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

Optimal. Leaf size=135 \[ -\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b d x \sqrt{c x-1} \sqrt{c x+1}}{24 c^3}-\frac{b d \cosh ^{-1}(c x)}{24 c^4}+\frac{1}{36} b c d x^5 \sqrt{c x-1} \sqrt{c x+1}-\frac{b d x^3 \sqrt{c x-1} \sqrt{c x+1}}{36 c} \]

[Out]

-(b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^3) - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) + (b*c*d*x^5*Sq
rt[-1 + c*x]*Sqrt[1 + c*x])/36 - (b*d*ArcCosh[c*x])/(24*c^4) + (d*x^4*(a + b*ArcCosh[c*x]))/4 - (c^2*d*x^6*(a
+ b*ArcCosh[c*x]))/6

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Rubi [A]  time = 0.139109, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {14, 5731, 12, 460, 100, 90, 52} \[ -\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b d x \sqrt{c x-1} \sqrt{c x+1}}{24 c^3}-\frac{b d \cosh ^{-1}(c x)}{24 c^4}+\frac{1}{36} b c d x^5 \sqrt{c x-1} \sqrt{c x+1}-\frac{b d x^3 \sqrt{c x-1} \sqrt{c x+1}}{36 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^3) - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) + (b*c*d*x^5*Sq
rt[-1 + c*x]*Sqrt[1 + c*x])/36 - (b*d*ArcCosh[c*x])/(24*c^4) + (d*x^4*(a + b*ArcCosh[c*x]))/4 - (c^2*d*x^6*(a
+ b*ArcCosh[c*x]))/6

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5731

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

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

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 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-c^2 d 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} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d x^4 \left (3-2 c^2 x^2\right )}{12 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{12} (b c d) \int \frac{x^4 \left (3-2 c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{36} b c d x^5 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} (b c d) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b d x^3 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{36} b c d x^5 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(b d) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{36 c}\\ &=-\frac{b d x^3 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{36} b c d x^5 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(b d) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{12 c}\\ &=-\frac{b d x \sqrt{-1+c x} \sqrt{1+c x}}{24 c^3}-\frac{b d x^3 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{36} b c d x^5 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(b d) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{24 c^3}\\ &=-\frac{b d x \sqrt{-1+c x} \sqrt{1+c x}}{24 c^3}-\frac{b d x^3 \sqrt{-1+c x} \sqrt{1+c x}}{36 c}+\frac{1}{36} b c d x^5 \sqrt{-1+c x} \sqrt{1+c x}-\frac{b d \cosh ^{-1}(c x)}{24 c^4}+\frac{1}{4} d x^4 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{6} c^2 d x^6 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0920071, size = 166, normalized size = 1.23 \[ -\frac{1}{6} a c^2 d x^6+\frac{1}{4} a d x^4-\frac{1}{6} b c^2 d x^6 \cosh ^{-1}(c x)-\frac{b d x \sqrt{c x-1} \sqrt{c x+1}}{24 c^3}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{c x-1}}{\sqrt{c x+1}}\right )}{12 c^4}+\frac{1}{36} b c d x^5 \sqrt{c x-1} \sqrt{c x+1}-\frac{b d x^3 \sqrt{c x-1} \sqrt{c x+1}}{36 c}+\frac{1}{4} b d x^4 \cosh ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*d*x^4)/4 - (a*c^2*d*x^6)/6 - (b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^3) - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1
+ c*x])/(36*c) + (b*c*d*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/36 + (b*d*x^4*ArcCosh[c*x])/4 - (b*c^2*d*x^6*ArcCos
h[c*x])/6 - (b*d*ArcTanh[Sqrt[-1 + c*x]/Sqrt[1 + c*x]])/(12*c^4)

