3.72 \(\int \frac{a+b \tan ^{-1}(c x^2)}{x^4} \, dx\)

Optimal. Leaf size=159 \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{3 x^3}-\frac{b c^{3/2} \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2}}+\frac{b c^{3/2} \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2}}+\frac{b c^{3/2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{3 \sqrt{2}}-\frac{b c^{3/2} \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{3 \sqrt{2}}-\frac{2 b c}{3 x} \]

[Out]

(-2*b*c)/(3*x) - (a + b*ArcTan[c*x^2])/(3*x^3) + (b*c^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]) - (b*c^
(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]) - (b*c^(3/2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2])
+ (b*c^(3/2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2])

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Rubi [A]  time = 0.104142, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5033, 325, 297, 1162, 617, 204, 1165, 628} \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{3 x^3}-\frac{b c^{3/2} \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2}}+\frac{b c^{3/2} \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2}}+\frac{b c^{3/2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{c} x\right )}{3 \sqrt{2}}-\frac{b c^{3/2} \tan ^{-1}\left (\sqrt{2} \sqrt{c} x+1\right )}{3 \sqrt{2}}-\frac{2 b c}{3 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^2])/x^4,x]

[Out]

(-2*b*c)/(3*x) - (a + b*ArcTan[c*x^2])/(3*x^3) + (b*c^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]) - (b*c^
(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[c]*x])/(3*Sqrt[2]) - (b*c^(3/2)*Log[1 - Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2])
+ (b*c^(3/2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2])

Rule 5033

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

Mathematica [A]  time = 0.0517281, size = 177, normalized size = 1.11 \[ -\frac{a}{3 x^3}-\frac{b c^{3/2} \log \left (c x^2-\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2}}+\frac{b c^{3/2} \log \left (c x^2+\sqrt{2} \sqrt{c} x+1\right )}{6 \sqrt{2}}-\frac{b c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} x-\sqrt{2}}{\sqrt{2}}\right )}{3 \sqrt{2}}-\frac{b c^{3/2} \tan ^{-1}\left (\frac{2 \sqrt{c} x+\sqrt{2}}{\sqrt{2}}\right )}{3 \sqrt{2}}-\frac{b \tan ^{-1}\left (c x^2\right )}{3 x^3}-\frac{2 b c}{3 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a/(3*x^3) - (2*b*c)/(3*x) - (b*ArcTan[c*x^2])/(3*x^3) - (b*c^(3/2)*ArcTan[(-Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/
(3*Sqrt[2]) - (b*c^(3/2)*ArcTan[(Sqrt[2] + 2*Sqrt[c]*x)/Sqrt[2]])/(3*Sqrt[2]) - (b*c^(3/2)*Log[1 - Sqrt[2]*Sqr
t[c]*x + c*x^2])/(6*Sqrt[2]) + (b*c^(3/2)*Log[1 + Sqrt[2]*Sqrt[c]*x + c*x^2])/(6*Sqrt[2])

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Maple [A]  time = 0.029, size = 132, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b\arctan \left ( c{x}^{2} \right ) }{3\,{x}^{3}}}-{\frac{bc\sqrt{2}}{12}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) \left ({x}^{2}+\sqrt [4]{{c}^{-2}}x\sqrt{2}+\sqrt{{c}^{-2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{bc\sqrt{2}}{6}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{bc\sqrt{2}}{6}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{c}^{-2}}}}}-1 \right ){\frac{1}{\sqrt [4]{{c}^{-2}}}}}-{\frac{2\,bc}{3\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a/x^3-1/3*b/x^3*arctan(c*x^2)-1/12*b*c/(1/c^2)^(1/4)*2^(1/2)*ln((x^2-(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2
))/(x^2+(1/c^2)^(1/4)*x*2^(1/2)+(1/c^2)^(1/2)))-1/6*b*c/(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*x+1
)-1/6*b*c/(1/c^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/c^2)^(1/4)*x-1)-2/3*b*c/x

