3.6.8 \(\int \frac {x (a+b \cosh ^{-1}(c x))}{(d+e x^2)^3} \, dx\) [508]

Optimal. Leaf size=177 \[ \frac {b c x \left (1-c^2 x^2\right )}{8 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5957, 533, 390, 385, 214} \begin {gather*} -\frac {a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \tanh ^{-1}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{8 d^{3/2} e \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right )^{3/2}}+\frac {b c x \left (1-c^2 x^2\right )}{8 d \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]

[Out]

(b*c*x*(1 - c^2*x^2))/(8*d*(c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)) - (a + b*ArcCosh[c*x])/(4*e*(
d + e*x^2)^2) + (b*c*(2*c^2*d + e)*Sqrt[-1 + c^2*x^2]*ArcTanh[(Sqrt[c^2*d + e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])
])/(8*d^(3/2)*e*(c^2*d + e)^(3/2)*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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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 )^3} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )^2} \, dx}{4 e}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\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 )^2} \, dx}{4 e \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c x \left (1-c^2 x^2\right )}{8 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {\left (b c \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d e \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c x \left (1-c^2 x^2\right )}{8 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {\left (b c \left (2 c^2 d+e\right ) \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 )}{8 d e \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b c x \left (1-c^2 x^2\right )}{8 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \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 )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 0.63, size = 183, normalized size = 1.03 \begin {gather*} \frac {1}{8} \left (-\frac {\frac {2 a}{e}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac {2 b \cosh ^{-1}(c x)}{e \left (d+e x^2\right )^2}-\frac {b c \left (2 c^2 d+e\right ) \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 )}{d^{3/2} \left (-c^2 d-e\right )^{3/2} e \sqrt {-1+c^2 x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]

[Out]

(-(((2*a)/e + (b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(d*(c^2*d + e)))/(d + e*x^2)^2) - (2*b*ArcCosh[
c*x])/(e*(d + e*x^2)^2) - (b*c*(2*c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[(Sqrt[-(c^2*d) - e]*x)/(Sqrt[
d]*Sqrt[-1 + c^2*x^2])])/(d^(3/2)*(-(c^2*d) - e)^(3/2)*e*Sqrt[-1 + c^2*x^2]))/8

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2458\) vs. \(2(154)=308\).
time = 10.17, size = 2459, normalized size = 13.89

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2459\)
default	\(\text {Expression too large to display}\)	\(2459\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccosh(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(-1/4*a*c^6/e/(c^2*e*x^2+c^2*d)^2-1/4*b*c^6/e/(c^2*e*x^2+c^2*d)^2*arccosh(c*x)-1/8*b*c^10*e^2*(c*x+1)^(1
/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(e*c*x-(-c^2*d*e)^(1/2))/(e-(
-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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^2-1/8*b*c^10*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e
)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(e*c*x-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/
2)+e)^2/(c^2*x^2-1)^(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*x^2+1/8*b*c^10*e^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2
*d+e)/e)^(1/2)/(e*c*x-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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^2+1/8*b*c^10*e^3*
(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(e*c*x-(-c^2*d*e)^(
1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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*x^2-3/16*b*c^8*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c
*x+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(e*c*x-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-
c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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-3/16*b*c^8*e^4*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2
)/(-(c^2*d+e)/e)^(1/2)/(e*c*x-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2+3/16*b*
c^8*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(e*c*x-(-c^
2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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+3/16*b*c^8*e^4*(c*x+1)^(1/2)*(c*x-1)^(1/2)/
(e*c*x+(-c^2*d*e)^(1/2))/(-c^2*d*e)^(1/2)/(-(c^2*d+e)/e)^(1/2)/(e*c*x-(-c^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2
/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2-1/8*b*c^7*e^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(e*c*x-(-c
^2*d*e)^(1/2))/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2*x-1/16*b*c^6*e^4*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c
*x+(-c^2*d*e)^(1/2))/(e*c*x-(-c^2*d*e)^(1/2))/(-(c^2*d+e)/e)^(1/2)/(-c^2*d*e)^(1/2)/(e-(-c^2*d*e)^(1/2))^2/((-
c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))-1/16*b*c^6*e^5*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(e*c*x-(-c^2*d*e)
^(1/2))/(-(c^2*d+e)/e)^(1/2)/(-c^2*d*e)^(1/2)/d/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2+1/16*b*
c^6*e^4*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(e*c*x-(-c^2*d*e)^(1/2))/(-(c^2*d+e)/e)^(1/2)/(-c
^2*d*e)^(1/2)/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))+1/16*b*c^6*e^5*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e
*c*x+(-c^2*d*e)^(1/2))/(e*c*x-(-c^2*d*e)^(1/2))/(-(c^2*d+e)/e)^(1/2)/(-c^2*d*e)^(1/2)/d/(e-(-c^2*d*e)^(1/2))^2
/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(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)))*x^2-1/8*b*c^5*e^4*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(e*c*x+(-c^2*d*e)^(1/2))/(e*c*x-(-c
^2*d*e)^(1/2))/d/(e-(-c^2*d*e)^(1/2))^2/((-c^2*d*e)^(1/2)+e)^2*x)

