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]

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

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Rubi [A]  time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.33, 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]

-1/2*(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]))/e

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fricas [B]  time = 0.54, size = 537, normalized size = 4.75 \[ \left [-\frac {2 \, a c^{2} d^{2} - 2 \, {\left (b c^{2} d e + b e^{2}\right )} x^{2} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, a d e - {\left (b c e x^{2} + b c d\right )} \sqrt {c^{2} d^{2} + d e} \log \left (-\frac {2 \, c^{2} d^{2} - {\left (4 \, c^{4} d^{2} + 4 \, c^{2} d e + e^{2}\right )} x^{2} + d e - 2 \, \sqrt {c^{2} d^{2} + d e} {\left ({\left (2 \, c^{3} d + c e\right )} x^{2} - c d\right )} - 2 \, \sqrt {c^{2} x^{2} - 1} {\left (\sqrt {c^{2} d^{2} + d e} {\left (2 \, c^{2} d + e\right )} x + 2 \, {\left (c^{3} d^{2} + c d e\right )} x\right )}}{e x^{2} + d}\right ) - 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} - {\left (b c^{2} d e + b e^{2}\right )} x^{2} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a d e - {\left (b c e x^{2} + b c d\right )} \sqrt {-c^{2} d^{2} - d e} \arctan \left (\frac {\sqrt {-c^{2} d^{2} - d e} \sqrt {c^{2} x^{2} - 1} e x - \sqrt {-c^{2} d^{2} - d e} {\left (c e x^{2} + c d\right )}}{c^{2} d^{2} + d e}\right ) - {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \]

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)

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maple [B]  time = 0.04, size = 638, normalized size = 5.65 \[ -\frac {c^{2} a}{2 e \left (c^{2} x^{2} e +c^{2} d \right )}-\frac {c^{2} b \,\mathrm {arccosh}\left (c x \right )}{2 e \left (c^{2} x^{2} e +c^{2} d \right )}-\frac {c^{4} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{c x e -\sqrt {-c^{2} d e}}\right ) d}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}+\frac {c^{4} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c x e +\sqrt {-c^{2} d e}}\right ) d}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}-\frac {c^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{c x e -\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}+\frac {c^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c x e +\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}} \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, {\left (4 \, c \int \frac {1}{2 \, {\left (c^{3} e^{2} x^{5} - c d e x + {\left (c^{3} d e - c e^{2}\right )} x^{3} + {\left (c^{2} e^{2} x^{4} + {\left (c^{2} d e - e^{2}\right )} x^{2} - d e\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}\right )}}\,{d x} + \frac {c^{2} \log \left (e x^{2} + d\right )}{c^{2} d e + e^{2}} + \frac {2 \, {\left (c^{2} d + e\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) - {\left (c^{2} e x^{2} + c^{2} d\right )} \log \left (c x + 1\right ) - {\left (c^{2} e x^{2} + c^{2} d\right )} \log \left (c x - 1\right )}{c^{2} d^{2} e + d e^{2} + {\left (c^{2} d e^{2} + e^{3}\right )} x^{2}}\right )} b - \frac {a}{2 \, {\left (e^{2} x^{2} + d e\right )}} \]

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]

-1/4*(4*c*integrate(1/2/(c^3*e^2*x^5 - c*d*e*x + (c^3*d*e - c*e^2)*x^3 + (c^2*e^2*x^4 + (c^2*d*e - e^2)*x^2 -
d*e)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x) + c^2*log(e*x^2 + d)/(c^2*d*e + e^2) + (2*(c^2*d + e)*log(c*
x + sqrt(c*x + 1)*sqrt(c*x - 1)) - (c^2*e*x^2 + c^2*d)*log(c*x + 1) - (c^2*e*x^2 + c^2*d)*log(c*x - 1))/(c^2*d
^2*e + d*e^2 + (c^2*d*e^2 + e^3)*x^2))*b - 1/2*a/(e^2*x^2 + d*e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]

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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