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

Optimal. Leaf size=177 \[ \frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1} \left (49 c^2 d+30 e\right )}{1225 c^3}-\frac{4 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (49 c^2 d+30 e\right )}{3675 c^5}-\frac{8 b \sqrt{c x-1} \sqrt{c x+1} \left (49 c^2 d+30 e\right )}{3675 c^7}-\frac{b e x^6 \sqrt{c x-1} \sqrt{c x+1}}{49 c} \]

[Out]

(-8*b*(49*c^2*d + 30*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^7) - (4*b*(49*c^2*d + 30*e)*x^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x])/(3675*c^5) - (b*(49*c^2*d + 30*e)*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1225*c^3) - (b*e*x^6*Sqrt[-
1 + c*x]*Sqrt[1 + c*x])/(49*c) + (d*x^5*(a + b*ArcCosh[c*x]))/5 + (e*x^7*(a + b*ArcCosh[c*x]))/7

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Rubi [A]  time = 0.141917, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5786, 460, 100, 12, 74} \[ \frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1} \left (49 c^2 d+30 e\right )}{1225 c^3}-\frac{4 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (49 c^2 d+30 e\right )}{3675 c^5}-\frac{8 b \sqrt{c x-1} \sqrt{c x+1} \left (49 c^2 d+30 e\right )}{3675 c^7}-\frac{b e x^6 \sqrt{c x-1} \sqrt{c x+1}}{49 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b*(49*c^2*d + 30*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3675*c^7) - (4*b*(49*c^2*d + 30*e)*x^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x])/(3675*c^5) - (b*(49*c^2*d + 30*e)*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(1225*c^3) - (b*e*x^6*Sqrt[-
1 + c*x]*Sqrt[1 + c*x])/(49*c) + (d*x^5*(a + b*ArcCosh[c*x]))/5 + (e*x^7*(a + b*ArcCosh[c*x]))/7

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 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+e 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} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{35} (b c) \int \frac{x^5 \left (7 d+5 e x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b e x^6 \sqrt{-1+c x} \sqrt{1+c x}}{49 c}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{245} \left (b c \left (-49 d-\frac{30 e}{c^2}\right )\right ) \int \frac{x^5}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b \left (49 c^2 d+30 e\right ) x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c^3}-\frac{b e x^6 \sqrt{-1+c x} \sqrt{1+c x}}{49 c}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b \left (49 c^2 d+30 e\right )\right ) \int \frac{4 x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1225 c^3}\\ &=-\frac{b \left (49 c^2 d+30 e\right ) x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c^3}-\frac{b e x^6 \sqrt{-1+c x} \sqrt{1+c x}}{49 c}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (4 b \left (49 c^2 d+30 e\right )\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{1225 c^3}\\ &=-\frac{4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^5}-\frac{b \left (49 c^2 d+30 e\right ) x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c^3}-\frac{b e x^6 \sqrt{-1+c x} \sqrt{1+c x}}{49 c}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (4 b \left (49 c^2 d+30 e\right )\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3675 c^5}\\ &=-\frac{4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^5}-\frac{b \left (49 c^2 d+30 e\right ) x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c^3}-\frac{b e x^6 \sqrt{-1+c x} \sqrt{1+c x}}{49 c}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (8 b \left (49 c^2 d+30 e\right )\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3675 c^5}\\ &=-\frac{8 b \left (49 c^2 d+30 e\right ) \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^7}-\frac{4 b \left (49 c^2 d+30 e\right ) x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3675 c^5}-\frac{b \left (49 c^2 d+30 e\right ) x^4 \sqrt{-1+c x} \sqrt{1+c x}}{1225 c^3}-\frac{b e x^6 \sqrt{-1+c x} \sqrt{1+c x}}{49 c}+\frac{1}{5} d x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{7} e x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.106008, size = 122, normalized size = 0.69 \[ \frac{1}{35} a x^5 \left (7 d+5 e x^2\right )-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (3 c^6 \left (49 d x^4+25 e x^6\right )+2 c^4 \left (98 d x^2+45 e x^4\right )+8 c^2 \left (49 d+15 e x^2\right )+240 e\right )}{3675 c^7}+\frac{1}{35} b x^5 \cosh ^{-1}(c x) \left (7 d+5 e x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x^5*(7*d + 5*e*x^2))/35 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e + 8*c^2*(49*d + 15*e*x^2) + 2*c^4*(98*d*x^
2 + 45*e*x^4) + 3*c^6*(49*d*x^4 + 25*e*x^6)))/(3675*c^7) + (b*x^5*(7*d + 5*e*x^2)*ArcCosh[c*x])/35