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Maple [A]  time = 0.017, size = 160, normalized size = 1.2 \begin{align*} -{\frac{{c}^{2}da{x}^{6}}{6}}+{\frac{da{x}^{4}}{4}}-{\frac{{c}^{2}db{\rm arccosh} \left (cx\right ){x}^{6}}{6}}+{\frac{db{\rm arccosh} \left (cx\right ){x}^{4}}{4}}+{\frac{dbc{x}^{5}}{36}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{db{x}^{3}}{36\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{dbx}{24\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{db}{24\,{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^3*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x)

[Out]

-1/6*c^2*d*a*x^6+1/4*d*a*x^4-1/6*c^2*d*b*arccosh(c*x)*x^6+1/4*d*b*arccosh(c*x)*x^4+1/36*b*c*d*x^5*(c*x-1)^(1/2
)*(c*x+1)^(1/2)-1/36*b*d*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/24*b*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-1/24/c^4
*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))

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Maxima [A]  time = 1.09547, size = 297, normalized size = 2.2 \begin{align*} -\frac{1}{6} \, a c^{2} d x^{6} + \frac{1}{4} \, a d x^{4} - \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 c^{2} 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 d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a*c^2*d*x^6 + 1/4*a*d*x^4 - 1/288*(48*x^6*arccosh(c*x) - (8*sqrt(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*c^2*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*d

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Fricas [A]  time = 1.82995, size = 243, normalized size = 1.8 \begin{align*} -\frac{12 \, a c^{6} d x^{6} - 18 \, a c^{4} d x^{4} + 3 \,{\left (4 \, b c^{6} d x^{6} - 6 \, b c^{4} d x^{4} + b d\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (2 \, b c^{5} d x^{5} - 2 \, b c^{3} d x^{3} - 3 \, b c d x\right )} \sqrt{c^{2} x^{2} - 1}}{72 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(12*a*c^6*d*x^6 - 18*a*c^4*d*x^4 + 3*(4*b*c^6*d*x^6 - 6*b*c^4*d*x^4 + b*d)*log(c*x + sqrt(c^2*x^2 - 1))
- (2*b*c^5*d*x^5 - 2*b*c^3*d*x^3 - 3*b*c*d*x)*sqrt(c^2*x^2 - 1))/c^4

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Sympy [A]  time = 5.38028, size = 144, normalized size = 1.07 \begin{align*} \begin{cases} - \frac{a c^{2} d x^{6}}{6} + \frac{a d x^{4}}{4} - \frac{b c^{2} d x^{6} \operatorname{acosh}{\left (c x \right )}}{6} + \frac{b c d x^{5} \sqrt{c^{2} x^{2} - 1}}{36} + \frac{b d x^{4} \operatorname{acosh}{\left (c x \right )}}{4} - \frac{b d x^{3} \sqrt{c^{2} x^{2} - 1}}{36 c} - \frac{b d x \sqrt{c^{2} x^{2} - 1}}{24 c^{3}} - \frac{b d \operatorname{acosh}{\left (c x \right )}}{24 c^{4}} & \text{for}\: c \neq 0 \\\frac{d x^{4} \left (a + \frac{i \pi b}{2}\right )}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**2*d*x**6/6 + a*d*x**4/4 - b*c**2*d*x**6*acosh(c*x)/6 + b*c*d*x**5*sqrt(c**2*x**2 - 1)/36 + b*
d*x**4*acosh(c*x)/4 - b*d*x**3*sqrt(c**2*x**2 - 1)/(36*c) - b*d*x*sqrt(c**2*x**2 - 1)/(24*c**3) - b*d*acosh(c*
x)/(24*c**4), Ne(c, 0)), (d*x**4*(a + I*pi*b/2)/4, True))

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Giac [A]  time = 1.87797, size = 273, normalized size = 2.02 \begin{align*} -\frac{1}{6} \, a c^{2} d x^{6} + \frac{1}{4} \, a d x^{4} - \frac{1}{288} \,{\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 c^{2} d + \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a*c^2*d*x^6 + 1/4*a*d*x^4 - 1/288*(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*c^2*d + 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