\(\int \frac {x (a+b \sec ^{-1}(c x))}{(d+e x^2)^2} \, dx\) [98]

Optimal result

Integrand size = 19, antiderivative size = 131 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \arctan \left (\sqrt {-1+c^2 x^2}\right )}{2 d e \sqrt {c^2 x^2}}-\frac {b c x \arctan \left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 d+e} \sqrt {c^2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5344, 457, 88, 65, 211} \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \sec ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c x \arctan \left (\sqrt {c^2 x^2-1}\right )}{2 d e \sqrt {c^2 x^2}}-\frac {b c x \arctan \left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{\sqrt {c^2 d+e}}\right )}{2 d \sqrt {e} \sqrt {c^2 x^2} \sqrt {c^2 d+e}} \]

[In]

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

[Out]

-1/2*(a + b*ArcSec[c*x])/(e*(d + e*x^2)) + (b*c*x*ArcTan[Sqrt[-1 + c^2*x^2]])/(2*d*e*Sqrt[c^2*x^2]) - (b*c*x*A
rcTan[(Sqrt[e]*Sqrt[-1 + c^2*x^2])/Sqrt[c^2*d + e]])/(2*d*Sqrt[e]*Sqrt[c^2*d + e]*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 88

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.52 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.18 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {-\frac {2 a}{d+e x^2}-\frac {2 b \sec ^{-1}(c x)}{d+e x^2}-\frac {2 b \arcsin \left (\frac {1}{c x}\right )}{d}+\frac {b \sqrt {e} \log \left (\frac {4 i d e+4 c d \sqrt {e} \left (c \sqrt {d}-i \sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x}{b \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d-e}}+\frac {b \sqrt {e} \log \left (\frac {-4 i d e+4 c d \sqrt {e} \left (c \sqrt {d}+i \sqrt {-c^2 d-e} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x}{b \sqrt {-c^2 d-e} \left (\sqrt {d}-i \sqrt {e} x\right )}\right )}{d \sqrt {-c^2 d-e}}}{4 e} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(264\) vs. \(2(112)=224\).

Time = 6.50 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.02

method	result	size
parts	\(-\frac {a}{2 e \left (e \,x^{2}+d \right )}+\frac {b \left (-\frac {c^{4} \operatorname {arcsec}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c \sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e -2 \sqrt {-c^{2} d e}\, c x -2 e}{c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\)	\(265\)
derivativedivides	\(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\)	\(276\)
default	\(\frac {-\frac {a \,c^{4}}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+b \,c^{4} \left (-\frac {\operatorname {arcsec}\left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\sqrt {c^{2} x^{2}-1}\, \left (2 \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {-\frac {c^{2} d +e}{e}}-\ln \left (-\frac {2 \left (\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x -e \right )}{-c e x +\sqrt {-c^{2} d e}}\right )-\ln \left (-\frac {2 \left (-\sqrt {c^{2} x^{2}-1}\, \sqrt {-\frac {c^{2} d +e}{e}}\, e +\sqrt {-c^{2} d e}\, c x +e \right )}{c e x +\sqrt {-c^{2} d e}}\right )\right )}{4 e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x d \sqrt {-\frac {c^{2} d +e}{e}}}\right )}{c^{2}}\)	\(276\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.93 \[ \int \frac {x \left (a+b \sec ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\left [-\frac {2 \, a c^{2} d^{2} + 2 \, a d e + \sqrt {-c^{2} d e - e^{2}} {\left (b e x^{2} + b d\right )} \log \left (\frac {c^{2} e x^{2} - c^{2} d + 2 \, \sqrt {-c^{2} d e - e^{2}} \sqrt {c^{2} x^{2} - 1} - 2 \, e}{e x^{2} + d}\right ) + 2 \, {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arcsec}\left (c x\right ) - 4 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{4 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} + a d e + \sqrt {c^{2} d e + e^{2}} {\left (b e x^{2} + b d\right )} \arctan \left (\frac {\sqrt {c^{2} d e + e^{2}} \sqrt {c^{2} x^{2} - 1}}{c^{2} d + e}\right ) + {\left (b c^{2} d^{2} + b d e\right )} \operatorname {arcsec}\left (c x\right ) - 2 \, {\left (b c^{2} d^{2} + b d e + {\left (b c^{2} d e + b e^{2}\right )} x^{2}\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{2 \, {\left (c^{2} d^{3} e + d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} + d e^{3}\right )} x^{2}\right )}}\right ] \]

[In]

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

[Out]

[-1/4*(2*a*c^2*d^2 + 2*a*d*e + sqrt(-c^2*d*e - e^2)*(b*e*x^2 + b*d)*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)) + 2*(b*c^2*d^2 + b*d*e)*arcsec(c*x) - 4*(b*c^2*d^2 + b*d*e + (b*c^
2*d*e + b*e^2)*x^2)*arctan(-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 + a*d*e + sqrt(c^2*d*e + e^2)*(b*e*x^2 + b*d)*arctan(sqrt(c^2*d*e + e^2)*sqrt(c^2*x^2 - 1)/(c^2*d +
e)) + (b*c^2*d^2 + b*d*e)*arcsec(c*x) - 2*(b*c^2*d^2 + b*d*e + (b*c^2*d*e + b*e^2)*x^2)*arctan(-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)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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