3.105 \(\int \frac{x (a+b \csc ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=131 \[ -\frac{a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac{b c x \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{2 d e \sqrt{c^2 x^2}}+\frac{b c x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right )}{2 d \sqrt{e} \sqrt{c^2 x^2} \sqrt{c^2 d+e}} \]

[Out]

-(a + b*ArcCsc[c*x])/(2*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*Arc
Tan[(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])

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Rubi [A]  time = 0.121013, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {5237, 446, 86, 63, 205} \[ -\frac{a+b \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac{b c x \tan ^{-1}\left (\sqrt{c^2 x^2-1}\right )}{2 d e \sqrt{c^2 x^2}}+\frac{b c x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right )}{2 d \sqrt{e} \sqrt{c^2 x^2} \sqrt{c^2 d+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + b*ArcCsc[c*x])/(2*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*Arc
Tan[(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 5237

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(x_)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p +
1)*(a + b*ArcCsc[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]

Rule 446

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 86

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 63

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 205

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

Rubi steps

\begin{align*} \int \frac{x \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\frac{a+b \csc ^{-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 \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac{(b c x) \operatorname{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 \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{(b c x) \operatorname{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) \operatorname{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 \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac{(b x) \operatorname{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) \operatorname{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 \csc ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac{b c x \tan ^{-1}\left (\sqrt{-1+c^2 x^2}\right )}{2 d e \sqrt{c^2 x^2}}+\frac{b c x \tan ^{-1}\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]  time = 0.725949, size = 286, normalized size = 2.18 \[ -\frac{\frac{2 a}{d+e x^2}+\frac{b \sqrt{e} \log \left (\frac{4 i d e-4 c d \sqrt{e} x \left (c \sqrt{d}+i \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}\right )}{b \sqrt{c^2 (-d)-e} \left (\sqrt{d}-i \sqrt{e} x\right )}\right )}{d \sqrt{c^2 (-d)-e}}+\frac{b \sqrt{e} \log \left (\frac{4 i \left (-d e+c d \sqrt{e} x \left (\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}+i c \sqrt{d}\right )\right )}{b \sqrt{c^2 (-d)-e} \left (\sqrt{d}+i \sqrt{e} x\right )}\right )}{d \sqrt{c^2 (-d)-e}}+\frac{2 b \csc ^{-1}(c x)}{d+e x^2}-\frac{2 b \sin ^{-1}\left (\frac{1}{c x}\right )}{d}}{4 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((2*a)/(d + e*x^2) + (2*b*ArcCsc[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) + c*d*Sqrt[e]*(I*c*Sqrt[d] + 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)

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Maple [B]  time = 0.276, size = 354, normalized size = 2.7 \begin{align*} -{\frac{{c}^{2}a}{2\,e \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}-{\frac{b{c}^{2}{\rm arccsc} \left (cx\right )}{2\,e \left ({c}^{2}e{x}^{2}+{c}^{2}d \right ) }}+{\frac{b}{2\,ecxd}\sqrt{{c}^{2}{x}^{2}-1}\arctan \left ({\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}} \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}}-{\frac{b}{4\,ecxd}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( 2\,{\frac{1}{ecx+\sqrt{-{c}^{2}ed}} \left ( \sqrt{-{\frac{{c}^{2}d+e}{e}}}\sqrt{{c}^{2}{x}^{2}-1}e-\sqrt{-{c}^{2}ed}cx-e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d+e}{e}}}}}}-{\frac{b}{4\,ecxd}\sqrt{{c}^{2}{x}^{2}-1}\ln \left ( -2\,{\frac{1}{-ecx+\sqrt{-{c}^{2}ed}} \left ( \sqrt{-{\frac{{c}^{2}d+e}{e}}}\sqrt{{c}^{2}{x}^{2}-1}e+\sqrt{-{c}^{2}ed}cx-e \right ) } \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}{\frac{1}{\sqrt{-{\frac{{c}^{2}d+e}{e}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left ({\left (c^{2} e^{2} x^{2} + c^{2} d e\right )} \int \frac{x e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{2} x^{4} +{\left (c^{2} e^{2} x^{4} +{\left (c^{2} d e - e^{2}\right )} x^{2} - d e\right )}{\left (c x + 1\right )}{\left (c x - 1\right )} +{\left (c^{2} d e - e^{2}\right )} x^{2} - d e}\,{d x} + \arctan \left (1, \sqrt{c x + 1} \sqrt{c x - 1}\right )\right )} b}{2 \,{\left (e^{2} x^{2} + d e\right )}} - \frac{a}{2 \,{\left (e^{2} x^{2} + d e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsc(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) + arctan2(
1, sqrt(c*x + 1)*sqrt(c*x - 1)))*b/(e^2*x^2 + d*e) - 1/2*a/(e^2*x^2 + d*e)

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Fricas [A]  time = 3.46322, size = 833, normalized size = 6.36 \begin{align*} \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{arccsc}\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{arccsc}\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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsc(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)*arccsc(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)*arccsc(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)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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