3.5.99 \(\int \frac {x (a+b \cosh ^{-1}(c x))}{(d+e x^2)^2} \, dx\) [499]

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

[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 = 0.06, 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 = {5957, 533, 385, 214} \begin {gather*} \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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a + b*ArcCosh[c*x])/(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 214

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

Rule 385

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 533

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)^FracPa
rt[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 5957

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 ) \text {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 = 0.21, size = 123, normalized size = 1.09 \begin {gather*} -\frac {\frac {a}{d+e x^2}+\frac {b \cosh ^{-1}(c x)}{d+e x^2}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \text {ArcTan}\left (\frac {\sqrt {-c^2 d-e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{\sqrt {d} \sqrt {-c^2 d-e} \sqrt {-1+c^2 x^2}}}{2 e} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(645\) vs. \(2(96)=192\).
time = 7.67, size = 646, normalized size = 5.72

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {b \,c^{4} \mathrm {arccosh}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{6} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) d}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}-\frac {b \,c^{6} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) d}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}+\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}-\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}}{c^{2}}\)	\(646\)
default	\(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {b \,c^{4} \mathrm {arccosh}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {b \,c^{6} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) d}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}-\frac {b \,c^{6} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) d}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}+\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x +\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}-\frac {b \,c^{4} \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\frac {2 \sqrt {-\frac {c^{2} d +e}{e}}\, \sqrt {c^{2} x^{2}-1}\, e +2 \sqrt {-c^{2} d e}\, c x -2 e}{e c x -\sqrt {-c^{2} d e}}\right ) e}{4 \sqrt {c^{2} x^{2}-1}\, \left (\sqrt {-c^{2} d e}+e \right ) \left (e -\sqrt {-c^{2} d e}\right ) \sqrt {-c^{2} d e}\, \sqrt {-\frac {c^{2} d +e}{e}}}}{c^{2}}\)	\(646\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(-1/2*a*c^4/e/(c^2*e*x^2+c^2*d)-1/2*b*c^4/e/(c^2*e*x^2+c^2*d)*arccosh(c*x)+1/4*b*c^6*(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*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x+(-c^2*d*e)^(1/2)))*d-1/4*b*c^6*(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*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x-(-c^2*d*e)^(1/2)))*d
+1/4*b*c^4*(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*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e-(-c^2*d*e)^(1/2)*c*x-e)/(e*c*x+(-c^2
*d*e)^(1/2)))*e-1/4*b*c^4*(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*d+e)/e)^(1/2)*(c^2*x^2-1)^(1/2)*e+(-c^2*d*e)^(1/2)*c*x-
e)/(e*c*x-(-c^2*d*e)^(1/2)))*e)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (94) = 188\).
time = 0.41, size = 978, normalized size = 8.65 \begin {gather*} \left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d \cosh \left (1\right ) + 2 \, a d \sinh \left (1\right ) - {\left (b c x^{2} \cosh \left (1\right ) + b c x^{2} \sinh \left (1\right ) + b c d\right )} \sqrt {c^{2} d^{2} + d \cosh \left (1\right ) + d \sinh \left (1\right )} \log \left (\frac {4 \, c^{4} d^{2} x^{2} - 2 \, c^{2} d^{2} + x^{2} \cosh \left (1\right )^{2} + x^{2} \sinh \left (1\right )^{2} + {\left (4 \, c^{2} d x^{2} - d\right )} \cosh \left (1\right ) + {\left (4 \, c^{2} d x^{2} + 2 \, x^{2} \cosh \left (1\right ) - d\right )} \sinh \left (1\right ) + 2 \, {\left (2 \, c^{3} d x^{2} + c x^{2} \cosh \left (1\right ) + c x^{2} \sinh \left (1\right ) - c d + {\left (2 \, c^{2} d x + x \cosh \left (1\right ) + x \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {c^{2} d^{2} + d \cosh \left (1\right ) + d \sinh \left (1\right )} + 4 \, {\left (c^{3} d^{2} x + c d x \cosh \left (1\right ) + c d x \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d}\right ) - 2 \, {\left (b c^{2} d x^{2} \cosh \left (1\right ) + b x^{2} \cosh \left (1\right )^{2} + b x^{2} \sinh \left (1\right )^{2} + {\left (b c^{2} d x^{2} + 2 \, b x^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 2 \, {\left (b c^{2} d^{2} + b x^{2} \cosh \left (1\right )^{2} + b x^{2} \sinh \left (1\right )^{2} + {\left (b c^{2} d x^{2} + b d\right )} \cosh \left (1\right ) + {\left (b c^{2} d x^{2} + 2 \, b x^{2} \cosh \left (1\right ) + b d\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} \cosh \left (1\right ) + d x^{2} \cosh \left (1\right )^{3} + d x^{2} \sinh \left (1\right )^{3} + {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \cosh \left (1\right )^{2} + {\left (c^{2} d^{2} x^{2} + 3 \, d x^{2} \cosh \left (1\right ) + d^{2}\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d^{3} + 3 \, d x^{2} \cosh \left (1\right )^{2} + 2 \, {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}}, -\frac {a c^{2} d^{2} + a d \cosh \left (1\right ) + a d \sinh \left (1\right ) - {\left (b c x^{2} \cosh \left (1\right ) + b c x^{2} \sinh \left (1\right ) + b c d\right )} \sqrt {-c^{2} d^{2} - d \cosh \left (1\right ) - d \sinh \left (1\right )} \arctan \left (\frac {\sqrt {-c^{2} d^{2} - d \cosh \left (1\right ) - d \sinh \left (1\right )} \sqrt {c^{2} x^{2} - 1} {\left (x \cosh \left (1\right ) + x \sinh \left (1\right )\right )} - \sqrt {-c^{2} d^{2} - d \cosh \left (1\right ) - d \sinh \left (1\right )} {\left (c x^{2} \cosh \left (1\right ) + c x^{2} \sinh \left (1\right ) + c d\right )}}{c^{2} d^{2} + d \cosh \left (1\right ) + d \sinh \left (1\right )}\right ) - {\left (b c^{2} d x^{2} \cosh \left (1\right ) + b x^{2} \cosh \left (1\right )^{2} + b x^{2} \sinh \left (1\right )^{2} + {\left (b c^{2} d x^{2} + 2 \, b x^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c^{2} d^{2} + b x^{2} \cosh \left (1\right )^{2} + b x^{2} \sinh \left (1\right )^{2} + {\left (b c^{2} d x^{2} + b d\right )} \cosh \left (1\right ) + {\left (b c^{2} d x^{2} + 2 \, b x^{2} \cosh \left (1\right ) + b d\right )} \sinh \left (1\right )\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{2} d^{3} \cosh \left (1\right ) + d x^{2} \cosh \left (1\right )^{3} + d x^{2} \sinh \left (1\right )^{3} + {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \cosh \left (1\right )^{2} + {\left (c^{2} d^{2} x^{2} + 3 \, d x^{2} \cosh \left (1\right ) + d^{2}\right )} \sinh \left (1\right )^{2} + {\left (c^{2} d^{3} + 3 \, d x^{2} \cosh \left (1\right )^{2} + 2 \, {\left (c^{2} d^{2} x^{2} + d^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}}\right ] \end {gather*}

