\(\int (d+e x^2) (a+b \text {arccosh}(c x))^2 \, dx\) [527]

Optimal result

Integrand size = 18, antiderivative size = 168 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d x+\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {4 b e \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c^3}-\frac {2 b e x^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{9 c}+d x (a+b \text {arccosh}(c x))^2+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5909, 5879, 5915, 8, 5883, 5939, 30} \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=-\frac {4 b e \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c^3}+d x (a+b \text {arccosh}(c x))^2-\frac {2 b d \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c}+\frac {1}{3} e x^3 (a+b \text {arccosh}(c x))^2-\frac {2 b e x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{9 c}+\frac {4 b^2 e x}{9 c^2}+2 b^2 d x+\frac {2}{27} b^2 e x^3 \]

[In]

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

[Out]

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

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 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5883

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

Rule 5909

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

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]

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.04 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (9 d+e x^2\right )\right )+2 b^2 c x \left (6 e+c^2 \left (27 d+e x^2\right )\right )-6 b \left (-3 a c^3 x \left (3 d+e x^2\right )+b \sqrt {-1+c x} \sqrt {1+c x} \left (2 e+c^2 \left (9 d+e x^2\right )\right )\right ) \text {arccosh}(c x)+9 b^2 c^3 x \left (3 d+e x^2\right ) \text {arccosh}(c x)^2}{27 c^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.21

method	result	size
parts	\(a^{2} \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b^{2} \left (\frac {e \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27 c^{2}}+d \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\frac {c \,\operatorname {arccosh}\left (c x \right ) x^{3} e}{3}+\operatorname {arccosh}\left (c x \right ) d c x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9 c^{2}}\right )}{c}\)	\(203\)
derivativedivides	\(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\)	\(217\)
default	\(\frac {\frac {a^{2} \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b^{2} \left (d \,c^{2} \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )+\frac {e \left (9 \operatorname {arccosh}\left (c x \right )^{2} x^{3} c^{3}-6 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}-12 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c^{3} x^{3}+12 c x \right )}{27}\right )}{c^{2}}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arccosh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (c^{2} e \,x^{2}+9 c^{2} d +2 e \right )}{9}\right )}{c^{2}}}{c}\)	\(217\)

[In]

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

[Out]

a^2*(1/3*x^3*e+d*x)+b^2/c*(1/27*e*(9*arccosh(c*x)^2*x^3*c^3-6*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c^2*x^2
-12*arccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)+2*c^3*x^3+12*c*x)/c^2+d*(arccosh(c*x)^2*x*c-2*arccosh(c*x)*(c*x-1
)^(1/2)*(c*x+1)^(1/2)+2*c*x))+2*a*b/c*(1/3*c*arccosh(c*x)*x^3*e+arccosh(c*x)*d*c*x-1/9/c^2*(c*x-1)^(1/2)*(c*x+
1)^(1/2)*(c^2*e*x^2+9*c^2*d+2*e))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.24 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \, {\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 3 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d + 4 \, b^{2} c e\right )} x + 6 \, {\left (3 \, a b c^{3} e x^{3} + 9 \, a b c^{3} d x - {\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 6 \, {\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d + 2 \, a b e\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, c^{3}} \]

[In]

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

[Out]

1/27*((9*a^2 + 2*b^2)*c^3*e*x^3 + 9*(b^2*c^3*e*x^3 + 3*b^2*c^3*d*x)*log(c*x + sqrt(c^2*x^2 - 1))^2 + 3*(9*(a^2
+ 2*b^2)*c^3*d + 4*b^2*c*e)*x + 6*(3*a*b*c^3*e*x^3 + 9*a*b*c^3*d*x - (b^2*c^2*e*x^2 + 9*b^2*c^2*d + 2*b^2*e)*
sqrt(c^2*x^2 - 1))*log(c*x + sqrt(c^2*x^2 - 1)) - 6*(a*b*c^2*e*x^2 + 9*a*b*c^2*d + 2*a*b*e)*sqrt(c^2*x^2 - 1))
/c^3

Sympy [F]

\[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x^{2}\right )\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.30 \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} e x^{3} \operatorname {arcosh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} e x^{3} + b^{2} d x \operatorname {arcosh}\left (c x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} a b e - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} - 1} \operatorname {arcosh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} a b d}{c} \]

[In]

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

[Out]

1/3*b^2*e*x^3*arccosh(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arccosh(c*x)^2 + 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 - 2/27*(3*c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1
)/c^4)*arccosh(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*e + 2*b^2*d*(x - sqrt(c^2*x^2 - 1)*arccosh(c*x)/c) + a^2*d*x +
2*(c*x*arccosh(c*x) - sqrt(c^2*x^2 - 1))*a*b*d/c

Giac [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((e*x^2+d)*(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

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right ) (a+b \text {arccosh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,\left (e\,x^2+d\right ) \,d x \]

[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)