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

Optimal. Leaf size=181 \[ \frac{\left (1-a^2 x^2\right ) \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{15 a^5 \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 d \left (1-a^2 x^2\right )^2 \left (5 a^2 c+3 d\right )}{45 a^5 \sqrt{a x-1} \sqrt{a x+1}}+\frac{d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt{a x-1} \sqrt{a x+1}}+c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x) \]

[Out]

((15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*(1 - a^2*x^2))/(15*a^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (2*d*(5*a^2*c + 3*d)
*(1 - a^2*x^2)^2)/(45*a^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (d^2*(1 - a^2*x^2)^3)/(25*a^5*Sqrt[-1 + a*x]*Sqrt[1
+ a*x]) + c^2*x*ArcCosh[a*x] + (2*c*d*x^3*ArcCosh[a*x])/3 + (d^2*x^5*ArcCosh[a*x])/5

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Rubi [A]  time = 0.188672, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {194, 5705, 12, 520, 1247, 698} \[ \frac{\left (1-a^2 x^2\right ) \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{15 a^5 \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 d \left (1-a^2 x^2\right )^2 \left (5 a^2 c+3 d\right )}{45 a^5 \sqrt{a x-1} \sqrt{a x+1}}+\frac{d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt{a x-1} \sqrt{a x+1}}+c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^2*ArcCosh[a*x],x]

[Out]

((15*a^4*c^2 + 10*a^2*c*d + 3*d^2)*(1 - a^2*x^2))/(15*a^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (2*d*(5*a^2*c + 3*d)
*(1 - a^2*x^2)^2)/(45*a^5*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (d^2*(1 - a^2*x^2)^3)/(25*a^5*Sqrt[-1 + a*x]*Sqrt[1
+ a*x]) + c^2*x*ArcCosh[a*x] + (2*c*d*x^3*ArcCosh[a*x])/3 + (d^2*x^5*ArcCosh[a*x])/5

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5705

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(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}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 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 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \left (c+d x^2\right )^2 \cosh ^{-1}(a x) \, dx &=c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x)-a \int \frac{x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{15 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac{1}{15} a \int \frac{x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{\sqrt{-1+a^2 x^2}} \, dx}{15 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{15 c^2+10 c d x+3 d^2 x^2}{\sqrt{-1+a^2 x}} \, dx,x,x^2\right )}{30 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{15 a^4 c^2+10 a^2 c d+3 d^2}{a^4 \sqrt{-1+a^2 x}}+\frac{2 d \left (5 a^2 c+3 d\right ) \sqrt{-1+a^2 x}}{a^4}+\frac{3 d^2 \left (-1+a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )}{30 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \left (1-a^2 x^2\right )}{15 a^5 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^2}{45 a^5 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{d^2 \left (1-a^2 x^2\right )^3}{25 a^5 \sqrt{-1+a x} \sqrt{1+a x}}+c^2 x \cosh ^{-1}(a x)+\frac{2}{3} c d x^3 \cosh ^{-1}(a x)+\frac{1}{5} d^2 x^5 \cosh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.145899, size = 103, normalized size = 0.57 \[ \cosh ^{-1}(a x) \left (c^2 x+\frac{2}{3} c d x^3+\frac{d^2 x^5}{5}\right )-\frac{\sqrt{a x-1} \sqrt{a x+1} \left (a^4 \left (225 c^2+50 c d x^2+9 d^2 x^4\right )+4 a^2 d \left (25 c+3 d x^2\right )+24 d^2\right )}{225 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(24*d^2 + 4*a^2*d*(25*c + 3*d*x^2) + a^4*(225*c^2 + 50*c*d*x^2 + 9*d^2*x^4)))/(
225*a^5) + (c^2*x + (2*c*d*x^3)/3 + (d^2*x^5)/5)*ArcCosh[a*x]

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Maple [A]  time = 0.011, size = 113, normalized size = 0.6 \begin{align*}{\frac{1}{a} \left ({\frac{a{\rm arccosh} \left (ax\right ){x}^{5}{d}^{2}}{5}}+{\frac{2\,a{\rm arccosh} \left (ax\right )cd{x}^{3}}{3}}+{\rm arccosh} \left (ax\right ){c}^{2}ax-{\frac{9\,{a}^{4}{d}^{2}{x}^{4}+50\,{a}^{4}cd{x}^{2}+225\,{a}^{4}{c}^{2}+12\,{a}^{2}{d}^{2}{x}^{2}+100\,{a}^{2}cd+24\,{d}^{2}}{225\,{a}^{4}}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(1/5*a*arccosh(a*x)*x^5*d^2+2/3*a*arccosh(a*x)*c*d*x^3+arccosh(a*x)*c^2*a*x-1/225/a^4*(a*x-1)^(1/2)*(a*x+1
)^(1/2)*(9*a^4*d^2*x^4+50*a^4*c*d*x^2+225*a^4*c^2+12*a^2*d^2*x^2+100*a^2*c*d+24*d^2))

