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

Optimal. Leaf size=191 \[ \frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{96 c^3}-\frac{b \left (24 c^2 d^2 e^2+8 c^4 d^4+3 e^4\right ) \cosh ^{-1}(c x)}{32 c^4 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (d+e x)^3}{16 c}-\frac{7 b d \sqrt{c x-1} \sqrt{c x+1} (d+e x)^2}{48 c} \]

[Out]

(-7*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(48*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^3)/(16*c)
- (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d*(19*c^2*d^2 + 16*e^2) + e*(26*c^2*d^2 + 9*e^2)*x))/(96*c^3) - (b*(8*c^
4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*ArcCosh[c*x])/(32*c^4*e) + ((d + e*x)^4*(a + b*ArcCosh[c*x]))/(4*e)

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Rubi [A]  time = 0.144522, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5802, 100, 153, 147, 52} \[ \frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (e x \left (26 c^2 d^2+9 e^2\right )+4 d \left (19 c^2 d^2+16 e^2\right )\right )}{96 c^3}-\frac{b \left (24 c^2 d^2 e^2+8 c^4 d^4+3 e^4\right ) \cosh ^{-1}(c x)}{32 c^4 e}-\frac{b \sqrt{c x-1} \sqrt{c x+1} (d+e x)^3}{16 c}-\frac{7 b d \sqrt{c x-1} \sqrt{c x+1} (d+e x)^2}{48 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^2)/(48*c) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x)^3)/(16*c)
- (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d*(19*c^2*d^2 + 16*e^2) + e*(26*c^2*d^2 + 9*e^2)*x))/(96*c^3) - (b*(8*c^
4*d^4 + 24*c^2*d^2*e^2 + 3*e^4)*ArcCosh[c*x])/(32*c^4*e) + ((d + e*x)^4*(a + b*ArcCosh[c*x]))/(4*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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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)^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 e}-\frac{(b c) \int \frac{(d+e x)^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 e}\\ &=-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^3}{16 c}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 e}-\frac{b \int \frac{(d+e x)^2 \left (4 c^2 d^2+3 e^2+7 c^2 d e x\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c e}\\ &=-\frac{7 b d \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{48 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^3}{16 c}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 e}-\frac{b \int \frac{(d+e x) \left (c^2 d \left (12 c^2 d^2+23 e^2\right )+c^2 e \left (26 c^2 d^2+9 e^2\right ) x\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{48 c^3 e}\\ &=-\frac{7 b d \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{48 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^3}{16 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 e}-\frac{\left (b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right )\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 c^3 e}\\ &=-\frac{7 b d \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^2}{48 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} (d+e x)^3}{16 c}-\frac{b \sqrt{-1+c x} \sqrt{1+c x} \left (4 d \left (19 c^2 d^2+16 e^2\right )+e \left (26 c^2 d^2+9 e^2\right ) x\right )}{96 c^3}-\frac{b \left (8 c^4 d^4+24 c^2 d^2 e^2+3 e^4\right ) \cosh ^{-1}(c x)}{32 c^4 e}+\frac{(d+e x)^4 \left (a+b \cosh ^{-1}(c x)\right )}{4 e}\\ \end{align*}

Mathematica [A]  time = 0.290879, size = 193, normalized size = 1.01 \[ \frac{24 a c^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-b c \sqrt{c x-1} \sqrt{c x+1} \left (c^2 \left (72 d^2 e x+96 d^3+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )+24 b c^4 x \cosh ^{-1}(c x) \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-9 b e \left (8 c^2 d^2+e^2\right ) \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )}{96 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(e^2*(64*d + 9*e*x)
+ c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 24*b*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x
^3)*ArcCosh[c*x] - 9*b*e*(8*c^2*d^2 + e^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(96*c^4)

