3.1.22 \(\int (d+e x)^2 (a+b \cosh ^{-1}(c x))^2 \, dx\) [22]

Optimal. Leaf size=259 \[ 2 b^2 d^2 x+\frac {4 b^2 e^2 x}{9 c^2}+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {4 b e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac {b d e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {2 b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac {d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e} \]

[Out]

2*b^2*d^2*x+4/9*b^2*e^2*x/c^2+1/2*b^2*d*e*x^2+2/27*b^2*e^2*x^3-1/3*d^3*(a+b*arccosh(c*x))^2/e-1/2*d*e*(a+b*arc
cosh(c*x))^2/c^2+1/3*(e*x+d)^3*(a+b*arccosh(c*x))^2/e-2*b*d^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
-4/9*b*e^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3-b*d*e*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)
^(1/2)/c-2/9*b*e^2*x^2*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c

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Rubi [A]
time = 0.76, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5963, 5975, 5893, 5915, 8, 5939, 30} \begin {gather*} -\frac {4 b e^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac {d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac {2 b d^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {b d e x \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac {2 b e^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}+\frac {4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*d^2*x + (4*b^2*e^2*x)/(9*c^2) + (b^2*d*e*x^2)/2 + (2*b^2*e^2*x^3)/27 - (2*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*(a + b*ArcCosh[c*x]))/c - (4*b*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3) - (b*d*e*x*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (2*b*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCos
h[c*x]))/(9*c) - (d^3*(a + b*ArcCosh[c*x])^2)/(3*e) - (d*e*(a + b*ArcCosh[c*x])^2)/(2*c^2) + ((d + e*x)^3*(a +
b*ArcCosh[c*x])^2)/(3*e)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5893

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*Arc
Cosh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && NeQ[n
, -1]

Rule 5915

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sy
mbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Dist[b*
(n/(2*c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Int[(1 + c*x)^(p + 1/2)*(-
1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c
*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]

Rule 5939

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1
*e2*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)
^p*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 +
e2*x)^p/(-1 + c*x)^p], Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IG
tQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rule 5963

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 5975

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g
_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x
)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m,
0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] |
| EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))

Rubi steps

\begin {align*} \int (d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac {(2 b c) \int \frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}\\ &=\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac {(2 b c) \int \left (\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d^2 e x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 d e^2 x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{3 e}\\ &=\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\left (2 b c d^2\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (2 b c d^3\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 e}-(2 b c d e) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{3} \left (2 b c e^2\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {2 b d^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {b d e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {2 b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}+\left (2 b^2 d^2\right ) \int 1 \, dx+\left (b^2 d e\right ) \int x \, dx-\frac {(b d e) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c}+\frac {1}{9} \left (2 b^2 e^2\right ) \int x^2 \, dx-\frac {\left (4 b e^2\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c}\\ &=2 b^2 d^2 x+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {4 b e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac {b d e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {2 b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac {d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}+\frac {\left (4 b^2 e^2\right ) \int 1 \, dx}{9 c^2}\\ &=2 b^2 d^2 x+\frac {4 b^2 e^2 x}{9 c^2}+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {4 b e^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3}-\frac {b d e x \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c}-\frac {2 b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c}-\frac {d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}-\frac {d e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 c^2}+\frac {(d+e x)^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 e}\\ \end {align*}

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Mathematica [A]
time = 0.41, size = 360, normalized size = 1.39 \begin {gather*} a^2 d^2 x+2 b^2 d^2 x+\frac {4 b^2 e^2 x}{9 c^2}+a^2 d e x^2+\frac {1}{2} b^2 d e x^2+\frac {1}{3} a^2 e^2 x^3+\frac {2}{27} b^2 e^2 x^3-\frac {2 a b d^2 \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {4 a b e^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3}-\frac {a b d e x \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {2 a b e^2 x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c}-\frac {b \left (-6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right ) \cosh ^{-1}(c x)}{9 c^3}+\frac {1}{6} b^2 \left (-\frac {3 d e}{c^2}+2 x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \cosh ^{-1}(c x)^2-\frac {a b d e \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*d^2*x + 2*b^2*d^2*x + (4*b^2*e^2*x)/(9*c^2) + a^2*d*e*x^2 + (b^2*d*e*x^2)/2 + (a^2*e^2*x^3)/3 + (2*b^2*e^2
*x^3)/27 - (2*a*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (4*a*b*e^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c^3) - (a*
b*d*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (2*a*b*e^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(9*c) - (b*(-6*a*c^3*x*
(3*d^2 + 3*d*e*x + e^2*x^2) + 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)))*Arc
Cosh[c*x])/(9*c^3) + (b^2*((-3*d*e)/c^2 + 2*x*(3*d^2 + 3*d*e*x + e^2*x^2))*ArcCosh[c*x]^2)/6 - (a*b*d*e*Log[c*
x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/c^2

