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]

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

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Rubi [A]  time = 0.10, antiderivative size = 159, normalized size of antiderivative = 1.00, 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 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 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 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 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.06, 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]

-1/3*a/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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fricas [B]  time = 0.45, size = 389, normalized size = 2.45 \[ \frac {4 \, \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {1}{4}} x^{3} \arctan \left (-\frac {b^{4} c^{6} + \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {1}{4}} b^{3} c^{5} x - \sqrt {2} \sqrt {b^{6} c^{10} x^{2} + \sqrt {b^{4} c^{6}} b^{4} c^{6} + \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {3}{4}} b^{3} c^{5} x} \left (b^{4} c^{6}\right )^{\frac {1}{4}}}{b^{4} c^{6}}\right ) + 4 \, \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {1}{4}} x^{3} \arctan \left (\frac {b^{4} c^{6} - \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {1}{4}} b^{3} c^{5} x + \sqrt {2} \sqrt {b^{6} c^{10} x^{2} + \sqrt {b^{4} c^{6}} b^{4} c^{6} - \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {3}{4}} b^{3} c^{5} x} \left (b^{4} c^{6}\right )^{\frac {1}{4}}}{b^{4} c^{6}}\right ) + \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {1}{4}} x^{3} \log \left (b^{6} c^{10} x^{2} + \sqrt {b^{4} c^{6}} b^{4} c^{6} + \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {3}{4}} b^{3} c^{5} x\right ) - \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {1}{4}} x^{3} \log \left (b^{6} c^{10} x^{2} + \sqrt {b^{4} c^{6}} b^{4} c^{6} - \sqrt {2} \left (b^{4} c^{6}\right )^{\frac {3}{4}} b^{3} c^{5} x\right ) - 8 \, b c x^{2} - 4 \, b \arctan \left (c x^{2}\right ) - 4 \, a}{12 \, x^{3}} \]

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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giac [A]  time = 5.36, size = 159, normalized size = 1.00 \[ -\frac {1}{12} \, b c^{3} {\left (\frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} + \frac {2 \, \sqrt {2} \sqrt {{\left | c \right |}} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \frac {\sqrt {2}}{\sqrt {{\left | c \right |}}}\right )} \sqrt {{\left | c \right |}}\right )}{c^{2}} - \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} + \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}} + \frac {\sqrt {2} \sqrt {{\left | c \right |}} \log \left (x^{2} - \frac {\sqrt {2} x}{\sqrt {{\left | c \right |}}} + \frac {1}{{\left | c \right |}}\right )}{c^{2}}\right )} - \frac {2 \, b c x^{2} + b \arctan \left (c x^{2}\right ) + a}{3 \, x^{3}} \]

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

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maple [A]  time = 0.03, size = 132, normalized size = 0.83 \[ -\frac {a}{3 x^{3}}-\frac {b \arctan \left (c \,x^{2}\right )}{3 x^{3}}-\frac {2 b c}{3 x}-\frac {b c \sqrt {2}\, \ln \left (\frac {x^{2}-\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}{x^{2}+\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {1}{c^{2}}}}\right )}{12 \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-\frac {b c \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}+1\right )}{6 \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-\frac {b c \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}}-1\right )}{6 \left (\frac {1}{c^{2}}\right )^{\frac {1}{4}}} \]

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)-2/3*b*c/x-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)

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maxima [A]  time = 0.41, size = 142, normalized size = 0.89 \[ -\frac {1}{12} \, {\left ({\left (c^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right )}{c^{\frac {3}{2}}}\right )} + \frac {8}{x}\right )} c + \frac {4 \, \arctan \left (c x^{2}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \]

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*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*c*x + sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) + 2*sqrt(2)*arctan(1/2*sq
rt(2)*(2*c*x - sqrt(2)*sqrt(c))/sqrt(c))/c^(3/2) - sqrt(2)*log(c*x^2 + sqrt(2)*sqrt(c)*x + 1)/c^(3/2) + sqrt(2
)*log(c*x^2 - sqrt(2)*sqrt(c)*x + 1)/c^(3/2)) + 8/x)*c + 4*arctan(c*x^2)/x^3)*b - 1/3*a/x^3

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mupad [B]  time = 0.43, size = 63, normalized size = 0.40 \[ -\frac {2\,b\,c\,x^2+a}{3\,x^3}-\frac {b\,\mathrm {atan}\left (c\,x^2\right )}{3\,x^3}-\frac {{\left (-1\right )}^{1/4}\,b\,c^{3/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\right )}{3}-\frac {{\left (-1\right )}^{1/4}\,b\,c^{3/2}\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x^2))/x^4,x)

[Out]

