3.462 \(\int x^3 (d+e x^2) (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=161 \[ \frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (9 c^2 d+5 e\right )}{144 c^3}-\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac{b \left (9 c^2 d+5 e\right ) \cosh ^{-1}(c x)}{96 c^6}-\frac{b e x^5 \sqrt{c x-1} \sqrt{c x+1}}{36 c} \]

[Out]

-(b*(9*c^2*d + 5*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c^5) - (b*(9*c^2*d + 5*e)*x^3*Sqrt[-1 + c*x]*Sqrt[1 +
c*x])/(144*c^3) - (b*e*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) - (b*(9*c^2*d + 5*e)*ArcCosh[c*x])/(96*c^6) +
(d*x^4*(a + b*ArcCosh[c*x]))/4 + (e*x^6*(a + b*ArcCosh[c*x]))/6

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Rubi [A]  time = 0.136614, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5786, 460, 100, 12, 90, 52} \[ \frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (9 c^2 d+5 e\right )}{144 c^3}-\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left (9 c^2 d+5 e\right )}{96 c^5}-\frac{b \left (9 c^2 d+5 e\right ) \cosh ^{-1}(c x)}{96 c^6}-\frac{b e x^5 \sqrt{c x-1} \sqrt{c x+1}}{36 c} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

-(b*(9*c^2*d + 5*e)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c^5) - (b*(9*c^2*d + 5*e)*x^3*Sqrt[-1 + c*x]*Sqrt[1 +
c*x])/(144*c^3) - (b*e*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) - (b*(9*c^2*d + 5*e)*ArcCosh[c*x])/(96*c^6) +
(d*x^4*(a + b*ArcCosh[c*x]))/4 + (e*x^6*(a + b*ArcCosh[c*x]))/6

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 100

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^3 \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{24} (b c) \int \frac{x^4 \left (6 d+4 e x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b e x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{36} \left (b c \left (9 d+\frac{5 e}{c^2}\right )\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b \left (9 c^2 d+5 e\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{144 c^3}\\ &=-\frac{b \left (9 c^2 d+5 e\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{48 c^3}\\ &=-\frac{b \left (9 c^2 d+5 e\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{96 c^5}-\frac{b \left (9 c^2 d+5 e\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \left (9 c^2 d+5 e\right )\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{96 c^5}\\ &=-\frac{b \left (9 c^2 d+5 e\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{96 c^5}-\frac{b \left (9 c^2 d+5 e\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{144 c^3}-\frac{b e x^5 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}-\frac{b \left (9 c^2 d+5 e\right ) \cosh ^{-1}(c x)}{96 c^6}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{6} e x^6 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.166379, size = 140, normalized size = 0.87 \[ \frac{24 a c^6 x^4 \left (3 d+2 e x^2\right )-b c x \sqrt{c x-1} \sqrt{c x+1} \left (2 c^4 \left (9 d x^2+4 e x^4\right )+c^2 \left (27 d+10 e x^2\right )+15 e\right )+24 b c^6 x^4 \cosh ^{-1}(c x) \left (3 d+2 e x^2\right )-6 b \left (9 c^2 d+5 e\right ) \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )}{288 c^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

(24*a*c^6*x^4*(3*d + 2*e*x^2) - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15*e + c^2*(27*d + 10*e*x^2) + 2*c^4*(9*d*
x^2 + 4*e*x^4)) + 24*b*c^6*x^4*(3*d + 2*e*x^2)*ArcCosh[c*x] - 6*b*(9*c^2*d + 5*e)*ArcTanh[Sqrt[(-1 + c*x)/(1 +
c*x)]])/(288*c^6)

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Maple [A]  time = 0.019, size = 250, normalized size = 1.6 \begin{align*}{\frac{ae{x}^{6}}{6}}+{\frac{a{x}^{4}d}{4}}+{\frac{b{\rm arccosh} \left (cx\right )e{x}^{6}}{6}}+{\frac{b{\rm arccosh} \left (cx\right ){x}^{4}d}{4}}-{\frac{be{x}^{5}}{36\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{bd{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,be{x}^{3}}{144\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,bdx}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,bd}{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}}}}-{\frac{5\,bex}{96\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,be}{96\,{c}^{6}}\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^3*(e*x^2+d)*(a+b*arccosh(c*x)),x)

[Out]