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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)^3,x, algorithm="maxima")

[Out]

-1/8*(c^4*log(x^2*e + d)/(c^4*d^2*e + 2*c^2*d*e^2 + e^3) + 8*c*integrate(1/4/(c^3*x^7*e^3 + (2*c^3*d*e^2 - c*e
^3)*x^5 - c*d^2*x*e + (c^3*d^2*e - 2*c*d*e^2)*x^3 + (c^2*x^6*e^3 + (2*c^2*d*e^2 - e^3)*x^4 + (c^2*d^2*e - 2*d*
e^2)*x^2 - d^2*e)*e^(1/2*log(c*x + 1) + 1/2*log(c*x - 1))), x) - (c^4*d^2 + c^2*d*e + (c^4*d*e + c^2*e^2)*x^2
- 2*(c^4*d^2 + 2*c^2*d*e + e^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)) + (c^4*x^4*e^2 + 2*c^4*d*x^2*e + c^4*d^
2)*log(c*x + 1) + (c^4*x^4*e^2 + 2*c^4*d*x^2*e + c^4*d^2)*log(c*x - 1))/(c^4*d^4*e + 2*c^2*d^3*e^2 + (c^4*d^2*
e^3 + 2*c^2*d*e^4 + e^5)*x^4 + 2*(c^4*d^3*e^2 + 2*c^2*d^2*e^3 + d*e^4)*x^2 + d^2*e^3))*b - 1/4*a/(x^4*e^3 + 2*
d*x^2*e^2 + d^2*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1643 vs. \(2 (154) = 308\).
time = 0.48, size = 3391, normalized size = 19.16 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccosh(c*x))/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

