\(\int \frac {x (a+b \sec ^{-1}(c x))}{(d+e x^2)^3} \, dx\) [106]

Optimal result

Integrand size = 19, antiderivative size = 193 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{4 d^2 e \sqrt {c^2 x^2}}-\frac {b c \left (3 c^2 d+2 e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5344, 457, 105, 162, 65, 211} \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c x \arctan \left (\sqrt {c^2 x^2-1}\right )}{4 d^2 e \sqrt {c^2 x^2}}-\frac {b c x \left (3 c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \sqrt {c^2 x^2} \left (c^2 d+e\right )^{3/2}}-\frac {b c x \sqrt {c^2 x^2-1}}{8 d \sqrt {c^2 x^2} \left (c^2 d+e\right ) \left (d+e x^2\right )} \]

[In]

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

[Out]

-1/8*(b*c*x*Sqrt[-1 + c^2*x^2])/(d*(c^2*d + e)*Sqrt[c^2*x^2]*(d + e*x^2)) - (a + b*ArcSec[c*x])/(4*e*(d + e*x^
2)^2) + (b*c*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(4*d^2*e*Sqrt[c^2*x^2]) - (b*c*(3*c^2*d + 2*e)*x*ArcTan[(Sqrt[e]*Sq
rt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(8*d^2*Sqrt[e]*(c^2*d + e)^(3/2)*Sqrt[c^2*x^2])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 211

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

Rule 457

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

Rule 5344

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

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c x) \int \frac {1}{x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 e \sqrt {c^2 x^2}} \\ & = -\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} (d+e x)^2} \, dx,x,x^2\right )}{8 e \sqrt {c^2 x^2}} \\ & = -\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {(b c x) \text {Subst}\left (\int \frac {-c^2 d-e+\frac {1}{2} c^2 e x}{x \sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{8 d e \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c x) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{8 d^2 e \sqrt {c^2 x^2}}-\frac {\left (b c \left (3 c^2 d+2 e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} (d+e x)} \, dx,x,x^2\right )}{16 d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b x) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{4 c d^2 e \sqrt {c^2 x^2}}-\frac {\left (b \left (3 c^2 d+2 e\right ) x\right ) \text {Subst}\left (\int \frac {1}{d+\frac {e}{c^2}+\frac {e x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{8 c d^2 \left (c^2 d+e\right ) \sqrt {c^2 x^2}} \\ & = -\frac {b c x \sqrt {-1+c^2 x^2}}{8 d \left (c^2 d+e\right ) \sqrt {c^2 x^2} \left (d+e x^2\right )}-\frac {a+b \sec ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {b c x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{4 d^2 e \sqrt {c^2 x^2}}-\frac {b c \left (3 c^2 d+2 e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{8 d^2 \sqrt {e} \left (c^2 d+e\right )^{3/2} \sqrt {c^2 x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.72 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.00 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {1}{16} \left (-\frac {4 a}{e \left (d+e x^2\right )^2}-\frac {2 b c \sqrt {1-\frac {1}{c^2 x^2}} x}{d \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {4 b \sec ^{-1}(c x)}{e \left (d+e x^2\right )^2}-\frac {4 b \arcsin \left (\frac {1}{c x}\right )}{d^2 e}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (-\frac {16 d^2 \sqrt {-c^2 d-e} \sqrt {e} \left (\sqrt {e}+c \left (i c \sqrt {d}-\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (3 c^2 d+2 e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{d^2 \left (-c^2 d-e\right )^{3/2} \sqrt {e}}-\frac {b \left (3 c^2 d+2 e\right ) \log \left (\frac {16 i d^2 \sqrt {-c^2 d-e} \sqrt {e} \left (-\sqrt {e}+c \left (i c \sqrt {d}+\sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{b \left (3 c^2 d+2 e\right ) \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d^2 \left (-c^2 d-e\right )^{3/2} \sqrt {e}}\right ) \]

[In]

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

[Out]

((-4*a)/(e*(d + e*x^2)^2) - (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)/(d*(c^2*d + e)*(d + e*x^2)) - (4*b*ArcSec[c*x])/(e
*(d + e*x^2)^2) - (4*b*ArcSin[1/(c*x)])/(d^2*e) - (b*(3*c^2*d + 2*e)*Log[(-16*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(
Sqrt[e] + c*(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[
e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]) - (b*(3*c^2*d + 2*e)*Log[((16*I)*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(-
Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(Sqrt[d] + I*Sqrt[
e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]))/16

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(893\) vs. \(2(168)=336\).

