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

Optimal. Leaf size=98 \[ -\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2}+\frac{3 b d \cosh ^{-1}(c x)}{32 c^2}+\frac{b d x (c x-1)^{3/2} (c x+1)^{3/2}}{16 c}-\frac{3 b d x \sqrt{c x-1} \sqrt{c x+1}}{32 c} \]

[Out]

(-3*b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c) + (b*d*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(16*c) + (3*b*d*ArcC
osh[c*x])/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/(4*c^2)

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Rubi [A]  time = 0.0416606, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5716, 38, 52} \[ -\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2}+\frac{3 b d \cosh ^{-1}(c x)}{32 c^2}+\frac{b d x (c x-1)^{3/2} (c x+1)^{3/2}}{16 c}-\frac{3 b d x \sqrt{c x-1} \sqrt{c x+1}}{32 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(32*c) + (b*d*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(16*c) + (3*b*d*ArcC
osh[c*x])/(32*c^2) - (d*(1 - c^2*x^2)^2*(a + b*ArcCosh[c*x]))/(4*c^2)

Rule 5716

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*(-d)^p)/(2*c*(p + 1)), 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, d, e, p}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 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 x \left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2}+\frac{(b d) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{4 c}\\ &=\frac{b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2}-\frac{(3 b d) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx}{16 c}\\ &=-\frac{3 b d x \sqrt{-1+c x} \sqrt{1+c x}}{32 c}+\frac{b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2}+\frac{(3 b d) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 c}\\ &=-\frac{3 b d x \sqrt{-1+c x} \sqrt{1+c x}}{32 c}+\frac{b d x (-1+c x)^{3/2} (1+c x)^{3/2}}{16 c}+\frac{3 b d \cosh ^{-1}(c x)}{32 c^2}-\frac{d \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2}\\ \end{align*}

Mathematica [A]  time = 0.147541, size = 100, normalized size = 1.02 \[ -\frac{d \left (c x \left (8 a c x \left (c^2 x^2-2\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (5-2 c^2 x^2\right )\right )+8 b c^2 x^2 \left (c^2 x^2-2\right ) \cosh ^{-1}(c x)+10 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{32 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(d*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(5 - 2*c^2*x^2) + 8*a*c*x*(-2 + c^2*x^2)) + 8*b*c^2*x^2*(-2 + c^2*x^2
)*ArcCosh[c*x] + 10*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(32*c^2)

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Maple [A]  time = 0.013, size = 136, normalized size = 1.4 \begin{align*} -{\frac{{c}^{2}da{x}^{4}}{4}}+{\frac{da{x}^{2}}{2}}-{\frac{{c}^{2}db{\rm arccosh} \left (cx\right ){x}^{4}}{4}}+{\frac{db{\rm arccosh} \left (cx\right ){x}^{2}}{2}}+{\frac{dbc{x}^{3}}{16}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,dbx}{32\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{5\,db}{32\,{c}^{2}}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( cx+\sqrt{{c}^{2}{x}^{2}-1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*c^2*d*a*x^4+1/2*d*a*x^2-1/4*c^2*d*b*arccosh(c*x)*x^4+1/2*d*b*arccosh(c*x)*x^2+1/16*c*d*b*(c*x-1)^(1/2)*(c
*x+1)^(1/2)*x^3-5/32*b*d*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-5/32/c^2*d*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)^
(1/2)*ln(c*x+(c^2*x^2-1)^(1/2))

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Maxima [B]  time = 1.10897, size = 243, normalized size = 2.48 \begin{align*} -\frac{1}{4} \, a c^{2} d x^{4} - \frac{1}{32} \,{\left (8 \, x^{4} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{2 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} x}{c^{4}} + \frac{3 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b c^{2} d + \frac{1}{2} \, a d x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} + \frac{\log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*c^2*d*x^4 - 1/32*(8*x^4*arccosh(c*x) - (2*sqrt(c^2*x^2 - 1)*x^3/c^2 + 3*sqrt(c^2*x^2 - 1)*x/c^4 + 3*log
(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*c^2*d + 1/2*a*d*x^2 + 1/4*(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)*sqrt(c^2))/(sqrt(c^2)*c^2)))*b*d

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Fricas [A]  time = 1.835, size = 221, normalized size = 2.26 \begin{align*} -\frac{8 \, a c^{4} d x^{4} - 16 \, a c^{2} d x^{2} +{\left (8 \, b c^{4} d x^{4} - 16 \, b c^{2} d x^{2} + 5 \, b d\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (2 \, b c^{3} d x^{3} - 5 \, b c d x\right )} \sqrt{c^{2} x^{2} - 1}}{32 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(8*a*c^4*d*x^4 - 16*a*c^2*d*x^2 + (8*b*c^4*d*x^4 - 16*b*c^2*d*x^2 + 5*b*d)*log(c*x + sqrt(c^2*x^2 - 1))
- (2*b*c^3*d*x^3 - 5*b*c*d*x)*sqrt(c^2*x^2 - 1))/c^2

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Sympy [A]  time = 1.64605, size = 124, normalized size = 1.27 \begin{align*} \begin{cases} - \frac{a c^{2} d x^{4}}{4} + \frac{a d x^{2}}{2} - \frac{b c^{2} d x^{4} \operatorname{acosh}{\left (c x \right )}}{4} + \frac{b c d x^{3} \sqrt{c^{2} x^{2} - 1}}{16} + \frac{b d x^{2} \operatorname{acosh}{\left (c x \right )}}{2} - \frac{5 b d x \sqrt{c^{2} x^{2} - 1}}{32 c} - \frac{5 b d \operatorname{acosh}{\left (c x \right )}}{32 c^{2}} & \text{for}\: c \neq 0 \\\frac{d x^{2} \left (a + \frac{i \pi b}{2}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**2*d*x**4/4 + a*d*x**2/2 - b*c**2*d*x**4*acosh(c*x)/4 + b*c*d*x**3*sqrt(c**2*x**2 - 1)/16 + b*
d*x**2*acosh(c*x)/2 - 5*b*d*x*sqrt(c**2*x**2 - 1)/(32*c) - 5*b*d*acosh(c*x)/(32*c**2), Ne(c, 0)), (d*x**2*(a +
I*pi*b/2)/2, True))

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Giac [B]  time = 1.49737, size = 243, normalized size = 2.48 \begin{align*} -\frac{1}{4} \, a c^{2} d x^{4} - \frac{1}{32} \,{\left (8 \, x^{4} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1} x{\left (\frac{2 \, x^{2}}{c^{2}} + \frac{3}{c^{4}}\right )} - \frac{3 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{4}{\left | c \right |}}\right )} c\right )} b c^{2} d + \frac{1}{2} \, a d x^{2} + \frac{1}{4} \,{\left (2 \, x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x}{c^{2}} - \frac{\log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{2}{\left | c \right |}}\right )}\right )} b d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*c^2*d*x^4 - 1/32*(8*x^4*log(c*x + sqrt(c^2*x^2 - 1)) - (sqrt(c^2*x^2 - 1)*x*(2*x^2/c^2 + 3/c^4) - 3*log
(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^4*abs(c)))*c)*b*c^2*d + 1/2*a*d*x^2 + 1/4*(2*x^2*log(c*x + sqrt(c^2*x^
2 - 1)) - c*(sqrt(c^2*x^2 - 1)*x/c^2 - log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^2*abs(c))))*b*d