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

Optimal. Leaf size=110 \[ \frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac{b e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac{b^2 e (c+d x)^2}{4 d} \]

[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)

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Rubi [A]  time = 0.250706, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5866, 12, 5662, 5759, 5676, 30} \[ \frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{2 d}-\frac{b e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac{e \left (a+b \cosh ^{-1}(c+d x)\right )^2}{4 d}+\frac{b^2 e (c+d x)^2}{4 d} \]

Antiderivative was successfully verified.

[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 5866

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
]

Rule 12

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

Rule 5662

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))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5759

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

Rule 5676

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

Rule 30

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

Rubi steps

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

Mathematica [A]  time = 0.217252, size = 167, normalized size = 1.52 \[ \frac{e \left ((c+d x) \left (2 a^2 (c+d x)-2 a b \sqrt{c+d x-1} \sqrt{c+d x+1}+b^2 (c+d x)\right )-2 a b \log \left (\sqrt{c+d x-1} \sqrt{c+d x+1}+c+d x\right )-2 b (c+d x) \cosh ^{-1}(c+d x) \left (b \sqrt{c+d x-1} \sqrt{c+d x+1}-2 a (c+d x)\right )+b^2 \left (2 c^2+4 c d x+2 d^2 x^2-1\right ) \cosh ^{-1}(c+d x)^2\right )}{4 d} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.036, size = 334, normalized size = 3. \begin{align*}{\frac{{a}^{2}e{x}^{2}d}{2}}+x{a}^{2}ce+{\frac{{a}^{2}{c}^{2}e}{2\,d}}+{\frac{de{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}{x}^{2}}{2}}+e{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}xc+{\frac{e{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}{c}^{2}}{2\,d}}-{\frac{e{b}^{2}{\rm arccosh} \left (dx+c\right )x}{2}\sqrt{dx+c-1}\sqrt{dx+c+1}}-{\frac{e{b}^{2}{\rm arccosh} \left (dx+c\right )c}{2\,d}\sqrt{dx+c-1}\sqrt{dx+c+1}}-{\frac{e{b}^{2} \left ({\rm arccosh} \left (dx+c\right ) \right ) ^{2}}{4\,d}}+{\frac{{b}^{2}de{x}^{2}}{4}}+{\frac{e{b}^{2}xc}{2}}+{\frac{e{b}^{2}{c}^{2}}{4\,d}}+d{\rm arccosh} \left (dx+c\right ){x}^{2}abe+2\,{\rm arccosh} \left (dx+c\right )xabce+{\frac{{\rm arccosh} \left (dx+c\right )ab{c}^{2}e}{d}}-{\frac{xabe}{2}\sqrt{dx+c-1}\sqrt{dx+c+1}}-{\frac{abce}{2\,d}\sqrt{dx+c-1}\sqrt{dx+c+1}}-{\frac{eab}{2\,d}\sqrt{dx+c-1}\sqrt{dx+c+1}\ln \left ( dx+c+\sqrt{ \left ( dx+c \right ) ^{2}-1} \right ){\frac{1}{\sqrt{ \left ( dx+c \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.38381, size = 533, normalized size = 4.85 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 1.18923, size = 335, normalized size = 3.05 \begin{align*} \begin{cases} a^{2} c e x + \frac{a^{2} d e x^{2}}{2} + \frac{a b c^{2} e \operatorname{acosh}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname{acosh}{\left (c + d x \right )} - \frac{a b c e \sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1}}{2 d} + a b d e x^{2} \operatorname{acosh}{\left (c + d x \right )} - \frac{a b e x \sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1}}{2} - \frac{a b e \operatorname{acosh}{\left (c + d x \right )}}{2 d} + \frac{b^{2} c^{2} e \operatorname{acosh}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname{acosh}^{2}{\left (c + d x \right )} + \frac{b^{2} c e x}{2} - \frac{b^{2} c e \sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname{acosh}{\left (c + d x \right )}}{2 d} + \frac{b^{2} d e x^{2} \operatorname{acosh}^{2}{\left (c + d x \right )}}{2} + \frac{b^{2} d e x^{2}}{4} - \frac{b^{2} e x \sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname{acosh}{\left (c + d x \right )}}{2} - \frac{b^{2} e \operatorname{acosh}^{2}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\c e x \left (a + b \operatorname{acosh}{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*c*e*x + a**2*d*e*x**2/2 + a*b*c**2*e*acosh(c + d*x)/d + 2*a*b*c*e*x*acosh(c + d*x) - a*b*c*e*s
qrt(c**2 + 2*c*d*x + d**2*x**2 - 1)/(2*d) + a*b*d*e*x**2*acosh(c + d*x) - a*b*e*x*sqrt(c**2 + 2*c*d*x + d**2*x
**2 - 1)/2 - a*b*e*acosh(c + d*x)/(2*d) + b**2*c**2*e*acosh(c + d*x)**2/(2*d) + b**2*c*e*x*acosh(c + d*x)**2 +
b**2*c*e*x/2 - b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/(2*d) + b**2*d*e*x**2*acosh(c + d
*x)**2/2 + b**2*d*e*x**2/4 - b**2*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*acosh(c + d*x)/2 - b**2*e*acosh(c +
d*x)**2/(4*d), Ne(d, 0)), (c*e*x*(a + b*acosh(c))**2, True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}{\left (b \operatorname{arcosh}\left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)