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

Optimal. Leaf size=75 \[ \frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac{b e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{4 d}-\frac{b e \cosh ^{-1}(c+d x)}{4 d} \]

[Out]

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

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Rubi [A]  time = 0.0389998, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5866, 12, 5662, 90, 52} \[ \frac{e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{2 d}-\frac{b e \sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)}{4 d}-\frac{b e \cosh ^{-1}(c+d x)}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(4*d) - (b*e*ArcCosh[c + d*x])/(4*d) + (e*(c + d*x)^2*(a
+ b*ArcCosh[c + d*x]))/(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 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

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

Mathematica [A]  time = 0.154843, size = 81, normalized size = 1.08 \[ \frac{e \left (\frac{1}{2} (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac{1}{4} b \left (\sqrt{c+d x-1} \sqrt{c+d x+1} (c+d x)+2 \tanh ^{-1}\left (\sqrt{\frac{c+d x-1}{c+d x+1}}\right )\right )\right )}{d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.004, size = 162, normalized size = 2.2 \begin{align*}{\frac{a{x}^{2}de}{2}}+xace+{\frac{a{c}^{2}e}{2\,d}}+{\frac{d{\rm arccosh} \left (dx+c\right ){x}^{2}be}{2}}+{\rm arccosh} \left (dx+c\right )xbce+{\frac{b{\rm arccosh} \left (dx+c\right ){c}^{2}e}{2\,d}}-{\frac{bxe}{4}\sqrt{dx+c-1}\sqrt{dx+c+1}}-{\frac{bce}{4\,d}\sqrt{dx+c-1}\sqrt{dx+c+1}}-{\frac{eb}{4\,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)),x)

[Out]

1/2*a*x^2*d*e+x*a*c*e+1/2/d*a*c^2*e+1/2*d*arccosh(d*x+c)*x^2*b*e+arccosh(d*x+c)*x*b*c*e+1/2/d*arccosh(d*x+c)*b
*c^2*e-1/4*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*x*b*e-1/4/d*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*b*c*e-1/4/d*e*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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.4183, size = 257, normalized size = 3.43 \begin{align*} \frac{2 \, a d^{2} e x^{2} + 4 \, a c d e x +{\left (2 \, b d^{2} e x^{2} + 4 \, b c d e x +{\left (2 \, b c^{2} - b\right )} e\right )} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) -{\left (b d e x + 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)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.454334, size = 148, normalized size = 1.97 \begin{align*} \begin{cases} a c e x + \frac{a d e x^{2}}{2} + \frac{b c^{2} e \operatorname{acosh}{\left (c + d x \right )}}{2 d} + b c e x \operatorname{acosh}{\left (c + d x \right )} - \frac{b c e \sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1}}{4 d} + \frac{b d e x^{2} \operatorname{acosh}{\left (c + d x \right )}}{2} - \frac{b e x \sqrt{c^{2} + 2 c d x + d^{2} x^{2} - 1}}{4} - \frac{b e \operatorname{acosh}{\left (c + d x \right )}}{4 d} & \text{for}\: d \neq 0 \\c e x \left (a + b \operatorname{acosh}{\left (c \right )}\right ) & \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)),x)

[Out]

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

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Giac [B]  time = 1.80385, size = 331, normalized size = 4.41 \begin{align*} \frac{1}{4} \,{\left (2 \, a d x^{2} - 4 \,{\left (d{\left (\frac{c \log \left ({\left | -c d -{\left (x{\left | d \right |} - \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{\left | d \right |} \right |}\right )}{d{\left | d \right |}} + \frac{\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c +{\left (2 \, x^{2} \log \left (d x + c + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) -{\left (\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}{\left (\frac{x}{d^{2}} - \frac{3 \, c}{d^{3}}\right )} - \frac{{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d -{\left (x{\left | d \right |} - \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{\left | d \right |} \right |}\right )}{d^{2}{\left | d \right |}}\right )} d\right )} b d + 4 \, a c x\right )} e \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)),x, algorithm="giac")

[Out]

1/4*(2*a*d*x^2 - 4*(d*(c*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d*abs(d)) + s
qrt(d^2*x^2 + 2*c*d*x + c^2 - 1)/d^2) - x*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)))*b*c + (2*x^2*log(d
*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)) - (sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(x/d^2 - 3*c/d^3) - (2*c^2 +
1)*log(abs(-c*d - (x*abs(d) - sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*abs(d)))/(d^2*abs(d)))*d)*b*d + 4*a*c*x)*e