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(526\) vs. \(2(227)=454\).
time = 1.64, size = 527, normalized size = 2.03 \[-\frac {b^{2} d e}{4 c^{2}}-\frac {2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{3} \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 e \sqrt {c^{2} x^{2}-1}}-\frac {2 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2} e^{2}}{9 c}-\frac {2 a b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}\, x^{2}}{9 c}-\frac {a b e \sqrt {c x -1}\, \sqrt {c x +1}\, d x}{c}+2 a b e \,\mathrm {arccosh}\left (c x \right ) x^{2} d -\frac {4 a b \,e^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3}}-\frac {2 a b \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2}}{c}-\frac {2 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, d^{2}}{c}-\frac {4 b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, e^{2}}{9 c^{3}}-\frac {a b e \sqrt {c x -1}\, \sqrt {c x +1}\, d \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{c^{2} \sqrt {c^{2} x^{2}-1}}+\frac {2 a b \,\mathrm {arccosh}\left (c x \right ) d^{3}}{3 e}+2 b^{2} d^{2} x +2 a b \,\mathrm {arccosh}\left (c x \right ) x \,d^{2}+\frac {2 a b \,e^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{3}+b^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{2} d e -\frac {b^{2} \mathrm {arccosh}\left (c x \right )^{2} d e}{2 c^{2}}-\frac {b^{2} \mathrm {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}\, x d e}{c}+\frac {4 b^{2} e^{2} x}{9 c^{2}}+\frac {b^{2} d e \,x^{2}}{2}+\frac {a^{2} d^{3}}{3 e}+\frac {b^{2} \mathrm {arccosh}\left (c x \right )^{2} x^{3} e^{2}}{3}+b^{2} \mathrm {arccosh}\left (c x \right )^{2} x \,d^{2}+a^{2} e \,x^{2} d +\frac {2 b^{2} e^{2} x^{3}}{27}+a^{2} x \,d^{2}+\frac {a^{2} e^{2} x^{3}}{3}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/c^2*b^2*d*e-2/3*a*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))-2/9/c*b
^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^2*e^2-2/9/c*a*b*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^2-1/c*a*b*e*(c
*x-1)^(1/2)*(c*x+1)^(1/2)*d*x+2*a*b*e*arccosh(c*x)*x^2*d-4/9/c^3*a*b*e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)-2/c*a*b*(
c*x-1)^(1/2)*(c*x+1)^(1/2)*d^2-2/c*b^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*d^2-4/9/c^3*b^2*arccosh(c*x)*(
c*x-1)^(1/2)*(c*x+1)^(1/2)*e^2-1/c^2*a*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))+2/3*a*b/e*arccosh(c*x)*d^3+2*b^2*d^2*x+2*a*b*arccosh(c*x)*x*d^2+2/3*a*b*e^2*arccosh(c*x)*x^3+b^2*arccos
h(c*x)^2*x^2*d*e-1/2/c^2*b^2*arccosh(c*x)^2*d*e-1/c*b^2*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*d*e+4/9*b^2
*e^2*x/c^2+1/2*b^2*d*e*x^2+1/3*a^2/e*d^3+1/3*b^2*arccosh(c*x)^2*x^3*e^2+b^2*arccosh(c*x)^2*x*d^2+a^2*e*x^2*d+2
/27*b^2*e^2*x^3+a^2*x*d^2+1/3*a^2*e^2*x^3

