3.6.0 ∫ x(a+barccosh(cx)) (d+ex2)3 dx [508]

Integrand size = 19, antiderivative size = 177 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {b c x \left (1-c^2 x^2\right )}{8 d \left (c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+e\right ) \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{8 d^{3/2} e \left (c^2 d+e\right )^{3/2} \sqrt {-1+c x} \sqrt {1+c x}} \]

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

Rubi [A] (veriﬁed)

Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5957, 533, 390, 385, 214} \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {a+b \text {arccosh}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c \sqrt {c^2 x^2-1} \left (2 c^2 d+e\right ) \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{8 d^{3/2} e \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right )^{3/2}}+\frac {b c x \left (1-c^2 x^2\right )}{8 d \sqrt {c x-1} \sqrt {c x+1} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

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

(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])

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

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]

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]

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])

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*} \text {integral}& = -\frac {a+b \text {arccosh}(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 \text {arccosh}(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 \text {arccosh}(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 \text {arccosh}(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 \text {arccosh}(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} \text {arctanh}\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*}

Mathematica [A] (veriﬁed)

Time = 0.83 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.03 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {1}{8} \left (-\frac {\frac {2 a}{e}+\frac {b c x \sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )}{d \left (c^2 d+e\right )}}{\left (d+e x^2\right )^2}-\frac {2 b \text {arccosh}(c x)}{e \left (d+e x^2\right )^2}-\frac {b c \left (2 c^2 d+e\right ) \sqrt {-1+c x} \sqrt {1+c x} \arctan \left (\frac {\sqrt {-c^2 d-e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{d^{3/2} \left (-c^2 d-e\right )^{3/2} e \sqrt {-1+c^2 x^2}}\right ) \]

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

(-(((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

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(1125\) vs. \(2(154)=308\).

Time = 0.62 (sec) , antiderivative size = 1126, normalized size of antiderivative = 6.36


method	result	size

parts	\(\text {Expression too large to display}\)	\(1126\)
derivativedivides	\(\text {Expression too large to display}\)	\(1149\)
default	\(\text {Expression too large to display}\)	\(1149\)

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

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

c^2*d*e)^(1/2)*(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*x+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)))*c^2*x^2*e^3+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)))*c^2*d*e^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^3*c^2*x^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)))*c^2*d*e^2-2*e^2*(-c^2*d*e)^(1/2)*(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1

/2)*c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(-c^2*d*e)^(1/2)/(e*c*x-(-c^2*d*e)^(1/2))/(-(c^2*d+e)/e)^(1/2)/(e*c*x+(-c

^2*d*e)^(1/2))/d/(-(-c^2*d*e)^(1/2)+e)^2/((-c^2*d*e)^(1/2)+e)^2/(c^2*x^2-1)^(1/2))

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (150) = 300\).

Time = 0.38 (sec) , antiderivative size = 1233, normalized size of antiderivative = 6.97 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Too large to display} \]

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

[-1/16*(2*(2*a + b)*c^4*d^4 + 2*(4*a + b)*c^2*d^3*e + 4*a*d^2*e^2 + 2*(b*c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 4*(b

*c^4*d^3*e + b*c^2*d^2*e^2)*x^2 - (2*b*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e

+ b*c*d*e^2)*x^2)*sqrt(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)) - 4*((b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2

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

*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(-c*x + sqrt(c^2*x^2 - 1)) + 2*s

qrt(c^2*x^2 - 1)*((b*c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x))/(c^4*d^6*e + 2*c^2*d^5*e^2

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

8*((2*a + b)*c^4*d^4 + (4*a + b)*c^2*d^3*e + 2*a*d^2*e^2 + (b*c^4*d^2*e^2 + b*c^2*d*e^3)*x^4 + 2*(b*c^4*d^3*e

+ b*c^2*d^2*e^2)*x^2 - (2*b*c^3*d^3 + b*c*d^2*e + (2*b*c^3*d*e^2 + b*c*e^3)*x^4 + 2*(2*b*c^3*d^2*e + b*c*d*e^2

)*x^2)*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)) - 2*((b*c^4*d^2*e^2 + 2*b*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2

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

*c^2*d*e^3 + b*e^4)*x^4 + 2*(b*c^4*d^3*e + 2*b*c^2*d^2*e^2 + b*d*e^3)*x^2)*log(-c*x + sqrt(c^2*x^2 - 1)) + sqr

t(c^2*x^2 - 1)*((b*c^3*d^2*e^2 + b*c*d*e^3)*x^3 + (b*c^3*d^3*e + b*c*d^2*e^2)*x))/(c^4*d^6*e + 2*c^2*d^5*e^2 +

d^4*e^3 + (c^4*d^4*e^3 + 2*c^2*d^3*e^4 + d^2*e^5)*x^4 + 2*(c^4*d^5*e^2 + 2*c^2*d^4*e^3 + d^3*e^4)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

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

Maxima [F]

\[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

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

-1/8*(c^4*log(e*x^2 + d)/(c^4*d^2*e + 2*c^2*d*e^2 + e^3) + 8*c*integrate(1/4/(c^3*e^3*x^7 + (2*c^3*d*e^2 - c*e

^3)*x^5 - c*d^2*e*x + (c^3*d^2*e - 2*c*d*e^2)*x^3 + (c^2*e^3*x^6 + (2*c^2*d*e^2 - e^3)*x^4 - d^2*e + (c^2*d^2*

e - 2*d*e^2)*x^2)*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*e^2*x^4 + 2*c^4*d*e*x^2 + c^4*d^

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

(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))*b - 1/4*a/(e^3*x^4 + 2*

Giac [F]

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

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

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

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

\(\int \frac {x (a+b \text {arccosh}(c x))}{(d+e x^2)^3} \, dx\) [508]

Optimal result