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Maxima [B]  time = 1.52514, size = 369, normalized size = 2.32 \begin{align*} \frac{1}{12} \,{\left ({\left (c^{2}{\left (\frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\sqrt{c^{2}} x^{2} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}} x + 1\right )}{{\left (c^{2}\right )}^{\frac{3}{4}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} + \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{c^{2}} \sqrt{-\sqrt{c^{2}}}} - \frac{\sqrt{2} \log \left (\frac{2 \, \sqrt{c^{2}} x - \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}{2 \, \sqrt{c^{2}} x + \sqrt{2} \sqrt{-\sqrt{c^{2}}} - \sqrt{2}{\left (c^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{c^{2}} \sqrt{-\sqrt{c^{2}}}}\right )} - \frac{8}{x}\right )} c - \frac{4 \, \arctan \left (c x^{2}\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((c^2*(sqrt(2)*log(sqrt(c^2)*x^2 + sqrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(3/4) - sqrt(2)*log(sqrt(c^2)*x^2 - s
qrt(2)*(c^2)^(1/4)*x + 1)/(c^2)^(3/4) - sqrt(2)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^2)) + sqrt(2)*(c^2)^
(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-sqrt(c^2)) + sqrt(2)*(c^2)^(1/4)))/(sqrt(c^2)*sqrt(-sqrt(c^2))) - sqrt(2
)*log((2*sqrt(c^2)*x - sqrt(2)*sqrt(-sqrt(c^2)) - sqrt(2)*(c^2)^(1/4))/(2*sqrt(c^2)*x + sqrt(2)*sqrt(-sqrt(c^2
)) - sqrt(2)*(c^2)^(1/4)))/(sqrt(c^2)*sqrt(-sqrt(c^2)))) - 8/x)*c - 4*arctan(c*x^2)/x^3)*b - 1/3*a/x^3

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Fricas [B]  time = 2.8576, size = 900, normalized size = 5.66 \begin{align*} \frac{4 \, \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{1}{4}} x^{3} \arctan \left (-\frac{b^{4} c^{6} + \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{1}{4}} b^{3} c^{5} x - \sqrt{2} \sqrt{b^{6} c^{10} x^{2} + \sqrt{b^{4} c^{6}} b^{4} c^{6} + \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{3}{4}} b^{3} c^{5} x} \left (b^{4} c^{6}\right )^{\frac{1}{4}}}{b^{4} c^{6}}\right ) + 4 \, \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{1}{4}} x^{3} \arctan \left (\frac{b^{4} c^{6} - \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{1}{4}} b^{3} c^{5} x + \sqrt{2} \sqrt{b^{6} c^{10} x^{2} + \sqrt{b^{4} c^{6}} b^{4} c^{6} - \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{3}{4}} b^{3} c^{5} x} \left (b^{4} c^{6}\right )^{\frac{1}{4}}}{b^{4} c^{6}}\right ) + \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{1}{4}} x^{3} \log \left (b^{6} c^{10} x^{2} + \sqrt{b^{4} c^{6}} b^{4} c^{6} + \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{3}{4}} b^{3} c^{5} x\right ) - \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{1}{4}} x^{3} \log \left (b^{6} c^{10} x^{2} + \sqrt{b^{4} c^{6}} b^{4} c^{6} - \sqrt{2} \left (b^{4} c^{6}\right )^{\frac{3}{4}} b^{3} c^{5} x\right ) - 8 \, b c x^{2} - 4 \, b \arctan \left (c x^{2}\right ) - 4 \, a}{12 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*sqrt(2)*(b^4*c^6)^(1/4)*x^3*arctan(-(b^4*c^6 + sqrt(2)*(b^4*c^6)^(1/4)*b^3*c^5*x - sqrt(2)*sqrt(b^6*c^
10*x^2 + sqrt(b^4*c^6)*b^4*c^6 + sqrt(2)*(b^4*c^6)^(3/4)*b^3*c^5*x)*(b^4*c^6)^(1/4))/(b^4*c^6)) + 4*sqrt(2)*(b
^4*c^6)^(1/4)*x^3*arctan((b^4*c^6 - sqrt(2)*(b^4*c^6)^(1/4)*b^3*c^5*x + sqrt(2)*sqrt(b^6*c^10*x^2 + sqrt(b^4*c
^6)*b^4*c^6 - sqrt(2)*(b^4*c^6)^(3/4)*b^3*c^5*x)*(b^4*c^6)^(1/4))/(b^4*c^6)) + sqrt(2)*(b^4*c^6)^(1/4)*x^3*log
(b^6*c^10*x^2 + sqrt(b^4*c^6)*b^4*c^6 + sqrt(2)*(b^4*c^6)^(3/4)*b^3*c^5*x) - sqrt(2)*(b^4*c^6)^(1/4)*x^3*log(b
^6*c^10*x^2 + sqrt(b^4*c^6)*b^4*c^6 - sqrt(2)*(b^4*c^6)^(3/4)*b^3*c^5*x) - 8*b*c*x^2 - 4*b*arctan(c*x^2) - 4*a
)/x^3

