\(\int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx\) [23]

Optimal result

Integrand size = 16, antiderivative size = 150 \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c}-\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 e}-\frac {e (a+b \text {arccosh}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e} \]

[Out]

2*b^2*d*x+1/4*b^2*e*x^2-1/2*d^2*(a+b*arccosh(c*x))^2/e-1/4*e*(a+b*arccosh(c*x))^2/c^2+1/2*(e*x+d)^2*(a+b*arcco
sh(c*x))^2/e-2*b*d*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/2*b*e*x*(a+b*arccosh(c*x))*(c*x-1)^(1/2)
*(c*x+1)^(1/2)/c

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5963, 5975, 5893, 5915, 8, 5939, 30} \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=-\frac {e (a+b \text {arccosh}(c x))^2}{4 c^2}-\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e}-\frac {2 b d \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c}-\frac {b e x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 c}+2 b^2 d x+\frac {1}{4} b^2 e x^2 \]

[In]

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

[Out]

2*b^2*d*x + (b^2*e*x^2)/4 - (2*b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/c - (b*e*x*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c) - (d^2*(a + b*ArcCosh[c*x])^2)/(2*e) - (e*(a + b*ArcCosh[c*x])^2)/
(4*c^2) + ((d + e*x)^2*(a + b*ArcCosh[c*x])^2)/(2*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*} \text {integral}& = \frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e}-\frac {(b c) \int \frac {(d+e x)^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e} \\ & = \frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e}-\frac {(b c) \int \left (\frac {d^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 d e x (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {e^2 x^2 (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}\right ) \, dx}{e} \\ & = \frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e}-(2 b c d) \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {\left (b c d^2\right ) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e}-(b c e) \int \frac {x^2 (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 {b e x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c}-\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e}+\left (2 b^2 d\right ) \int 1 \, dx+\frac {1}{2} \left (b^2 e\right ) \int x \, dx-\frac {(b e) \int \frac {a+b \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c} \\ & = 2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{2 c}-\frac {d^2 (a+b \text {arccosh}(c x))^2}{2 e}-\frac {e (a+b \text {arccosh}(c x))^2}{4 c^2}+\frac {(d+e x)^2 (a+b \text {arccosh}(c x))^2}{2 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.16 \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=\frac {c \left (2 a^2 c x (2 d+e x)-2 a b \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)+b^2 c x (8 d+e x)\right )-2 b c \left (-2 a c x (2 d+e x)+b \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)\right ) \text {arccosh}(c x)+b^2 \left (4 c^2 d x+e \left (-1+2 c^2 x^2\right )\right ) \text {arccosh}(c x)^2-2 a b e \log \left (c x+\sqrt {-1+c x} \sqrt {1+c x}\right )}{4 c^2} \]

[In]

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

[Out]

(c*(2*a^2*c*x*(2*d + e*x) - 2*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d + e*x) + b^2*c*x*(8*d + e*x)) - 2*b*c*(-2*
a*c*x*(2*d + e*x) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d + e*x))*ArcCosh[c*x] + b^2*(4*c^2*d*x + e*(-1 + 2*c^2*
x^2))*ArcCosh[c*x]^2 - 2*a*b*e*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(4*c^2)

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.49

method	result	size
parts	\(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4 c}+d \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) d c x +\frac {c \,\operatorname {arccosh}\left (c x \right ) x^{2} e}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{4 c \sqrt {c^{2} x^{2}-1}}\right )}{c}\)	\(224\)
derivativedivides	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}+d c \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) c^{2} x d +\frac {\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{4 \sqrt {c^{2} x^{2}-1}}\right )}{c}}{c}\)	\(236\)
default	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\frac {e \left (2 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}-2 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x -\operatorname {arccosh}\left (c x \right )^{2}+c^{2} x^{2}\right )}{4}+d c \left (\operatorname {arccosh}\left (c x \right )^{2} x c -2 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+2 c x \right )\right )}{c}+\frac {2 a b \left (\operatorname {arccosh}\left (c x \right ) c^{2} x d +\frac {\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+e c x \sqrt {c^{2} x^{2}-1}+e \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{4 \sqrt {c^{2} x^{2}-1}}\right )}{c}}{c}\)	\(236\)

[In]

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

[Out]

a^2*(1/2*e*x^2+d*x)+b^2/c*(1/4*e*(2*arccosh(c*x)^2*x^2*c^2-2*(c*x+1)^(1/2)*arccosh(c*x)*(c*x-1)^(1/2)*c*x-arcc
osh(c*x)^2+c^2*x^2)/c+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*(arccos
h(c*x)*d*c*x+1/2*c*arccosh(c*x)*x^2*e-1/4/c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(4*d*c*(c^2*x^2-1)^(1/2)+e*c*x*(c^2*x^
2-1)^(1/2)+e*ln(c*x+(c^2*x^2-1)^(1/2)))/(c^2*x^2-1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.23 \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} c^{2} e x^{2} + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x + {\left (2 \, b^{2} c^{2} e x^{2} + 4 \, b^{2} c^{2} d x - b^{2} e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} + 2 \, {\left (2 \, a b c^{2} e x^{2} + 4 \, a b c^{2} d x - a b e - {\left (b^{2} c e x + 4 \, b^{2} c d\right )} \sqrt {c^{2} x^{2} - 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (a b c e x + 4 \, a b c d\right )} \sqrt {c^{2} x^{2} - 1}}{4 \, c^{2}} \]

[In]

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

[Out]

1/4*((2*a^2 + b^2)*c^2*e*x^2 + 4*(a^2 + 2*b^2)*c^2*d*x + (2*b^2*c^2*e*x^2 + 4*b^2*c^2*d*x - b^2*e)*log(c*x + s
qrt(c^2*x^2 - 1))^2 + 2*(2*a*b*c^2*e*x^2 + 4*a*b*c^2*d*x - a*b*e - (b^2*c*e*x + 4*b^2*c*d)*sqrt(c^2*x^2 - 1))*
log(c*x + sqrt(c^2*x^2 - 1)) - 2*(a*b*c*e*x + 4*a*b*c*d)*sqrt(c^2*x^2 - 1))/c^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

b^2*d*x*arccosh(c*x)^2 + 1/2*a^2*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)*c)/c^3))*a*b*e + 1/2*(x^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2 - 2*integrate((c^3*x^4
+ sqrt(c*x + 1)*sqrt(c*x - 1)*c^2*x^3 - c*x^2)*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))*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. \[ \int (d+e x) (a+b \text {arccosh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[In]

int((a + b*acosh(c*x))^2*(d + e*x),x)

[Out]

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