[-1/16*(2*b*c^2*d*x^4*cosh(1)^3 + 2*b*c^2*d*x^4*sinh(1)^3 + 2*(2*a + b)*c^4*d^4 + 2*(b*c^4*d^2*x^4 + 2*b*c^2*d
^2*x^2 + 2*a*d^2)*cosh(1)^2 + 2*(b*c^4*d^2*x^4 + 3*b*c^2*d*x^4*cosh(1) + 2*b*c^2*d^2*x^2 + 2*a*d^2)*sinh(1)^2
- (b*c*x^4*cosh(1)^3 + b*c*x^4*sinh(1)^3 + 2*b*c^3*d^3 + 2*(b*c^3*d*x^4 + b*c*d*x^2)*cosh(1)^2 + (2*b*c^3*d*x^
4 + 3*b*c*x^4*cosh(1) + 2*b*c*d*x^2)*sinh(1)^2 + (4*b*c^3*d^2*x^2 + b*c*d^2)*cosh(1) + (4*b*c^3*d^2*x^2 + 3*b*
c*x^4*cosh(1)^2 + b*c*d^2 + 4*(b*c^3*d*x^4 + b*c*d*x^2)*cosh(1))*sinh(1))*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)*cosh(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*(2*b*c^4*d^3*x^2 + (4*a + b)*c^2*d^3)*cosh(1) - 4
*(2*b*c^4*d^3*x^2*cosh(1) + b*x^4*cosh(1)^4 + b*x^4*sinh(1)^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)^3 + 2*(b*c^2
*d*x^4 + 2*b*x^4*cosh(1) + b*d*x^2)*sinh(1)^3 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2)*cosh(1)^2 + (b*c^4*d^2*x^4 +
4*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + 6*(b*c^2*d*x^4 + b*d*x^2)*cosh(1))*sinh(1)^2 + 2*(b*c^4*d^3*x^2 + 2*b*x
^4*cosh(1)^3 + 3*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2)*cosh(1))*sinh(1))*log(c
*x + sqrt(c^2*x^2 - 1)) - 4*(b*c^4*d^4 + b*x^4*cosh(1)^4 + b*x^4*sinh(1)^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)
^3 + 2*(b*c^2*d*x^4 + 2*b*x^4*cosh(1) + b*d*x^2)*sinh(1)^3 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*cosh(1)
^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + b*d^2 + 6*(b*c^2*d*x^4 + b*d*x^2)*cosh(1))*sinh(1)
^2 + 2*(b*c^4*d^3*x^2 + b*c^2*d^3)*cosh(1) + 2*(b*c^4*d^3*x^2 + 2*b*x^4*cosh(1)^3 + b*c^2*d^3 + 3*(b*c^2*d*x^4
+ b*d*x^2)*cosh(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*cosh(1))*sinh(1))*log(-c*x + sqrt(c^2*x^2 -
1)) + 2*(2*b*c^4*d^3*x^2 + 3*b*c^2*d*x^4*cosh(1)^2 + (4*a + b)*c^2*d^3 + 2*(b*c^4*d^2*x^4 + 2*b*c^2*d^2*x^2 +
2*a*d^2)*cosh(1))*sinh(1) + 2*(b*c^3*d^3*x*cosh(1) + b*c*d*x^3*cosh(1)^3 + b*c*d*x^3*sinh(1)^3 + (b*c^3*d^2*x^
3 + b*c*d^2*x)*cosh(1)^2 + (b*c^3*d^2*x^3 + 3*b*c*d*x^3*cosh(1) + b*c*d^2*x)*sinh(1)^2 + (b*c^3*d^3*x + 3*b*c*
d*x^3*cosh(1)^2 + 2*(b*c^3*d^2*x^3 + b*c*d^2*x)*cosh(1))*sinh(1))*sqrt(c^2*x^2 - 1))/(c^4*d^6*cosh(1) + d^2*x^
4*cosh(1)^5 + d^2*x^4*sinh(1)^5 + 2*(c^2*d^3*x^4 + d^3*x^2)*cosh(1)^4 + (2*c^2*d^3*x^4 + 5*d^2*x^4*cosh(1) + 2
*d^3*x^2)*sinh(1)^4 + (c^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4)*cosh(1)^3 + (c^4*d^4*x^4 + 4*c^2*d^4*x^2 + 10*d^2*x^
4*cosh(1)^2 + d^4 + 8*(c^2*d^3*x^4 + d^3*x^2)*cosh(1))*sinh(1)^3 + 2*(c^4*d^5*x^2 + c^2*d^5)*cosh(1)^2 + (2*c^
4*d^5*x^2 + 10*d^2*x^4*cosh(1)^3 + 2*c^2*d^5 + 12*(c^2*d^3*x^4 + d^3*x^2)*cosh(1)^2 + 3*(c^4*d^4*x^4 + 4*c^2*d
^4*x^2 + d^4)*cosh(1))*sinh(1)^2 + (c^4*d^6 + 5*d^2*x^4*cosh(1)^4 + 8*(c^2*d^3*x^4 + d^3*x^2)*cosh(1)^3 + 3*(c