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Sympy [A]  time = 55.5763, size = 704, normalized size = 4.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/(3*x**3), Eq(c, 0)), (-(a - oo*I*b)/(3*x**3), Eq(c, -I/x**2)), (-(a + oo*I*b)/(3*x**3), Eq(c, I/
x**2)), (-6*a*x**4/(18*x**7 + 18*x**3/c**2) - 6*a/(18*c**2*x**7 + 18*x**3) + 4*I*b*c**24*x**4*(c**(-2))**(27/2
)/(18*x**7/c**3 + 18*x**3/c**5) - 6*(-1)**(3/4)*b*c**18*x**7*(c**(-2))**(39/4)*atan((-1)**(3/4)*x/(c**(-2))**(
1/4))/(18*x**7/c**3 + 18*x**3/c**5) - 4*I*b*c**16*x**4*(c**(-2))**(19/2)/(18*x**7/c**3 + 18*x**3/c**5) - 6*(-1
)**(3/4)*b*c**16*x**3*(c**(-2))**(39/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(18*x**7/c**3 + 18*x**3/c**5) + 6
*(-1)**(3/4)*b*c**10*x**7*(c**(-2))**(23/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(18*x**7/c**3 + 18*x**3/c**5
) - 3*(-1)**(3/4)*b*c**10*x**7*(c**(-2))**(23/4)*log(x**2 + I*sqrt(c**(-2)))/(18*x**7/c**3 + 18*x**3/c**5) + 6
*(-1)**(3/4)*b*c**8*x**3*(c**(-2))**(23/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(18*x**7/c**3 + 18*x**3/c**5)
- 3*(-1)**(3/4)*b*c**8*x**3*(c**(-2))**(23/4)*log(x**2 + I*sqrt(c**(-2)))/(18*x**7/c**3 + 18*x**3/c**5) + 6*(
-1)**(1/4)*b*c**5*x**7*(c**(-2))**(13/4)*atan(c*x**2)/(18*x**7/c**3 + 18*x**3/c**5) + 6*(-1)**(1/4)*b*c**3*x**
3*(c**(-2))**(13/4)*atan(c*x**2)/(18*x**7/c**3 + 18*x**3/c**5) - 12*b*x**6/(18*x**7/c + 18*x**3/c**3) - 6*b*x*
*4*atan(c*x**2)/(18*x**7 + 18*x**3/c**2) - 12*b*x**2/(18*c*x**7 + 18*x**3/c) - 6*b*atan(c*x**2)/(18*c**2*x**7
+ 18*x**3), True))

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Giac [A]  time = 1.26766, size = 215, normalized size = 1.35 \begin{align*} -\frac{1}{12} \, b c^{3}{\left (\frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{2}} + \frac{2 \, \sqrt{2} \sqrt{{\left | c \right |}} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \frac{\sqrt{2}}{\sqrt{{\left | c \right |}}}\right )} \sqrt{{\left | c \right |}}\right )}{c^{2}} - \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} + \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{2}} + \frac{\sqrt{2} \sqrt{{\left | c \right |}} \log \left (x^{2} - \frac{\sqrt{2} x}{\sqrt{{\left | c \right |}}} + \frac{1}{{\left | c \right |}}\right )}{c^{2}}\right )} - \frac{2 \, b c x^{2} + b \arctan \left (c x^{2}\right ) + a}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*b*c^3*(2*sqrt(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x + sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/c^2 + 2*sqrt
(2)*sqrt(abs(c))*arctan(1/2*sqrt(2)*(2*x - sqrt(2)/sqrt(abs(c)))*sqrt(abs(c)))/c^2 - sqrt(2)*sqrt(abs(c))*log(
x^2 + sqrt(2)*x/sqrt(abs(c)) + 1/abs(c))/c^2 + sqrt(2)*sqrt(abs(c))*log(x^2 - sqrt(2)*x/sqrt(abs(c)) + 1/abs(c
))/c^2) - 1/3*(2*b*c*x^2 + b*arctan(c*x^2) + a)/x^3