1/6*a*e*x^6+1/4*a*x^4*d+1/6*b*arccosh(c*x)*e*x^6+1/4*b*arccosh(c*x)*x^4*d-1/36*b*e*x^5*(c*x-1)^(1/2)*(c*x+1)^(
1/2)/c-1/16*b*d*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-5/144/c^3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e*x^3-3/32*b*d*x*(c*
x-1)^(1/2)*(c*x+1)^(1/2)/c^3-3/32/c^4*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-5/96/c^5*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e*x-5/96/c^6*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 = 1.1482, size = 289, normalized size = 1.8 \begin{align*} \frac{1}{6} \, a e x^{6} + \frac{1}{4} \, a d x^{4} + \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 d + \frac{1}{288} \,{\left (48 \, x^{6} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} - 1} x}{c^{6}} + \frac{15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

1/6*a*e*x^6 + 1/4*a*d*x^4 + 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*d + 1/288*(48*x^6*arccosh(c*x) - (8*s
qrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt
(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^6))*c)*b*e

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Fricas [A]  time = 2.31987, size = 308, normalized size = 1.91 \begin{align*} \frac{48 \, a c^{6} e x^{6} + 72 \, a c^{6} d x^{4} + 3 \,{\left (16 \, b c^{6} e x^{6} + 24 \, b c^{6} d x^{4} - 9 \, b c^{2} d - 5 \, b e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (8 \, b c^{5} e x^{5} + 2 \,{\left (9 \, b c^{5} d + 5 \, b c^{3} e\right )} x^{3} + 3 \,{\left (9 \, b c^{3} d + 5 \, b c e\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{288 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

1/288*(48*a*c^6*e*x^6 + 72*a*c^6*d*x^4 + 3*(16*b*c^6*e*x^6 + 24*b*c^6*d*x^4 - 9*b*c^2*d - 5*b*e)*log(c*x + sqr
t(c^2*x^2 - 1)) - (8*b*c^5*e*x^5 + 2*(9*b*c^5*d + 5*b*c^3*e)*x^3 + 3*(9*b*c^3*d + 5*b*c*e)*x)*sqrt(c^2*x^2 - 1
))/c^6

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Sympy [A]  time = 6.39488, size = 212, normalized size = 1.32 \begin{align*} \begin{cases} \frac{a d x^{4}}{4} + \frac{a e x^{6}}{6} + \frac{b d x^{4} \operatorname{acosh}{\left (c x \right )}}{4} + \frac{b e x^{6} \operatorname{acosh}{\left (c x \right )}}{6} - \frac{b d x^{3} \sqrt{c^{2} x^{2} - 1}}{16 c} - \frac{b e x^{5} \sqrt{c^{2} x^{2} - 1}}{36 c} - \frac{3 b d x \sqrt{c^{2} x^{2} - 1}}{32 c^{3}} - \frac{5 b e x^{3} \sqrt{c^{2} x^{2} - 1}}{144 c^{3}} - \frac{3 b d \operatorname{acosh}{\left (c x \right )}}{32 c^{4}} - \frac{5 b e x \sqrt{c^{2} x^{2} - 1}}{96 c^{5}} - \frac{5 b e \operatorname{acosh}{\left (c x \right )}}{96 c^{6}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d x^{4}}{4} + \frac{e x^{6}}{6}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x**2+d)*(a+b*acosh(c*x)),x)

[Out]

Piecewise((a*d*x**4/4 + a*e*x**6/6 + b*d*x**4*acosh(c*x)/4 + b*e*x**6*acosh(c*x)/6 - b*d*x**3*sqrt(c**2*x**2 -
1)/(16*c) - b*e*x**5*sqrt(c**2*x**2 - 1)/(36*c) - 3*b*d*x*sqrt(c**2*x**2 - 1)/(32*c**3) - 5*b*e*x**3*sqrt(c**
2*x**2 - 1)/(144*c**3) - 3*b*d*acosh(c*x)/(32*c**4) - 5*b*e*x*sqrt(c**2*x**2 - 1)/(96*c**5) - 5*b*e*acosh(c*x)
/(96*c**6), Ne(c, 0)), ((a + I*pi*b/2)*(d*x**4/4 + e*x**6/6), True))

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Giac [A]  time = 1.34462, size = 267, normalized size = 1.66 \begin{align*} \frac{1}{4} \, a d x^{4} + \frac{1}{32} \,{\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 d + \frac{1}{288} \,{\left (48 \, a x^{6} +{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} + \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x - \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

1/4*a*d*x^4 + 1/32*(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*d + 1/288*(48*a*x^6 + (48*x^6*log(c*x + sqrt(c^2*x^2 - 1))
- (sqrt(c^2*x^2 - 1)*(2*x^2*(4*x^2/c^2 + 5/c^4) + 15/c^6)*x - 15*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^6*
abs(c)))*c)*b)*e