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Maple [B]  time = 0.006, size = 408, normalized size = 2.1 \begin{align*}{\frac{a{e}^{3}{x}^{4}}{4}}+ad{e}^{2}{x}^{3}+{\frac{3\,a{d}^{2}e{x}^{2}}{2}}+ax{d}^{3}+{\frac{a{d}^{4}}{4\,e}}+{\frac{b{\rm arccosh} \left (cx\right ){e}^{3}{x}^{4}}{4}}+b{\rm arccosh} \left (cx\right )d{e}^{2}{x}^{3}+{\frac{3\,b{\rm arccosh} \left (cx\right ){d}^{2}e{x}^{2}}{2}}+b{\rm arccosh} \left (cx\right )x{d}^{3}+{\frac{b{\rm arccosh} \left (cx\right ){d}^{4}}{4\,e}}-{\frac{b{d}^{4}}{4\,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{e}^{3}{x}^{3}}{16\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{x}^{2}d{e}^{2}}{3\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,be{d}^{2}x}{4\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{b{d}^{3}}{c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,be{d}^{2}}{4\,{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{3\,b{e}^{3}x}{32\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{2\,b{e}^{2}d}{3\,{c}^{3}}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{3\,b{e}^{3}}{32\,{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((e*x+d)^3*(a+b*arccosh(c*x)),x)

[Out]

1/4*a*e^3*x^4+a*d*e^2*x^3+3/2*a*d^2*e*x^2+a*x*d^3+1/4*a/e*d^4+1/4*b*arccosh(c*x)*e^3*x^4+b*arccosh(c*x)*d*e^2*
x^3+3/2*b*arccosh(c*x)*d^2*e*x^2+b*arccosh(c*x)*x*d^3+1/4*b/e*arccosh(c*x)*d^4-1/4*b/e*(c*x-1)^(1/2)*(c*x+1)^(
1/2)/(c^2*x^2-1)^(1/2)*d^4*ln(c*x+(c^2*x^2-1)^(1/2))-1/16/c*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*e^3*x^3-1/3*b/c*(c*x
-1)^(1/2)*(c*x+1)^(1/2)*x^2*d*e^2-3/4*b*d^2*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^3-3/4/c^2*b*e*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^(1/2)*d^2*ln(c*x+(c^2*x^2-1)^(1/2))-3/32/c^3*b*(c*x-
1)^(1/2)*(c*x+1)^(1/2)*e^3*x-2/3*b/c^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d*e^2-3/32/c^4*b*e^3*(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 = 1.19032, size = 382, normalized size = 2. \begin{align*} \frac{1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac{3}{2} \, a d^{2} e x^{2} + \frac{3}{4} \,{\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^{2} e + \frac{1}{3} \,{\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 d e^{2} + \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 e^{3} + a d^{3} x + \frac{{\left (c x \operatorname{arcosh}\left (c x\right ) - \sqrt{c^{2} x^{2} - 1}\right )} b d^{3}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + 3/4*(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^2*e + 1/3*(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*d*e^2 + 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*s
qrt(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*e^3 + a*d^3*x +
(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*b*d^3/c

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Fricas [A]  time = 2.36774, size = 468, normalized size = 2.45 \begin{align*} \frac{24 \, a c^{4} e^{3} x^{4} + 96 \, a c^{4} d e^{2} x^{3} + 144 \, a c^{4} d^{2} e x^{2} + 96 \, a c^{4} d^{3} x + 3 \,{\left (8 \, b c^{4} e^{3} x^{4} + 32 \, b c^{4} d e^{2} x^{3} + 48 \, b c^{4} d^{2} e x^{2} + 32 \, b c^{4} d^{3} x - 24 \, b c^{2} d^{2} e - 3 \, b e^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (6 \, b c^{3} e^{3} x^{3} + 32 \, b c^{3} d e^{2} x^{2} + 96 \, b c^{3} d^{3} + 64 \, b c d e^{2} + 9 \,{\left (8 \, b c^{3} d^{2} e + b c e^{3}\right )} x\right )} \sqrt{c^{2} x^{2} - 1}}{96 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(24*a*c^4*e^3*x^4 + 96*a*c^4*d*e^2*x^3 + 144*a*c^4*d^2*e*x^2 + 96*a*c^4*d^3*x + 3*(8*b*c^4*e^3*x^4 + 32*b
*c^4*d*e^2*x^3 + 48*b*c^4*d^2*e*x^2 + 32*b*c^4*d^3*x - 24*b*c^2*d^2*e - 3*b*e^3)*log(c*x + sqrt(c^2*x^2 - 1))
- (6*b*c^3*e^3*x^3 + 32*b*c^3*d*e^2*x^2 + 96*b*c^3*d^3 + 64*b*c*d*e^2 + 9*(8*b*c^3*d^2*e + b*c*e^3)*x)*sqrt(c^
2*x^2 - 1))/c^4

