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

Optimal. Leaf size=113 \[ \frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{2 \sqrt{d} e \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}-\frac{a+b \cosh ^{-1}(c x)}{2 e \left (d+e x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0941429, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {5788, 519, 377, 208} \[ \frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left (\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{2 \sqrt{d} e \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}-\frac{a+b \cosh ^{-1}(c x)}{2 e \left (d+e x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5788

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

Rule 519

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1*a2 + b1*b2*x^n)^FracP
art[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.307635, size = 123, normalized size = 1.09 \[ -\frac{\frac{a}{d+e x^2}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (\frac{x \sqrt{c^2 (-d)-e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right )}{\sqrt{d} \sqrt{c^2 x^2-1} \sqrt{c^2 (-d)-e}}+\frac{b \cosh ^{-1}(c x)}{d+e x^2}}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.047, size = 638, normalized size = 5.7 \begin{align*} -{\frac{{c}^{2}a}{2\,e \left ({x}^{2}{c}^{2}e+{c}^{2}d \right ) }}-{\frac{{c}^{2}b{\rm arccosh} \left (cx\right )}{2\,e \left ({x}^{2}{c}^{2}e+{c}^{2}d \right ) }}-{\frac{b{c}^{4}d}{4}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( 2\,{\frac{1}{cxe-\sqrt{-{c}^{2}de}} \left ( \sqrt{{c}^{2}{x}^{2}-1}\sqrt{-{\frac{{c}^{2}d+e}{e}}}e+\sqrt{-{c}^{2}de}cx-e \right ) } \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \left ( \sqrt{-{c}^{2}de}+e \right ) ^{-1} \left ( e-\sqrt{-{c}^{2}de} \right ) ^{-1}{\frac{1}{\sqrt{-{c}^{2}de}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d+e}{e}}}}}}+{\frac{b{c}^{4}d}{4}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( -2\,{\frac{1}{cxe+\sqrt{-{c}^{2}de}} \left ( -\sqrt{{c}^{2}{x}^{2}-1}\sqrt{-{\frac{{c}^{2}d+e}{e}}}e+\sqrt{-{c}^{2}de}cx+e \right ) } \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \left ( \sqrt{-{c}^{2}de}+e \right ) ^{-1} \left ( e-\sqrt{-{c}^{2}de} \right ) ^{-1}{\frac{1}{\sqrt{-{c}^{2}de}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d+e}{e}}}}}}-{\frac{{c}^{2}be}{4}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( 2\,{\frac{1}{cxe-\sqrt{-{c}^{2}de}} \left ( \sqrt{{c}^{2}{x}^{2}-1}\sqrt{-{\frac{{c}^{2}d+e}{e}}}e+\sqrt{-{c}^{2}de}cx-e \right ) } \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \left ( \sqrt{-{c}^{2}de}+e \right ) ^{-1} \left ( e-\sqrt{-{c}^{2}de} \right ) ^{-1}{\frac{1}{\sqrt{-{c}^{2}de}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d+e}{e}}}}}}+{\frac{{c}^{2}be}{4}\sqrt{cx-1}\sqrt{cx+1}\ln \left ( -2\,{\frac{1}{cxe+\sqrt{-{c}^{2}de}} \left ( -\sqrt{{c}^{2}{x}^{2}-1}\sqrt{-{\frac{{c}^{2}d+e}{e}}}e+\sqrt{-{c}^{2}de}cx+e \right ) } \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \left ( \sqrt{-{c}^{2}de}+e \right ) ^{-1} \left ( e-\sqrt{-{c}^{2}de} \right ) ^{-1}{\frac{1}{\sqrt{-{c}^{2}de}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d+e}{e}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.33244, size = 1122, normalized size = 9.93 \begin{align*} \left [-\frac{2 \, a c^{2} d^{2} - 2 \,{\left (b c^{2} d e + b e^{2}\right )} x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 2 \, a d e -{\left (b c e x^{2} + b c d\right )} \sqrt{c^{2} d^{2} + d e} \log \left (-\frac{2 \, c^{2} d^{2} -{\left (4 \, c^{4} d^{2} + 4 \, c^{2} d e + e^{2}\right )} x^{2} + d e - 2 \, \sqrt{c^{2} d^{2} + d e}{\left ({\left (2 \, c^{3} d + c e\right )} x^{2} - c d\right )} - 2 \, \sqrt{c^{2} x^{2} - 1}{\left (\sqrt{c^{2} d^{2} + d e}{\left (2 \, c^{2} d + e\right )} x + 2 \,{\left (c^{3} d^{2} + c d e\right )} x\right )}}{e x^{2} + d}\right ) - 2 \,{\left (b c^{2} d^{2} + b d e +{\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{4 \,{\left (c^{2} d^{3} e + d^{2} e^{2} +{\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac{a c^{2} d^{2} -{\left (b c^{2} d e + b e^{2}\right )} x^{2} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + a d e -{\left (b c e x^{2} + b c d\right )} \sqrt{-c^{2} d^{2} - d e} \arctan \left (\frac{\sqrt{-c^{2} d^{2} - d e} \sqrt{c^{2} x^{2} - 1} e x - \sqrt{-c^{2} d^{2} - d e}{\left (c e x^{2} + c d\right )}}{c^{2} d^{2} + d e}\right ) -{\left (b c^{2} d^{2} + b d e +{\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (-c x + \sqrt{c^{2} x^{2} - 1}\right )}{2 \,{\left (c^{2} d^{3} e + d^{2} e^{2} +{\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*c^2*d^2 - 2*(b*c^2*d*e + b*e^2)*x^2*log(c*x + sqrt(c^2*x^2 - 1)) + 2*a*d*e - (b*c*e*x^2 + b*c*d)*sq
rt(c^2*d^2 + d*e)*log(-(2*c^2*d^2 - (4*c^4*d^2 + 4*c^2*d*e + e^2)*x^2 + d*e - 2*sqrt(c^2*d^2 + d*e)*((2*c^3*d
+ c*e)*x^2 - c*d) - 2*sqrt(c^2*x^2 - 1)*(sqrt(c^2*d^2 + d*e)*(2*c^2*d + e)*x + 2*(c^3*d^2 + c*d*e)*x))/(e*x^2
+ d)) - 2*(b*c^2*d^2 + b*d*e + (b*c^2*d*e + b*e^2)*x^2)*log(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*e + d^2*e^2 +
(c^2*d^2*e^2 + d*e^3)*x^2), -1/2*(a*c^2*d^2 - (b*c^2*d*e + b*e^2)*x^2*log(c*x + sqrt(c^2*x^2 - 1)) + a*d*e - (
b*c*e*x^2 + b*c*d)*sqrt(-c^2*d^2 - d*e)*arctan((sqrt(-c^2*d^2 - d*e)*sqrt(c^2*x^2 - 1)*e*x - sqrt(-c^2*d^2 - d
*e)*(c*e*x^2 + c*d))/(c^2*d^2 + d*e)) - (b*c^2*d^2 + b*d*e + (b*c^2*d*e + b*e^2)*x^2)*log(-c*x + sqrt(c^2*x^2
- 1)))/(c^2*d^3*e + d^2*e^2 + (c^2*d^2*e^2 + d*e^3)*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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