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Maple [A]  time = 0.029, size = 133, normalized size = 0.8 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{7}{x}^{5}d}{5}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccosh} \left (cx\right )e{c}^{7}{x}^{7}}{7}}+{\frac{{\rm arccosh} \left (cx\right ){c}^{7}{x}^{5}d}{5}}-{\frac{75\,{c}^{6}e{x}^{6}+147\,{c}^{6}d{x}^{4}+90\,{c}^{4}e{x}^{4}+196\,{c}^{4}d{x}^{2}+120\,{x}^{2}{c}^{2}e+392\,{c}^{2}d+240\,e}{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*(e*x^2+d)*(a+b*arccosh(c*x)),x)

[Out]

1/c^5*(a/c^2*(1/7*e*c^7*x^7+1/5*c^7*x^5*d)+b/c^2*(1/7*arccosh(c*x)*e*c^7*x^7+1/5*arccosh(c*x)*c^7*x^5*d-1/3675
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(75*c^6*e*x^6+147*c^6*d*x^4+90*c^4*e*x^4+196*c^4*d*x^2+120*c^2*e*x^2+392*c^2*d+24
0*e)))

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Maxima [A]  time = 1.16551, size = 240, normalized size = 1.36 \begin{align*} \frac{1}{7} \, a e x^{7} + \frac{1}{5} \, a d x^{5} + \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 + \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 e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*e*x^7 + 1/5*a*d*x^5 + 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 + 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*e

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Fricas [A]  time = 2.49662, size = 329, normalized size = 1.86 \begin{align*} \frac{525 \, a c^{7} e x^{7} + 735 \, a c^{7} d x^{5} + 105 \,{\left (5 \, b c^{7} e x^{7} + 7 \, b c^{7} d x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (75 \, b c^{6} e x^{6} + 3 \,{\left (49 \, b c^{6} d + 30 \, b c^{4} e\right )} x^{4} + 392 \, b c^{2} d + 4 \,{\left (49 \, b c^{4} d + 30 \, b c^{2} e\right )} x^{2} + 240 \, b e\right )} \sqrt{c^{2} x^{2} - 1}}{3675 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3675*(525*a*c^7*e*x^7 + 735*a*c^7*d*x^5 + 105*(5*b*c^7*e*x^7 + 7*b*c^7*d*x^5)*log(c*x + sqrt(c^2*x^2 - 1)) -
(75*b*c^6*e*x^6 + 3*(49*b*c^6*d + 30*b*c^4*e)*x^4 + 392*b*c^2*d + 4*(49*b*c^4*d + 30*b*c^2*e)*x^2 + 240*b*e)*
sqrt(c^2*x^2 - 1))/c^7

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Sympy [A]  time = 9.52757, size = 230, normalized size = 1.3 \begin{align*} \begin{cases} \frac{a d x^{5}}{5} + \frac{a e x^{7}}{7} + \frac{b d x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{b e x^{7} \operatorname{acosh}{\left (c x \right )}}{7} - \frac{b d x^{4} \sqrt{c^{2} x^{2} - 1}}{25 c} - \frac{b e x^{6} \sqrt{c^{2} x^{2} - 1}}{49 c} - \frac{4 b d x^{2} \sqrt{c^{2} x^{2} - 1}}{75 c^{3}} - \frac{6 b e x^{4} \sqrt{c^{2} x^{2} - 1}}{245 c^{3}} - \frac{8 b d \sqrt{c^{2} x^{2} - 1}}{75 c^{5}} - \frac{8 b e x^{2} \sqrt{c^{2} x^{2} - 1}}{245 c^{5}} - \frac{16 b e \sqrt{c^{2} x^{2} - 1}}{245 c^{7}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (\frac{d x^{5}}{5} + \frac{e x^{7}}{7}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d*x**5/5 + a*e*x**7/7 + b*d*x**5*acosh(c*x)/5 + b*e*x**7*acosh(c*x)/7 - b*d*x**4*sqrt(c**2*x**2 -
1)/(25*c) - b*e*x**6*sqrt(c**2*x**2 - 1)/(49*c) - 4*b*d*x**2*sqrt(c**2*x**2 - 1)/(75*c**3) - 6*b*e*x**4*sqrt(
c**2*x**2 - 1)/(245*c**3) - 8*b*d*sqrt(c**2*x**2 - 1)/(75*c**5) - 8*b*e*x**2*sqrt(c**2*x**2 - 1)/(245*c**5) -
16*b*e*sqrt(c**2*x**2 - 1)/(245*c**7), Ne(c, 0)), ((a + I*pi*b/2)*(d*x**5/5 + e*x**7/7), True))

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Giac [A]  time = 1.27582, size = 232, normalized size = 1.31 \begin{align*} \frac{1}{5} \, a d x^{5} + \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 + \frac{1}{245} \,{\left (35 \, a x^{7} +{\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\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*d*x^5 + 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 + 1/245*(35*a*x^7 + (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)*e