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

Optimal. Leaf size=132 \[ \frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac{b d \left (\frac{3 e^2}{c^2}+2 d^2\right ) \cosh ^{-1}(c x)}{6 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (d+e x)^2}{9 c} \]

[Out]

-(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(9*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) + 5
*c^2*d*e*x))/(18*c^3) - (b*d*(2*d^2 + (3*e^2)/c^2)*ArcCosh[c*x])/(6*e) + ((d + e*x)^3*(a + b*ArcCosh[c*x]))/(3
*e)

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Rubi [A]  time = 0.104595, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5802, 100, 147, 52} \[ \frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac{b d \left (\frac{3 e^2}{c^2}+2 d^2\right ) \cosh ^{-1}(c x)}{6 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (d+e x)^2}{9 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(9*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*(4*c^2*d^2 + e^2) + 5
*c^2*d*e*x))/(18*c^3) - (b*d*(2*d^2 + (3*e^2)/c^2)*ArcCosh[c*x])/(6*e) + ((d + e*x)^3*(a + b*ArcCosh[c*x]))/(3
*e)

Rule 5802

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)
*(a + b*ArcCosh[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcCosh[c*x
])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 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 (d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{(b c) \int \frac{(d+e x)^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 e}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{9 c}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{b \int \frac{(d+e x) \left (3 c^2 d^2+2 e^2+5 c^2 d e x\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c e}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{9 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}-\frac{1}{6} \left (b d \left (\frac{2 c d^2}{e}+\frac{3 e}{c}\right )\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{9 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \left (4 \left (4 c^2 d^2+e^2\right )+5 c^2 d e x\right )}{18 c^3}-\frac{b d \left (2 d^2+\frac{3 e^2}{c^2}\right ) \cosh ^{-1}(c x)}{6 e}+\frac{(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 e}\\ \end{align*}

Mathematica [A]  time = 0.199151, size = 142, normalized size = 1.08 \[ a d^2 x+a d e x^2+\frac{1}{3} a e^2 x^3-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )}{18 c^3}-\frac{b d e \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )}{2 c^2}+\frac{1}{3} b x \cosh ^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d^2*x + a*d*e*x^2 + (a*e^2*x^3)/3 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x
^2)))/(18*c^3) + (b*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcCosh[c*x])/3 - (b*d*e*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c
*x]])/(2*c^2)

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Maple [B]  time = 0.006, size = 274, normalized size = 2.1 \begin{align*}{\frac{a{x}^{3}{e}^{2}}{3}}+a{x}^{2}de+ax{d}^{2}+{\frac{a{d}^{3}}{3\,e}}+{\frac{b{\rm arccosh} \left (cx\right ){x}^{3}{e}^{2}}{3}}+b{\rm arccosh} \left (cx\right ){x}^{2}de+b{\rm arccosh} \left (cx\right )x{d}^{2}+{\frac{b{d}^{3}{\rm arccosh} \left (cx\right )}{3\,e}}-{\frac{b{d}^{3}}{3\,e}\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{b{x}^{2}{e}^{2}}{9\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{bedx}{2\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{2}}{c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{bed}{2\,{c}^{2}}\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{2\,b{e}^{2}}{9\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3*e^2+a*x^2*d*e+a*x*d^2+1/3*a/e*d^3+1/3*b*arccosh(c*x)*x^3*e^2+b*arccosh(c*x)*x^2*d*e+b*arccosh(c*x)*x
*d^2+1/3*b/e*arccosh(c*x)*d^3-1/3*b/e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d^3*ln(c*x+(c^2*x^2-1)^(1/
2))-1/9*b/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^2*e^2-1/2*b*d*e*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/c*b*(c*x-1)^(1/2)*
(c*x+1)^(1/2)*d^2-1/2/c^2*b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d*ln(c*x+(c^2*x^2-1)^(1/2))-2/9*b/
c^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2