Time = 5.55 (sec) , antiderivative size = 894, normalized size of antiderivative = 4.63

method	result	size
parts	\(-\frac {a}{4 e \left (e \,x^{2}+d \right )^{2}}+\frac {b \left (-\frac {c^{6} \operatorname {arcsec}\left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {c \sqrt {c^{2} x^{2}-1}\, \left (4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e -2 \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 e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \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 e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e \right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x \,d^{2} \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (-c e x +\sqrt {-c^{2} d e}\right ) \left (c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\)	\(894\)
derivativedivides	\(\frac {-\frac {a \,c^{6}}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}-2 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \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 e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \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 e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x \,d^{2} \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (c e x +\sqrt {-c^{2} d e}\right ) \left (-c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\)	\(899\)
default	\(\frac {-\frac {a \,c^{6}}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+b \,c^{6} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d^{2}+4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{4} d e \,x^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d^{2}-3 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{4} d e \,x^{2}+2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, c^{2} d e +4 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}\, e^{2} c^{2} x^{2}-2 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \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 e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}-2 \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 e x +\sqrt {-c^{2} d e}}\right ) c^{2} d e -2 \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 e x +\sqrt {-c^{2} d e}}\right ) e^{2} c^{2} x^{2}\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x \,d^{2} \sqrt {-\frac {c^{2} d +e}{e}}\, \left (c^{2} d +e \right ) \left (c e x +\sqrt {-c^{2} d e}\right ) \left (-c e x +\sqrt {-c^{2} d e}\right )}\right )}{c^{2}}\)	\(899\)

[In]

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

[Out]

-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*arcsec(c*x)+1/16*c*(c^2*x^2-1)^(1/2)*(4*arctan(1/(c
^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*c^4*d*e*x^2+4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*c^4*d^2-3
*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*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^
2-3*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*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2
-3*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*e*x+(-c^2*d*e)^(1/2)))*c^4*d*e*x^
2-3*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*e*x+(-c^2*d*e)^(1/2)))*c^4*d^2+4
*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/2)*e^2*c^2*x^2+4*arctan(1/(c^2*x^2-1)^(1/2))*(-(c^2*d+e)/e)^(1/
2)*c^2*d*e+2*(c^2*x^2-1)^(1/2)*(-(c^2*d+e)/e)^(1/2)*c^2*d*e-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*e*x+(-c^2*d*e)^(1/2)))*e^2*c^2*x^2-2*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*e*x+(-c^2*d*e)^(1/2)))*c^2*d*e-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*e*x+(-c^2*d*e)^(1/2)))*e^2*c^2*x^2-2*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*e*x+(-c^2*d*e)^(1/2)))*c^2*d*e)/((c^2*x^2-1)/c^2/x^2)^(1/2)/x/d^2/(-(c^2*d+e)/e)^
(1/2)/(c^2*d+e)/(-c*e*x+(-c^2*d*e)^(1/2))/(c*e*x+(-c^2*d*e)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (165) = 330\).

Time = 0.49 (sec) , antiderivative size = 888, normalized size of antiderivative = 4.60 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\left [-\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + {\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d e - e^{2}} \log \left (\frac {c^{2} e x^{2} - c^{2} d + 2 \, \sqrt {-c^{2} d e - e^{2}} \sqrt {c^{2} x^{2} - 1} - 2 \, e}{e x^{2} + d}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{16 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} + 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} + {\left (3 \, b c^{2} d^{3} + {\left (3 \, b c^{2} d e^{2} + 2 \, b e^{3}\right )} x^{4} + 2 \, b d^{2} e + 2 \, {\left (3 \, b c^{2} d^{2} e + 2 \, b d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d e + e^{2}} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} d + e}\right ) + 2 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2}\right )} \operatorname {arcsec}\left (c x\right ) - 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{4} d^{2} e^{2} + 2 \, b c^{2} d e^{3} + b e^{4}\right )} x^{4} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c^{2} d^{3} e + b d^{2} e^{2} + {\left (b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{8 \, {\left (c^{4} d^{6} e + 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/16*(4*a*c^4*d^4 + 8*a*c^2*d^3*e + 4*a*d^2*e^2 + (3*b*c^2*d^3 + (3*b*c^2*d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e +
2*(3*b*c^2*d^2*e + 2*b*d*e^2)*x^2)*sqrt(-c^2*d*e - e^2)*log((c^2*e*x^2 - c^2*d + 2*sqrt(-c^2*d*e - e^2)*sqrt(
c^2*x^2 - 1) - 2*e)/(e*x^2 + d)) + 4*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*arcsec(c*x) - 8*(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)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + 2*(b*c^2*d^3*e + b*d^2*e^2 + (b*c^2*d^2*e^2 + b*d*e^3)*x^2)*sqrt(c
^2*x^2 - 1))/(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*c^4*d^4 + 4*a*c^2*d^3*e + 2*a*d^2*e^2 + (3*b*c^2*d^3 + (3*b*c^2*
d*e^2 + 2*b*e^3)*x^4 + 2*b*d^2*e + 2*(3*b*c^2*d^2*e + 2*b*d*e^2)*x^2)*sqrt(c^2*d*e + e^2)*arctan(sqrt(c^2*d*e
+ e^2)*sqrt(c^2*x^2 - 1)/(c^2*d + e)) + 2*(b*c^4*d^4 + 2*b*c^2*d^3*e + b*d^2*e^2)*arcsec(c*x) - 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)*arctan(-c*x + sqrt(c^2*x^2 - 1)) + (b*c^2*d^3*e + b*d^2*e^2 + (b*c^2*d^2*e^2 + b*d*e^3)*x^2)*sqr
t(c^2*x^2 - 1))/(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. \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

int((x*(a + b*acos(1/(c*x))))/(d + e*x^2)^3,x)

[Out]

int((x*(a + b*acos(1/(c*x))))/(d + e*x^2)^3, x)