- (a + 2*b*c*x^2)/(3*x^3) - (b*atan(c*x^2))/(3*x^3) - ((-1)^(1/4)*b*c^(3/2)*atan((-1)^(1/4)*c^(1/2)*x))/3 - ((
-1)^(1/4)*b*c^(3/2)*atan((-1)^(1/4)*c^(1/2)*x*1i)*1i)/3

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sympy [A]  time = 38.52, size = 590, normalized size = 3.71 \[ \begin {cases} - \frac {a}{3 x^{3}} & \text {for}\: c = 0 \\- \frac {a - \infty i b}{3 x^{3}} & \text {for}\: c = - \frac {i}{x^{2}} \\- \frac {a + \infty i b}{3 x^{3}} & \text {for}\: c = \frac {i}{x^{2}} \\- \frac {2 a x^{4}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {2 a}{6 c^{2} x^{7} + 6 x^{3}} + \frac {2 \left (-1\right )^{\frac {3}{4}} b c^{3} x^{7} \left (\frac {1}{c^{2}}\right )^{\frac {3}{4}} \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{c^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {\left (-1\right )^{\frac {3}{4}} b c^{3} x^{7} \left (\frac {1}{c^{2}}\right )^{\frac {3}{4}} \log {\left (x^{2} + i \sqrt {\frac {1}{c^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {2 \left (-1\right )^{\frac {3}{4}} b c^{3} x^{7} \left (\frac {1}{c^{2}}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{c^{2}}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} + \frac {2 \sqrt [4]{-1} b c^{2} x^{7} \sqrt [4]{\frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{2} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {4 b c x^{6}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} + \frac {2 \left (-1\right )^{\frac {3}{4}} b c x^{3} \left (\frac {1}{c^{2}}\right )^{\frac {3}{4}} \log {\left (x - \sqrt [4]{-1} \sqrt [4]{\frac {1}{c^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {\left (-1\right )^{\frac {3}{4}} b c x^{3} \left (\frac {1}{c^{2}}\right )^{\frac {3}{4}} \log {\left (x^{2} + i \sqrt {\frac {1}{c^{2}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {2 \left (-1\right )^{\frac {3}{4}} b c x^{3} \left (\frac {1}{c^{2}}\right )^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\left (-1\right )^{\frac {3}{4}} x}{\sqrt [4]{\frac {1}{c^{2}}}} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {2 b x^{4} \operatorname {atan}{\left (c x^{2} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} + \frac {2 \sqrt [4]{-1} b x^{3} \sqrt [4]{\frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{2} \right )}}{6 x^{7} + \frac {6 x^{3}}{c^{2}}} - \frac {4 b x^{2}}{6 c x^{7} + \frac {6 x^{3}}{c}} - \frac {2 b \operatorname {atan}{\left (c x^{2} \right )}}{6 c^{2} x^{7} + 6 x^{3}} & \text {otherwise} \end {cases} \]

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)), (-2*a*x**4/(6*x**7 + 6*x**3/c**2) - 2*a/(6*c**2*x**7 + 6*x**3) + 2*(-1)**(3/4)*b*c**3*x**7*(c**(-2))**
(3/4)*log(x - (-1)**(1/4)*(c**(-2))**(1/4))/(6*x**7 + 6*x**3/c**2) - (-1)**(3/4)*b*c**3*x**7*(c**(-2))**(3/4)*
log(x**2 + I*sqrt(c**(-2)))/(6*x**7 + 6*x**3/c**2) - 2*(-1)**(3/4)*b*c**3*x**7*(c**(-2))**(3/4)*atan((-1)**(3/
4)*x/(c**(-2))**(1/4))/(6*x**7 + 6*x**3/c**2) + 2*(-1)**(1/4)*b*c**2*x**7*(c**(-2))**(1/4)*atan(c*x**2)/(6*x**
7 + 6*x**3/c**2) - 4*b*c*x**6/(6*x**7 + 6*x**3/c**2) + 2*(-1)**(3/4)*b*c*x**3*(c**(-2))**(3/4)*log(x - (-1)**(
1/4)*(c**(-2))**(1/4))/(6*x**7 + 6*x**3/c**2) - (-1)**(3/4)*b*c*x**3*(c**(-2))**(3/4)*log(x**2 + I*sqrt(c**(-2
)))/(6*x**7 + 6*x**3/c**2) - 2*(-1)**(3/4)*b*c*x**3*(c**(-2))**(3/4)*atan((-1)**(3/4)*x/(c**(-2))**(1/4))/(6*x
**7 + 6*x**3/c**2) - 2*b*x**4*atan(c*x**2)/(6*x**7 + 6*x**3/c**2) + 2*(-1)**(1/4)*b*x**3*(c**(-2))**(1/4)*atan
(c*x**2)/(6*x**7 + 6*x**3/c**2) - 4*b*x**2/(6*c*x**7 + 6*x**3/c) - 2*b*atan(c*x**2)/(6*c**2*x**7 + 6*x**3), Tr
ue))

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