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Maxima [A]  time = 1.11011, size = 238, normalized size = 1.8 \begin{align*} \frac{1}{3} \, a e^{2} x^{3} + a d e x^{2} + \frac{1}{2} \,{\left (2 \, x^{2} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} + \frac{\log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d e + \frac{1}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b e^{2} + a d^{2} x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b d^{2}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*e^2*x^3 + a*d*e*x^2 + 1/2*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2*c^2*x + 2*sqrt(c^2*x^
2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^2)))*b*d*e + 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^
2*x^2 - 1)/c^4))*b*e^2 + a*d^2*x + (c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^2/c

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Fricas [A]  time = 2.31549, size = 324, normalized size = 2.45 \begin{align*} \frac{6 \, a c^{3} e^{2} x^{3} + 18 \, a c^{3} d e x^{2} + 18 \, a c^{3} d^{2} x + 3 \,{\left (2 \, b c^{3} e^{2} x^{3} + 6 \, b c^{3} d e x^{2} + 6 \, b c^{3} d^{2} x - 3 \, b c d e\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (2 \, b c^{2} e^{2} x^{2} + 9 \, b c^{2} d e x + 18 \, b c^{2} d^{2} + 4 \, b e^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{18 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(6*a*c^3*e^2*x^3 + 18*a*c^3*d*e*x^2 + 18*a*c^3*d^2*x + 3*(2*b*c^3*e^2*x^3 + 6*b*c^3*d*e*x^2 + 6*b*c^3*d^2
*x - 3*b*c*d*e)*log(c*x + sqrt(c^2*x^2 - 1)) - (2*b*c^2*e^2*x^2 + 9*b*c^2*d*e*x + 18*b*c^2*d^2 + 4*b*e^2)*sqrt
(c^2*x^2 - 1))/c^3

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Sympy [A]  time = 1.02843, size = 197, normalized size = 1.49 \begin{align*} \begin{cases} a d^{2} x + a d e x^{2} + \frac{a e^{2} x^{3}}{3} + b d^{2} x \operatorname{acosh}{\left (c x \right )} + b d e x^{2} \operatorname{acosh}{\left (c x \right )} + \frac{b e^{2} x^{3} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{b d^{2} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{b d e x \sqrt{c^{2} x^{2} - 1}}{2 c} - \frac{b e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{9 c} - \frac{b d e \operatorname{acosh}{\left (c x \right )}}{2 c^{2}} - \frac{2 b e^{2} \sqrt{c^{2} x^{2} - 1}}{9 c^{3}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x + a*d*e*x**2 + a*e**2*x**3/3 + b*d**2*x*acosh(c*x) + b*d*e*x**2*acosh(c*x) + b*e**2*x**3*a
cosh(c*x)/3 - b*d**2*sqrt(c**2*x**2 - 1)/c - b*d*e*x*sqrt(c**2*x**2 - 1)/(2*c) - b*e**2*x**2*sqrt(c**2*x**2 -
1)/(9*c) - b*d*e*acosh(c*x)/(2*c**2) - 2*b*e**2*sqrt(c**2*x**2 - 1)/(9*c**3), Ne(c, 0)), ((a + I*pi*b/2)*(d**2
*x + d*e*x**2 + e**2*x**3/3), True))

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Giac [A]  time = 1.35861, size = 266, normalized size = 2.02 \begin{align*}{\left (x \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{\sqrt{c^{2} x^{2} - 1}}{c}\right )} b d^{2} + a d^{2} x + \frac{1}{9} \,{\left (3 \, a x^{3} +{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b\right )} e^{2} + \frac{1}{2} \,{\left (2 \, a d x^{2} +{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} - \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b d\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*log(c*x + sqrt(c^2*x^2 - 1)) - sqrt(c^2*x^2 - 1)/c)*b*d^2 + a*d^2*x + 1/9*(3*a*x^3 + (3*x^3*log(c*x + sqrt(
c^2*x^2 - 1)) - ((c^2*x^2 - 1)^(3/2) + 3*sqrt(c^2*x^2 - 1))/c^3)*b)*e^2 + 1/2*(2*a*d*x^2 + (2*x^2*log(c*x + sq
rt(c^2*x^2 - 1)) - c*(sqrt(c^2*x^2 - 1)*x/c^2 - log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^2*abs(c))))*b*d)*e