x^{2} - 1}}{16 c} - \frac {b d \operatorname {acosh}{\left (c x \right )}}{4 c^{2}} - \frac {3 b e x \sqrt {c^{2} x^{2} - 1}}{32 c^{3}} - \frac {3 b e \operatorname {acosh}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d x^{2}}{2} + \frac {e x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d*x**2/2 + a*e*x**4/4 + b*d*x**2*acosh(c*x)/2 + b*e*x**4*acosh(c*x)/4 - b*d*x*sqrt(c**2*x**2 - 1)
/(4*c) - b*e*x**3*sqrt(c**2*x**2 - 1)/(16*c) - b*d*acosh(c*x)/(4*c**2) - 3*b*e*x*sqrt(c**2*x**2 - 1)/(32*c**3)
- 3*b*e*acosh(c*x)/(32*c**4), Ne(c, 0)), ((a + I*pi*b/2)*(d*x**2/2 + e*x**4/4), True))

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