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*a*d*cosh(1) + 2*a*d*sinh(1) - (b*c*x^2*cosh(1) + b*c*x^2*sinh(1) + b*c*d)*sqrt(c^2*d^2
+ d*cosh(1) + d*sinh(1))*log((4*c^4*d^2*x^2 - 2*c^2*d^2 + x^2*cosh(1)^2 + x^2*sinh(1)^2 + (4*c^2*d*x^2 - d)*co
sh(1) + (4*c^2*d*x^2 + 2*x^2*cosh(1) - d)*sinh(1) + 2*(2*c^3*d*x^2 + c*x^2*cosh(1) + c*x^2*sinh(1) - c*d + (2*
c^2*d*x + x*cosh(1) + x*sinh(1))*sqrt(c^2*x^2 - 1))*sqrt(c^2*d^2 + d*cosh(1) + d*sinh(1)) + 4*(c^3*d^2*x + c*d
*x*cosh(1) + c*d*x*sinh(1))*sqrt(c^2*x^2 - 1))/(x^2*cosh(1) + x^2*sinh(1) + d)) - 2*(b*c^2*d*x^2*cosh(1) + b*x
^2*cosh(1)^2 + b*x^2*sinh(1)^2 + (b*c^2*d*x^2 + 2*b*x^2*cosh(1))*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(b*
c^2*d^2 + b*x^2*cosh(1)^2 + b*x^2*sinh(1)^2 + (b*c^2*d*x^2 + b*d)*cosh(1) + (b*c^2*d*x^2 + 2*b*x^2*cosh(1) + b
*d)*sinh(1))*log(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^3*cosh(1) + d*x^2*cosh(1)^3 + d*x^2*sinh(1)^3 + (c^2*d^2*x^
2 + d^2)*cosh(1)^2 + (c^2*d^2*x^2 + 3*d*x^2*cosh(1) + d^2)*sinh(1)^2 + (c^2*d^3 + 3*d*x^2*cosh(1)^2 + 2*(c^2*d
^2*x^2 + d^2)*cosh(1))*sinh(1)), -1/2*(a*c^2*d^2 + a*d*cosh(1) + a*d*sinh(1) - (b*c*x^2*cosh(1) + b*c*x^2*sinh
(1) + b*c*d)*sqrt(-c^2*d^2 - d*cosh(1) - d*sinh(1))*arctan((sqrt(-c^2*d^2 - d*cosh(1) - d*sinh(1))*sqrt(c^2*x^
2 - 1)*(x*cosh(1) + x*sinh(1)) - sqrt(-c^2*d^2 - d*cosh(1) - d*sinh(1))*(c*x^2*cosh(1) + c*x^2*sinh(1) + c*d))
/(c^2*d^2 + d*cosh(1) + d*sinh(1))) - (b*c^2*d*x^2*cosh(1) + b*x^2*cosh(1)^2 + b*x^2*sinh(1)^2 + (b*c^2*d*x^2
+ 2*b*x^2*cosh(1))*sinh(1))*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c^2*d^2 + b*x^2*cosh(1)^2 + b*x^2*sinh(1)^2 + (b
*c^2*d*x^2 + b*d)*cosh(1) + (b*c^2*d*x^2 + 2*b*x^2*cosh(1) + b*d)*sinh(1))*log(-c*x + sqrt(c^2*x^2 - 1)))/(c^2
*d^3*cosh(1) + d*x^2*cosh(1)^3 + d*x^2*sinh(1)^3 + (c^2*d^2*x^2 + d^2)*cosh(1)^2 + (c^2*d^2*x^2 + 3*d*x^2*cosh
(1) + d^2)*sinh(1)^2 + (c^2*d^3 + 3*d*x^2*cosh(1)^2 + 2*(c^2*d^2*x^2 + d^2)*cosh(1))*sinh(1))]

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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