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Maxima [A]  time = 1.12145, size = 208, normalized size = 1.15 \begin{align*} -\frac{1}{225} \,{\left (\frac{9 \, \sqrt{a^{2} x^{2} - 1} d^{2} x^{4}}{a^{2}} + \frac{50 \, \sqrt{a^{2} x^{2} - 1} c d x^{2}}{a^{2}} + \frac{225 \, \sqrt{a^{2} x^{2} - 1} c^{2}}{a^{2}} + \frac{12 \, \sqrt{a^{2} x^{2} - 1} d^{2} x^{2}}{a^{4}} + \frac{100 \, \sqrt{a^{2} x^{2} - 1} c d}{a^{4}} + \frac{24 \, \sqrt{a^{2} x^{2} - 1} d^{2}}{a^{6}}\right )} a + \frac{1}{15} \,{\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \operatorname{arcosh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/225*(9*sqrt(a^2*x^2 - 1)*d^2*x^4/a^2 + 50*sqrt(a^2*x^2 - 1)*c*d*x^2/a^2 + 225*sqrt(a^2*x^2 - 1)*c^2/a^2 + 1
2*sqrt(a^2*x^2 - 1)*d^2*x^2/a^4 + 100*sqrt(a^2*x^2 - 1)*c*d/a^4 + 24*sqrt(a^2*x^2 - 1)*d^2/a^6)*a + 1/15*(3*d^
2*x^5 + 10*c*d*x^3 + 15*c^2*x)*arccosh(a*x)

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Fricas [A]  time = 2.1763, size = 269, normalized size = 1.49 \begin{align*} \frac{15 \,{\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (9 \, a^{4} d^{2} x^{4} + 225 \, a^{4} c^{2} + 100 \, a^{2} c d + 2 \,{\left (25 \, a^{4} c d + 6 \, a^{2} d^{2}\right )} x^{2} + 24 \, d^{2}\right )} \sqrt{a^{2} x^{2} - 1}}{225 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/225*(15*(3*a^5*d^2*x^5 + 10*a^5*c*d*x^3 + 15*a^5*c^2*x)*log(a*x + sqrt(a^2*x^2 - 1)) - (9*a^4*d^2*x^4 + 225*
a^4*c^2 + 100*a^2*c*d + 2*(25*a^4*c*d + 6*a^2*d^2)*x^2 + 24*d^2)*sqrt(a^2*x^2 - 1))/a^5

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Sympy [A]  time = 2.93482, size = 199, normalized size = 1.1 \begin{align*} \begin{cases} c^{2} x \operatorname{acosh}{\left (a x \right )} + \frac{2 c d x^{3} \operatorname{acosh}{\left (a x \right )}}{3} + \frac{d^{2} x^{5} \operatorname{acosh}{\left (a x \right )}}{5} - \frac{c^{2} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{2 c d x^{2} \sqrt{a^{2} x^{2} - 1}}{9 a} - \frac{d^{2} x^{4} \sqrt{a^{2} x^{2} - 1}}{25 a} - \frac{4 c d \sqrt{a^{2} x^{2} - 1}}{9 a^{3}} - \frac{4 d^{2} x^{2} \sqrt{a^{2} x^{2} - 1}}{75 a^{3}} - \frac{8 d^{2} \sqrt{a^{2} x^{2} - 1}}{75 a^{5}} & \text{for}\: a \neq 0 \\\frac{i \pi \left (c^{2} x + \frac{2 c d x^{3}}{3} + \frac{d^{2} x^{5}}{5}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**2*x*acosh(a*x) + 2*c*d*x**3*acosh(a*x)/3 + d**2*x**5*acosh(a*x)/5 - c**2*sqrt(a**2*x**2 - 1)/a -
2*c*d*x**2*sqrt(a**2*x**2 - 1)/(9*a) - d**2*x**4*sqrt(a**2*x**2 - 1)/(25*a) - 4*c*d*sqrt(a**2*x**2 - 1)/(9*a*
*3) - 4*d**2*x**2*sqrt(a**2*x**2 - 1)/(75*a**3) - 8*d**2*sqrt(a**2*x**2 - 1)/(75*a**5), Ne(a, 0)), (I*pi*(c**2
*x + 2*c*d*x**3/3 + d**2*x**5/5)/2, True))

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Giac [A]  time = 1.11683, size = 203, normalized size = 1.12 \begin{align*} \frac{1}{15} \,{\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{225 \, \sqrt{a^{2} x^{2} - 1} a^{4} c^{2} + 50 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} a^{2} c d + 150 \, \sqrt{a^{2} x^{2} - 1} a^{2} c d + 9 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} d^{2} + 30 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} d^{2} + 45 \, \sqrt{a^{2} x^{2} - 1} d^{2}}{225 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*d^2*x^5 + 10*c*d*x^3 + 15*c^2*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 1/225*(225*sqrt(a^2*x^2 - 1)*a^4*c^2 +
50*(a^2*x^2 - 1)^(3/2)*a^2*c*d + 150*sqrt(a^2*x^2 - 1)*a^2*c*d + 9*(a^2*x^2 - 1)^(5/2)*d^2 + 30*(a^2*x^2 - 1)
^(3/2)*d^2 + 45*sqrt(a^2*x^2 - 1)*d^2)/a^5