\(\int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx\) [107]

Optimal result

Integrand size = 21, antiderivative size = 110 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {b^2 e (c+d x)^2}{4 d}-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{2 d}-\frac {e (a+b \text {arccosh}(c+d x))^2}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5883, 5939, 5893, 30} \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{2 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))}{2 d}-\frac {e (a+b \text {arccosh}(c+d x))^2}{4 d}+\frac {b^2 e (c+d x)^2}{4 d} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

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

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 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 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 5996

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

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.52 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {e \left ((c+d x) \left (2 a^2 (c+d x)+b^2 (c+d x)-2 a b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right )-2 b (c+d x) \left (-2 a (c+d x)+b \sqrt {-1+c+d x} \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+b^2 \left (-1+2 c^2+4 c d x+2 d^2 x^2\right ) \text {arccosh}(c+d x)^2-2 a b \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{4 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(175\)
default	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(175\)
parts	\(e \,a^{2} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{2} \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{2}-\frac {\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{2}-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}\right )}{d}+\frac {2 e a b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\)	\(179\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (98) = 196\).

Time = 0.28 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.12 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x))^2 \, dx=\frac {{\left (2 \, a^{2} + b^{2}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d e x + {\left (2 \, b^{2} d^{2} e x^{2} + 4 \, b^{2} c d e x + {\left (2 \, b^{2} c^{2} - b^{2}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 2 \, {\left (2 \, a b d^{2} e x^{2} + 4 \, a b c d e x + {\left (2 \, a b c^{2} - a b\right )} e - {\left (b^{2} d e x + b^{2} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (a b d e x + a b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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