^4*d^4*x^4 + 4*c^2*d^4*x^2 + d^4)*cosh(1)^2 + 4*(c^4*d^5*x^2 + c^2*d^5)*cosh(1))*sinh(1)), -1/8*(b*c^2*d*x^4*c
osh(1)^3 + b*c^2*d*x^4*sinh(1)^3 + (2*a + b)*c^4*d^4 + (b*c^4*d^2*x^4 + 2*b*c^2*d^2*x^2 + 2*a*d^2)*cosh(1)^2 +
(b*c^4*d^2*x^4 + 3*b*c^2*d*x^4*cosh(1) + 2*b*c^2*d^2*x^2 + 2*a*d^2)*sinh(1)^2 - (b*c*x^4*cosh(1)^3 + b*c*x^4*
sinh(1)^3 + 2*b*c^3*d^3 + 2*(b*c^3*d*x^4 + b*c*d*x^2)*cosh(1)^2 + (2*b*c^3*d*x^4 + 3*b*c*x^4*cosh(1) + 2*b*c*d
*x^2)*sinh(1)^2 + (4*b*c^3*d^2*x^2 + b*c*d^2)*cosh(1) + (4*b*c^3*d^2*x^2 + 3*b*c*x^4*cosh(1)^2 + b*c*d^2 + 4*(
b*c^3*d*x^4 + b*c*d*x^2)*cosh(1))*sinh(1))*sqrt(-c^2*d^2 - d*cosh(1) - d*sinh(1))*arctan((sqrt(-c^2*d^2 - d*co
sh(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*c
osh(1) + c*x^2*sinh(1) + c*d))/(c^2*d^2 + d*cosh(1) + d*sinh(1))) + (2*b*c^4*d^3*x^2 + (4*a + b)*c^2*d^3)*cosh
(1) - 2*(2*b*c^4*d^3*x^2*cosh(1) + b*x^4*cosh(1)^4 + b*x^4*sinh(1)^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)^3 + 2
*(b*c^2*d*x^4 + 2*b*x^4*cosh(1) + b*d*x^2)*sinh(1)^3 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2)*cosh(1)^2 + (b*c^4*d^
2*x^4 + 4*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + 6*(b*c^2*d*x^4 + b*d*x^2)*cosh(1))*sinh(1)^2 + 2*(b*c^4*d^3*x^2
+ 2*b*x^4*cosh(1)^3 + 3*(b*c^2*d*x^4 + b*d*x^2)*cosh(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2)*cosh(1))*sinh(1)
)*log(c*x + sqrt(c^2*x^2 - 1)) - 2*(b*c^4*d^4 + b*x^4*cosh(1)^4 + b*x^4*sinh(1)^4 + 2*(b*c^2*d*x^4 + b*d*x^2)*
cosh(1)^3 + 2*(b*c^2*d*x^4 + 2*b*x^4*cosh(1) + b*d*x^2)*sinh(1)^3 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*
cosh(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + 6*b*x^4*cosh(1)^2 + b*d^2 + 6*(b*c^2*d*x^4 + b*d*x^2)*cosh(1))*
sinh(1)^2 + 2*(b*c^4*d^3*x^2 + b*c^2*d^3)*cosh(1) + 2*(b*c^4*d^3*x^2 + 2*b*x^4*cosh(1)^3 + b*c^2*d^3 + 3*(b*c^
2*d*x^4 + b*d*x^2)*cosh(1)^2 + (b*c^4*d^2*x^4 + 4*b*c^2*d^2*x^2 + b*d^2)*cosh(1))*sinh(1))*log(-c*x + sqrt(c^2
*x^2 - 1)) + (2*b*c^4*d^3*x^2 + 3*b*c^2*d*x^4*cosh(1)^2 + (4*a + b)*c^2*d^3 + 2*(b*c^4*d^2*x^4 + 2*b*c^2*d^2*x
^2 + 2*a*d^2)*cosh(1))*sinh(1) + (b*c^3*d^3*x*c...

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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 )^{3}}\, 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)**3,x)

[Out]

Integral(x*(a + b*acosh(c*x))/(d + e*x**2)**3, 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)^3,x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*x/(e*x^2 + d)^3, 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 )}^3} \,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)^3,x)

[Out]

int((x*(a + b*acosh(c*x)))/(d + e*x^2)^3, x)

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