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

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

[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

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Rubi [A]  time = 0.109358, antiderivative size = 122, normalized size of antiderivative = 1., 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 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]

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 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 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]

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*}

Mathematica [A]  time = 0.131741, 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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Maple [A]  time = 0.013, size = 202, normalized size = 1.7 \begin{align*}{\frac{a{x}^{4}e}{4}}+{\frac{da{x}^{2}}{2}}+{\frac{b{\rm arccosh} \left (cx\right ){x}^{4}e}{4}}+{\frac{db{\rm arccosh} \left (cx\right ){x}^{2}}{2}}-{\frac{be{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{bdx}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{bd}{4\,{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}}-{\frac{3\,bex}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,be}{32\,{c}^{4}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

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*d*a*x^2+1/4*b*arccosh(c*x)*x^4*e+1/2*d*b*arccosh(c*x)*x^2-1/16*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*d*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))-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 = 1.10462, size = 235, normalized size = 1.93 \begin{align*} \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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\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} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b e \end{align*}

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)*sqrt(c^2))/(sqrt(c^2)*c^2)))*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)*sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*e

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Fricas [A]  time = 2.42342, size = 255, normalized size = 2.09 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 1.89702, size = 160, normalized size = 1.31 \begin{align*} \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} \end{align*}

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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Giac [A]  time = 1.343, size = 238, normalized size = 1.95 \begin{align*} \frac{1}{2} \, a d x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} - \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b d + \frac{1}{32} \,{\left (8 \, a x^{4} +{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{c^{2}} + \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b\right )} e \end{align*}

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]

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