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*d^2*x*arccosh(c*x)^2 + 1/3*a^2*x^3*e^2 + a^2*d*x^2*e + 2*b^2*d^2*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) +
a^2*d^2*x + (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)*c)/c^3))*a*b*
d*e + 2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d^2/c + 2/9*(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))*a*b*e^2 + 1/3*(b^2*x^3*e^2 + 3*b^2*d*x^2*e)*log(c*x + sqrt(c*x + 1)*sqrt(c*x -
1))^2 - integrate(2/3*(b^2*c^3*x^5*e^2 + 3*b^2*c^3*d*x^4*e - b^2*c*x^3*e^2 - 3*b^2*c*d*x^2*e + (b^2*c^2*x^4*e
^2 + 3*b^2*c^2*d*x^3*e)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^3*x^3 + (c^2*x^
2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (225) = 450\).
time = 0.40, size = 618, normalized size = 2.39 \begin {gather*} \frac {27 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} d x^{2} \cosh \left (1\right ) + 54 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d^{2} x + 2 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} + 12 \, b^{2} c x\right )} \cosh \left (1\right )^{2} + 9 \, {\left (2 \, b^{2} c^{3} x^{3} \cosh \left (1\right )^{2} + 2 \, b^{2} c^{3} x^{3} \sinh \left (1\right )^{2} + 6 \, b^{2} c^{3} d^{2} x + 3 \, {\left (2 \, b^{2} c^{3} d x^{2} - b^{2} c d\right )} \cosh \left (1\right ) + {\left (4 \, b^{2} c^{3} x^{3} \cosh \left (1\right ) + 6 \, b^{2} c^{3} d x^{2} - 3 \, b^{2} c d\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} + 12 \, b^{2} c x\right )} \sinh \left (1\right )^{2} + 6 \, {\left (6 \, a b c^{3} x^{3} \cosh \left (1\right )^{2} + 6 \, a b c^{3} x^{3} \sinh \left (1\right )^{2} + 18 \, a b c^{3} d^{2} x + 9 \, {\left (2 \, a b c^{3} d x^{2} - a b c d\right )} \cosh \left (1\right ) + 3 \, {\left (4 \, a b c^{3} x^{3} \cosh \left (1\right ) + 6 \, a b c^{3} d x^{2} - 3 \, a b c d\right )} \sinh \left (1\right ) - {\left (9 \, b^{2} c^{2} d x \cosh \left (1\right ) + 18 \, b^{2} c^{2} d^{2} + 2 \, {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \sinh \left (1\right )^{2} + {\left (9 \, b^{2} c^{2} d x + 4 \, {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (27 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} d x^{2} + 4 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} + 12 \, b^{2} c x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right ) - 6 \, {\left (9 \, a b c^{2} d x \cosh \left (1\right ) + 18 \, a b c^{2} d^{2} + 2 \, {\left (a b c^{2} x^{2} + 2 \, a b\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a b c^{2} x^{2} + 2 \, a b\right )} \sinh \left (1\right )^{2} + {\left (9 \, a b c^{2} d x + 4 \, {\left (a b c^{2} x^{2} + 2 \, a b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{54 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(27*(2*a^2 + b^2)*c^3*d*x^2*cosh(1) + 54*(a^2 + 2*b^2)*c^3*d^2*x + 2*((9*a^2 + 2*b^2)*c^3*x^3 + 12*b^2*c*
x)*cosh(1)^2 + 9*(2*b^2*c^3*x^3*cosh(1)^2 + 2*b^2*c^3*x^3*sinh(1)^2 + 6*b^2*c^3*d^2*x + 3*(2*b^2*c^3*d*x^2 - b
^2*c*d)*cosh(1) + (4*b^2*c^3*x^3*cosh(1) + 6*b^2*c^3*d*x^2 - 3*b^2*c*d)*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1))^
2 + 2*((9*a^2 + 2*b^2)*c^3*x^3 + 12*b^2*c*x)*sinh(1)^2 + 6*(6*a*b*c^3*x^3*cosh(1)^2 + 6*a*b*c^3*x^3*sinh(1)^2
+ 18*a*b*c^3*d^2*x + 9*(2*a*b*c^3*d*x^2 - a*b*c*d)*cosh(1) + 3*(4*a*b*c^3*x^3*cosh(1) + 6*a*b*c^3*d*x^2 - 3*a*
b*c*d)*sinh(1) - (9*b^2*c^2*d*x*cosh(1) + 18*b^2*c^2*d^2 + 2*(b^2*c^2*x^2 + 2*b^2)*cosh(1)^2 + 2*(b^2*c^2*x^2
+ 2*b^2)*sinh(1)^2 + (9*b^2*c^2*d*x + 4*(b^2*c^2*x^2 + 2*b^2)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))*log(c*x + s
qrt(c^2*x^2 - 1)) + (27*(2*a^2 + b^2)*c^3*d*x^2 + 4*((9*a^2 + 2*b^2)*c^3*x^3 + 12*b^2*c*x)*cosh(1))*sinh(1) -
6*(9*a*b*c^2*d*x*cosh(1) + 18*a*b*c^2*d^2 + 2*(a*b*c^2*x^2 + 2*a*b)*cosh(1)^2 + 2*(a*b*c^2*x^2 + 2*a*b)*sinh(1
)^2 + (9*a*b*c^2*d*x + 4*(a*b*c^2*x^2 + 2*a*b)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))/c^3

