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

Optimal. Leaf size=151 \[ -\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{76 b d x^2 \sqrt{c x-1} \sqrt{c x+1}}{3675 c^3}-\frac{152 b d \sqrt{c x-1} \sqrt{c x+1}}{3675 c^5}+\frac{1}{49} b c d x^6 \sqrt{c x-1} \sqrt{c x+1}-\frac{19 b d x^4 \sqrt{c x-1} \sqrt{c x+1}}{1225 c} \]

[Out]

(-152*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^5) - (76*b*d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^3) - (1
9*b*d*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1225*c) + (b*c*d*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/49 + (d*x^5*(a + b
*ArcCosh[c*x]))/5 - (c^2*d*x^7*(a + b*ArcCosh[c*x]))/7

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Rubi [A]  time = 0.150428, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {14, 5731, 12, 460, 100, 74} \[ -\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{76 b d x^2 \sqrt{c x-1} \sqrt{c x+1}}{3675 c^3}-\frac{152 b d \sqrt{c x-1} \sqrt{c x+1}}{3675 c^5}+\frac{1}{49} b c d x^6 \sqrt{c x-1} \sqrt{c x+1}-\frac{19 b d x^4 \sqrt{c x-1} \sqrt{c x+1}}{1225 c} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

(-152*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^5) - (76*b*d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^3) - (1
9*b*d*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1225*c) + (b*c*d*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/49 + (d*x^5*(a + b
*ArcCosh[c*x]))/5 - (c^2*d*x^7*(a + b*ArcCosh[c*x]))/7

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 74

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

Rubi steps

\begin{align*} \int x^4 \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d x^5 \left (7-5 c^2 x^2\right )}{35 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{35} (b c d) \int \frac{x^5 \left (7-5 c^2 x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{49} b c d x^6 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{245} (19 b c d) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{19 b d x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c}+\frac{1}{49} b c d x^6 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(19 b d) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1225 c}\\ &=-\frac{19 b d x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c}+\frac{1}{49} b c d x^6 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(76 b d) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1225 c}\\ &=-\frac{76 b d x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^3}-\frac{19 b d x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c}+\frac{1}{49} b c d x^6 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(76 b d) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3675 c^3}\\ &=-\frac{76 b d x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^3}-\frac{19 b d x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c}+\frac{1}{49} b c d x^6 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{(152 b d) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3675 c^3}\\ &=-\frac{152 b d \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^5}-\frac{76 b d x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^3}-\frac{19 b d x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c}+\frac{1}{49} b c d x^6 \sqrt{-1+c x} \sqrt{1+c x}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{7} c^2 d x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.156148, size = 91, normalized size = 0.6 \[ \frac{d \left (-105 a x^5 \left (5 c^2 x^2-7\right )+\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (75 c^6 x^6-57 c^4 x^4-76 c^2 x^2-152\right )}{c^5}-105 b x^5 \left (5 c^2 x^2-7\right ) \cosh ^{-1}(c x)\right )}{3675} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]

[Out]

(d*(-105*a*x^5*(-7 + 5*c^2*x^2) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6
))/c^5 - 105*b*x^5*(-7 + 5*c^2*x^2)*ArcCosh[c*x]))/3675

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Maple [A]  time = 0.014, size = 98, normalized size = 0.7 \begin{align*}{\frac{1}{{c}^{5}} \left ( -da \left ({\frac{{c}^{7}{x}^{7}}{7}}-{\frac{{c}^{5}{x}^{5}}{5}} \right ) -db \left ({\frac{{\rm arccosh} \left (cx\right ){c}^{7}{x}^{7}}{7}}-{\frac{{\rm arccosh} \left (cx\right ){c}^{5}{x}^{5}}{5}}-{\frac{75\,{c}^{6}{x}^{6}-57\,{c}^{4}{x}^{4}-76\,{c}^{2}{x}^{2}-152}{3675}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^5*(-d*a*(1/7*c^7*x^7-1/5*c^5*x^5)-d*b*(1/7*arccosh(c*x)*c^7*x^7-1/5*arccosh(c*x)*c^5*x^5-1/3675*(c*x-1)^(1
/2)*(c*x+1)^(1/2)*(75*c^6*x^6-57*c^4*x^4-76*c^2*x^2-152)))

