3.239 \(\int \frac{\tan ^{-1}(a x)}{x^4 (c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=165 \[ -\frac{a \sqrt{a^2 c x^2+c}}{6 c^2 x^2}+\frac{5 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{11 a^3 \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{6 c^{3/2}}+\frac{a^3}{c \sqrt{a^2 c x^2+c}}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]

[Out]

a^3/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2])/(6*c^2*x^2) + (a^4*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]
) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x^3) + (5*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x) + (11*a^
3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/(6*c^(3/2))

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Rubi [A]  time = 0.494693, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4966, 4962, 266, 51, 63, 208, 4944, 4894} \[ -\frac{a \sqrt{a^2 c x^2+c}}{6 c^2 x^2}+\frac{5 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{11 a^3 \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )}{6 c^{3/2}}+\frac{a^3}{c \sqrt{a^2 c x^2+c}}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^(3/2)),x]

[Out]

a^3/(c*Sqrt[c + a^2*c*x^2]) - (a*Sqrt[c + a^2*c*x^2])/(6*c^2*x^2) + (a^4*x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2]
) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x^3) + (5*a^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(3*c^2*x) + (11*a^
3*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/(6*c^(3/2))

Rule 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[1/d, Int[
x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Dist[e/d, Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*
x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2*q] && LtQ[q, -1] && ILtQ[m, 0] &
& NeQ[p, -1]

Rule 4962

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] + (-Dist[(b*c*p)/(f*(m + 1)), Int[((f*
x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/Sqrt[d + e*x^2], x], x] - Dist[(c^2*(m + 2))/(f^2*(m + 1)), Int[((f*x)
^(m + 2)*(a + b*ArcTan[c*x])^p)/Sqrt[d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && G
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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 208

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

Rule 4944

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

Rule 4894

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[b/(c*d*Sqrt[d + e*x^2]),
x] + Simp[(x*(a + b*ArcTan[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]

Rubi steps

\begin{align*} \int \frac{\tan ^{-1}(a x)}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)}{x^4 \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x^3}+a^4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac{a \int \frac{1}{x^3 \sqrt{c+a^2 c x^2}} \, dx}{3 c}-\frac{\left (2 a^2\right ) \int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx}{3 c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=\frac{a^3}{c \sqrt{c+a^2 c x^2}}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{5 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x}+\frac{a \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{c+a^2 c x}} \, dx,x,x^2\right )}{6 c}-\frac{\left (2 a^3\right ) \int \frac{1}{x \sqrt{c+a^2 c x^2}} \, dx}{3 c}-\frac{a^3 \int \frac{1}{x \sqrt{c+a^2 c x^2}} \, dx}{c}\\ &=\frac{a^3}{c \sqrt{c+a^2 c x^2}}-\frac{a \sqrt{c+a^2 c x^2}}{6 c^2 x^2}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{5 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )}{12 c}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )}{3 c}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac{a^3}{c \sqrt{c+a^2 c x^2}}-\frac{a \sqrt{c+a^2 c x^2}}{6 c^2 x^2}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{5 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )}{6 c^2}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )}{3 c^2}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )}{c^2}\\ &=\frac{a^3}{c \sqrt{c+a^2 c x^2}}-\frac{a \sqrt{c+a^2 c x^2}}{6 c^2 x^2}+\frac{a^4 x \tan ^{-1}(a x)}{c \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x^3}+\frac{5 a^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 c^2 x}+\frac{11 a^3 \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )}{6 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.32687, size = 143, normalized size = 0.87 \[ \frac{\frac{a \left (5 a^2 x^2-1\right ) \sqrt{a^2 c x^2+c}}{a^2 x^4+x^2}+11 a^3 \sqrt{c} \log \left (\sqrt{c} \sqrt{a^2 c x^2+c}+c\right )+\frac{2 \left (8 a^4 x^4+4 a^2 x^2-1\right ) \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{a^2 x^5+x^3}-11 a^3 \sqrt{c} \log (x)}{6 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTan[a*x]/(x^4*(c + a^2*c*x^2)^(3/2)),x]

[Out]

((a*(-1 + 5*a^2*x^2)*Sqrt[c + a^2*c*x^2])/(x^2 + a^2*x^4) + (2*Sqrt[c + a^2*c*x^2]*(-1 + 4*a^2*x^2 + 8*a^4*x^4
)*ArcTan[a*x])/(x^3 + a^2*x^5) - 11*a^3*Sqrt[c]*Log[x] + 11*a^3*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c + a^2*c*x^2]])/
(6*c^2)

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Maple [C]  time = 0.655, size = 259, normalized size = 1.6 \begin{align*}{\frac{{a}^{3} \left ( \arctan \left ( ax \right ) +i \right ) \left ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( ax+i \right ) \left ( \arctan \left ( ax \right ) -i \right ){a}^{3}}{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{10\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-ax-2\,\arctan \left ( ax \right ) }{6\,{c}^{2}{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{11\,{a}^{3}}{6\,{c}^{2}}\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-1 \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+{\frac{11\,{a}^{3}}{6\,{c}^{2}}\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctan(a*x)/x^4/(a^2*c*x^2+c)^(3/2),x)

[Out]

1/2*a^3*(arctan(a*x)+I)*(a*x-I)*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)/c^2+1/2*(c*(a*x-I)*(a*x+I))^(1/2)*(a*x+I
)*(arctan(a*x)-I)*a^3/(a^2*x^2+1)/c^2+1/6*(10*arctan(a*x)*a^2*x^2-a*x-2*arctan(a*x))*(c*(a*x-I)*(a*x+I))^(1/2)
/x^3/c^2-11/6*a^3*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/c^2+11/6*a^3*l
n(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(a*x+I))^(1/2)/c^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.55897, size = 282, normalized size = 1.71 \begin{align*} \frac{11 \,{\left (a^{5} x^{5} + a^{3} x^{3}\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, \sqrt{a^{2} c x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (5 \, a^{3} x^{3} - a x + 2 \,{\left (8 \, a^{4} x^{4} + 4 \, a^{2} x^{2} - 1\right )} \arctan \left (a x\right )\right )} \sqrt{a^{2} c x^{2} + c}}{12 \,{\left (a^{2} c^{2} x^{5} + c^{2} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

1/12*(11*(a^5*x^5 + a^3*x^3)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(a^2*c*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(5*a^3*x^
3 - a*x + 2*(8*a^4*x^4 + 4*a^2*x^2 - 1)*arctan(a*x))*sqrt(a^2*c*x^2 + c))/(a^2*c^2*x^5 + c^2*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}{\left (a x \right )}}{x^{4} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atan(a*x)/x**4/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral(atan(a*x)/(x**4*(c*(a**2*x**2 + 1))**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctan(a*x)/x^4/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(arctan(a*x)/((a^2*c*x^2 + c)^(3/2)*x^4), x)