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

Optimal. Leaf size=136 \[ -\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}+\frac{5 b d^2 \cosh ^{-1}(c x)}{96 c^2}-\frac{b d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{36 c}+\frac{5 b d^2 x (c x-1)^{3/2} (c x+1)^{3/2}}{144 c}-\frac{5 b d^2 x \sqrt{c x-1} \sqrt{c x+1}}{96 c} \]

[Out]

(-5*b*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c) + (5*b*d^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(144*c) - (b*d
^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(36*c) + (5*b*d^2*ArcCosh[c*x])/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(a + b*
ArcCosh[c*x]))/(6*c^2)

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Rubi [A]  time = 0.0673495, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {5716, 38, 52} \[ -\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}+\frac{5 b d^2 \cosh ^{-1}(c x)}{96 c^2}-\frac{b d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{36 c}+\frac{5 b d^2 x (c x-1)^{3/2} (c x+1)^{3/2}}{144 c}-\frac{5 b d^2 x \sqrt{c x-1} \sqrt{c x+1}}{96 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*d^2*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(96*c) + (5*b*d^2*x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(144*c) - (b*d
^2*x*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(36*c) + (5*b*d^2*ArcCosh[c*x])/(96*c^2) - (d^2*(1 - c^2*x^2)^3*(a + b*
ArcCosh[c*x]))/(6*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 )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}-\frac{\left (b d^2\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \, dx}{6 c}\\ &=-\frac{b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}+\frac{\left (5 b d^2\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx}{36 c}\\ &=\frac{5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac{b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}-\frac{\left (5 b d^2\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx}{48 c}\\ &=-\frac{5 b d^2 x \sqrt{-1+c x} \sqrt{1+c x}}{96 c}+\frac{5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac{b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}+\frac{\left (5 b d^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{96 c}\\ &=-\frac{5 b d^2 x \sqrt{-1+c x} \sqrt{1+c x}}{96 c}+\frac{5 b d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}}{144 c}-\frac{b d^2 x (-1+c x)^{5/2} (1+c x)^{5/2}}{36 c}+\frac{5 b d^2 \cosh ^{-1}(c x)}{96 c^2}-\frac{d^2 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{6 c^2}\\ \end{align*}

Mathematica [A]  time = 0.217832, size = 126, normalized size = 0.93 \[ \frac{d^2 \left (c x \left (48 a c x \left (c^4 x^4-3 c^2 x^2+3\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (-8 c^4 x^4+26 c^2 x^2-33\right )\right )+48 b c^2 x^2 \left (c^4 x^4-3 c^2 x^2+3\right ) \cosh ^{-1}(c x)-66 b \tanh ^{-1}\left (\sqrt{\frac{c x-1}{c x+1}}\right )\right )}{288 c^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(d^2*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-33 + 26*c^2*x^2 - 8*c^4*x^4) + 48*a*c*x*(3 - 3*c^2*x^2 + c^4*x^4))
+ 48*b*c^2*x^2*(3 - 3*c^2*x^2 + c^4*x^4)*ArcCosh[c*x] - 66*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(288*c^2)

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Maple [A]  time = 0.013, size = 204, normalized size = 1.5 \begin{align*}{\frac{{c}^{4}{d}^{2}a{x}^{6}}{6}}-{\frac{{c}^{2}{d}^{2}a{x}^{4}}{2}}+{\frac{{d}^{2}a{x}^{2}}{2}}+{\frac{{c}^{4}{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{6}}{6}}-{\frac{{c}^{2}{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{4}}{2}}+{\frac{{d}^{2}b{\rm arccosh} \left (cx\right ){x}^{2}}{2}}-{\frac{{d}^{2}b{c}^{3}{x}^{5}}{36}\sqrt{cx-1}\sqrt{cx+1}}+{\frac{13\,{d}^{2}bc{x}^{3}}{144}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{11\,{d}^{2}bx}{96\,c}\sqrt{cx-1}\sqrt{cx+1}}-{\frac{11\,{d}^{2}b}{96\,{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)^2*(a+b*arccosh(c*x)),x)

[Out]

1/6*c^4*d^2*a*x^6-1/2*c^2*d^2*a*x^4+1/2*d^2*a*x^2+1/6*c^4*d^2*b*arccosh(c*x)*x^6-1/2*c^2*d^2*b*arccosh(c*x)*x^
4+1/2*d^2*b*arccosh(c*x)*x^2-1/36*c^3*d^2*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x^5+13/144*c*d^2*b*(c*x-1)^(1/2)*(c*x+
1)^(1/2)*x^3-11/96*b*d^2*x*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-11/96/c^2*d^2*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 = 1.21624, size = 424, normalized size = 3.12 \begin{align*} \frac{1}{6} \, a c^{4} d^{2} x^{6} - \frac{1}{2} \, a c^{2} d^{2} x^{4} + \frac{1}{288} \,{\left (48 \, x^{6} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{c^{2} x^{2} - 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{c^{2} x^{2} - 1} x}{c^{6}} + \frac{15 \, \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b c^{4} d^{2} - \frac{1}{16} \,{\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^{2} + \frac{1}{2} \, a d^{2} 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^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*c^4*d^2*x^6 - 1/2*a*c^2*d^2*x^4 + 1/288*(48*x^6*arccosh(c*x) - (8*sqrt(c^2*x^2 - 1)*x^5/c^2 + 10*sqrt(c^
2*x^2 - 1)*x^3/c^4 + 15*sqrt(c^2*x^2 - 1)*x/c^6 + 15*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c
^6))*c)*b*c^4*d^2 - 1/16*(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*lo
g(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*c^2*d^2 + 1/2*a*d^2*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^2