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Maxima [A]  time = 1.16226, size = 248, normalized size = 1.64 \begin{align*} -\frac{1}{7} \, a c^{2} d x^{7} + \frac{1}{5} \, a d x^{5} - \frac{1}{245} \,{\left (35 \, x^{7} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{2} d + \frac{1}{75} \,{\left (15 \, x^{5} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/7*a*c^2*d*x^7 + 1/5*a*d*x^5 - 1/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^2*x^2 -
1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*c^2*d + 1/75*(15*x^5*arccosh(c*x) -
(3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*d

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Fricas [A]  time = 1.7331, size = 266, normalized size = 1.76 \begin{align*} -\frac{525 \, a c^{7} d x^{7} - 735 \, a c^{5} d x^{5} + 105 \,{\left (5 \, b c^{7} d x^{7} - 7 \, b c^{5} d x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (75 \, b c^{6} d x^{6} - 57 \, b c^{4} d x^{4} - 76 \, b c^{2} d x^{2} - 152 \, b d\right )} \sqrt{c^{2} x^{2} - 1}}{3675 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

-1/3675*(525*a*c^7*d*x^7 - 735*a*c^5*d*x^5 + 105*(5*b*c^7*d*x^7 - 7*b*c^5*d*x^5)*log(c*x + sqrt(c^2*x^2 - 1))
- (75*b*c^6*d*x^6 - 57*b*c^4*d*x^4 - 76*b*c^2*d*x^2 - 152*b*d)*sqrt(c^2*x^2 - 1))/c^5

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Sympy [A]  time = 8.58666, size = 158, normalized size = 1.05 \begin{align*} \begin{cases} - \frac{a c^{2} d x^{7}}{7} + \frac{a d x^{5}}{5} - \frac{b c^{2} d x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{b c d x^{6} \sqrt{c^{2} x^{2} - 1}}{49} + \frac{b d x^{5} \operatorname{acosh}{\left (c x \right )}}{5} - \frac{19 b d x^{4} \sqrt{c^{2} x^{2} - 1}}{1225 c} - \frac{76 b d x^{2} \sqrt{c^{2} x^{2} - 1}}{3675 c^{3}} - \frac{152 b d \sqrt{c^{2} x^{2} - 1}}{3675 c^{5}} & \text{for}\: c \neq 0 \\\frac{d x^{5} \left (a + \frac{i \pi b}{2}\right )}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**2*d*x**7/7 + a*d*x**5/5 - b*c**2*d*x**7*acosh(c*x)/7 + b*c*d*x**6*sqrt(c**2*x**2 - 1)/49 + b*
d*x**5*acosh(c*x)/5 - 19*b*d*x**4*sqrt(c**2*x**2 - 1)/(1225*c) - 76*b*d*x**2*sqrt(c**2*x**2 - 1)/(3675*c**3) -
152*b*d*sqrt(c**2*x**2 - 1)/(3675*c**5), Ne(c, 0)), (d*x**5*(a + I*pi*b/2)/5, True))

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Giac [A]  time = 1.99931, size = 238, normalized size = 1.58 \begin{align*} -\frac{1}{7} \, a c^{2} d x^{7} + \frac{1}{5} \, a d x^{5} - \frac{1}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b c^{2} d + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} b d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

-1/7*a*c^2*d*x^7 + 1/5*a*d*x^5 - 1/245*(35*x^7*log(c*x + sqrt(c^2*x^2 - 1)) - (5*(c^2*x^2 - 1)^(7/2) + 21*(c^2
*x^2 - 1)^(5/2) + 35*(c^2*x^2 - 1)^(3/2) + 35*sqrt(c^2*x^2 - 1))/c^7)*b*c^2*d + 1/75*(15*x^5*log(c*x + sqrt(c^
2*x^2 - 1)) - (3*(c^2*x^2 - 1)^(5/2) + 10*(c^2*x^2 - 1)^(3/2) + 15*sqrt(c^2*x^2 - 1))/c^5)*b*d