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Sympy [A]  time = 2.06011, size = 323, normalized size = 1.69 \begin{align*} \begin{cases} a d^{3} x + \frac{3 a d^{2} e x^{2}}{2} + a d e^{2} x^{3} + \frac{a e^{3} x^{4}}{4} + b d^{3} x \operatorname{acosh}{\left (c x \right )} + \frac{3 b d^{2} e x^{2} \operatorname{acosh}{\left (c x \right )}}{2} + b d e^{2} x^{3} \operatorname{acosh}{\left (c x \right )} + \frac{b e^{3} x^{4} \operatorname{acosh}{\left (c x \right )}}{4} - \frac{b d^{3} \sqrt{c^{2} x^{2} - 1}}{c} - \frac{3 b d^{2} e x \sqrt{c^{2} x^{2} - 1}}{4 c} - \frac{b d e^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{3 c} - \frac{b e^{3} x^{3} \sqrt{c^{2} x^{2} - 1}}{16 c} - \frac{3 b d^{2} e \operatorname{acosh}{\left (c x \right )}}{4 c^{2}} - \frac{2 b d e^{2} \sqrt{c^{2} x^{2} - 1}}{3 c^{3}} - \frac{3 b e^{3} x \sqrt{c^{2} x^{2} - 1}}{32 c^{3}} - \frac{3 b e^{3} \operatorname{acosh}{\left (c x \right )}}{32 c^{4}} & \text{for}\: c \neq 0 \\\left (a + \frac{i \pi b}{2}\right ) \left (d^{3} x + \frac{3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac{e^{3} x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x + 3*a*d**2*e*x**2/2 + a*d*e**2*x**3 + a*e**3*x**4/4 + b*d**3*x*acosh(c*x) + 3*b*d**2*e*x**
2*acosh(c*x)/2 + b*d*e**2*x**3*acosh(c*x) + b*e**3*x**4*acosh(c*x)/4 - b*d**3*sqrt(c**2*x**2 - 1)/c - 3*b*d**2
*e*x*sqrt(c**2*x**2 - 1)/(4*c) - b*d*e**2*x**2*sqrt(c**2*x**2 - 1)/(3*c) - b*e**3*x**3*sqrt(c**2*x**2 - 1)/(16
*c) - 3*b*d**2*e*acosh(c*x)/(4*c**2) - 2*b*d*e**2*sqrt(c**2*x**2 - 1)/(3*c**3) - 3*b*e**3*x*sqrt(c**2*x**2 - 1
)/(32*c**3) - 3*b*e**3*acosh(c*x)/(32*c**4), Ne(c, 0)), ((a + I*pi*b/2)*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**
3 + e**3*x**4/4), True))

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Giac [A]  time = 1.4652, size = 401, normalized size = 2.1 \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^{3} + a d^{3} x + \frac{1}{32} \,{\left (8 \, a x^{4} +{\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\right )} e^{3} + \frac{1}{3} \,{\left (3 \, a d 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 d\right )} e^{2} + \frac{3}{4} \,{\left (2 \, a d^{2} 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^{2}\right )} e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(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^3 + a*d^3*x + 1/32*(8*a*x^4 + (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)*e^3 + 1/3*(3*a*d*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*d)*e^2 + 3/4*(2*a*d^2*x^2 + (2*x^2*log(c*x + sqrt(c^2*x^2 - 1)) - c*(sqrt(c^2*x^2 - 1)*x/c^2 - l
og(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^2*abs(c))))*b*d^2)*e