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Sympy [C] Result contains complex when optimal does not.
time = 0.33, size = 461, normalized size = 1.78 \begin {gather*} \begin {cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac {a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname {acosh}{\left (c x \right )} + 2 a b d e x^{2} \operatorname {acosh}{\left (c x \right )} + \frac {2 a b e^{2} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} - \frac {2 a b d^{2} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {a b d e x \sqrt {c^{2} x^{2} - 1}}{c} - \frac {2 a b e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {a b d e \operatorname {acosh}{\left (c x \right )}}{c^{2}} - \frac {4 a b e^{2} \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname {acosh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname {acosh}^{2}{\left (c x \right )} + \frac {b^{2} d e x^{2}}{2} + \frac {b^{2} e^{2} x^{3} \operatorname {acosh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} e^{2} x^{3}}{27} - \frac {2 b^{2} d^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {b^{2} d e x \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{c} - \frac {2 b^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c} - \frac {b^{2} d e \operatorname {acosh}^{2}{\left (c x \right )}}{2 c^{2}} + \frac {4 b^{2} e^{2} x}{9 c^{2}} - \frac {4 b^{2} e^{2} \sqrt {c^{2} x^{2} - 1} \operatorname {acosh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right )^{2} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*x + a**2*d*e*x**2 + a**2*e**2*x**3/3 + 2*a*b*d**2*x*acosh(c*x) + 2*a*b*d*e*x**2*acosh(c*x
) + 2*a*b*e**2*x**3*acosh(c*x)/3 - 2*a*b*d**2*sqrt(c**2*x**2 - 1)/c - a*b*d*e*x*sqrt(c**2*x**2 - 1)/c - 2*a*b*
e**2*x**2*sqrt(c**2*x**2 - 1)/(9*c) - a*b*d*e*acosh(c*x)/c**2 - 4*a*b*e**2*sqrt(c**2*x**2 - 1)/(9*c**3) + b**2
*d**2*x*acosh(c*x)**2 + 2*b**2*d**2*x + b**2*d*e*x**2*acosh(c*x)**2 + b**2*d*e*x**2/2 + b**2*e**2*x**3*acosh(c
*x)**2/3 + 2*b**2*e**2*x**3/27 - 2*b**2*d**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/c - b**2*d*e*x*sqrt(c**2*x**2 - 1)
*acosh(c*x)/c - 2*b**2*e**2*x**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(9*c) - b**2*d*e*acosh(c*x)**2/(2*c**2) + 4*b*
*2*e**2*x/(9*c**2) - 4*b**2*e**2*sqrt(c**2*x**2 - 1)*acosh(c*x)/(9*c**3), Ne(c, 0)), ((a + I*pi*b/2)**2*(d**2*
x + d*e*x**2 + e**2*x**3/3), True))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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