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Fricas [A]  time = 1.81814, size = 328, normalized size = 2.41 \begin{align*} \frac{48 \, a c^{6} d^{2} x^{6} - 144 \, a c^{4} d^{2} x^{4} + 144 \, a c^{2} d^{2} x^{2} + 3 \,{\left (16 \, b c^{6} d^{2} x^{6} - 48 \, b c^{4} d^{2} x^{4} + 48 \, b c^{2} d^{2} x^{2} - 11 \, b d^{2}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (8 \, b c^{5} d^{2} x^{5} - 26 \, b c^{3} d^{2} x^{3} + 33 \, b c d^{2} x\right )} \sqrt{c^{2} x^{2} - 1}}{288 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(48*a*c^6*d^2*x^6 - 144*a*c^4*d^2*x^4 + 144*a*c^2*d^2*x^2 + 3*(16*b*c^6*d^2*x^6 - 48*b*c^4*d^2*x^4 + 48*
b*c^2*d^2*x^2 - 11*b*d^2)*log(c*x + sqrt(c^2*x^2 - 1)) - (8*b*c^5*d^2*x^5 - 26*b*c^3*d^2*x^3 + 33*b*c*d^2*x)*s
qrt(c^2*x^2 - 1))/c^2

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Sympy [A]  time = 6.11155, size = 197, normalized size = 1.45 \begin{align*} \begin{cases} \frac{a c^{4} d^{2} x^{6}}{6} - \frac{a c^{2} d^{2} x^{4}}{2} + \frac{a d^{2} x^{2}}{2} + \frac{b c^{4} d^{2} x^{6} \operatorname{acosh}{\left (c x \right )}}{6} - \frac{b c^{3} d^{2} x^{5} \sqrt{c^{2} x^{2} - 1}}{36} - \frac{b c^{2} d^{2} x^{4} \operatorname{acosh}{\left (c x \right )}}{2} + \frac{13 b c d^{2} x^{3} \sqrt{c^{2} x^{2} - 1}}{144} + \frac{b d^{2} x^{2} \operatorname{acosh}{\left (c x \right )}}{2} - \frac{11 b d^{2} x \sqrt{c^{2} x^{2} - 1}}{96 c} - \frac{11 b d^{2} \operatorname{acosh}{\left (c x \right )}}{96 c^{2}} & \text{for}\: c \neq 0 \\\frac{d^{2} 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)**2*(a+b*acosh(c*x)),x)

[Out]

Piecewise((a*c**4*d**2*x**6/6 - a*c**2*d**2*x**4/2 + a*d**2*x**2/2 + b*c**4*d**2*x**6*acosh(c*x)/6 - b*c**3*d*
*2*x**5*sqrt(c**2*x**2 - 1)/36 - b*c**2*d**2*x**4*acosh(c*x)/2 + 13*b*c*d**2*x**3*sqrt(c**2*x**2 - 1)/144 + b*
d**2*x**2*acosh(c*x)/2 - 11*b*d**2*x*sqrt(c**2*x**2 - 1)/(96*c) - 11*b*d**2*acosh(c*x)/(96*c**2), Ne(c, 0)), (
d**2*x**2*(a + I*pi*b/2)/2, True))

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Giac [B]  time = 1.66007, size = 406, normalized size = 2.99 \begin{align*} \frac{1}{6} \, a c^{4} d^{2} x^{6} - \frac{1}{2} \, a c^{2} d^{2} x^{4} + \frac{1}{288} \,{\left (48 \, x^{6} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (\sqrt{c^{2} x^{2} - 1}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c^{2}} + \frac{5}{c^{4}}\right )} + \frac{15}{c^{6}}\right )} x - \frac{15 \, \log \left ({\left | -x{\left | c \right |} + \sqrt{c^{2} x^{2} - 1} \right |}\right )}{c^{6}{\left | c \right |}}\right )} c\right )} b c^{4} d^{2} - \frac{1}{16} \,{\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^{2} + \frac{1}{2} \, a d^{2} 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^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*c^4*d^2*x^6 - 1/2*a*c^2*d^2*x^4 + 1/288*(48*x^6*log(c*x + sqrt(c^2*x^2 - 1)) - (sqrt(c^2*x^2 - 1)*(2*x^2
*(4*x^2/c^2 + 5/c^4) + 15/c^6)*x - 15*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^6*abs(c)))*c)*b*c^4*d^2 - 1/1
6*(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^2 + 1/2*a*d^2*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^2