3.112 \(\int \frac {a+b \tan ^{-1}(c x^3)}{x^5} \, dx\)

Optimal. Leaf size=174 \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )-\frac {1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )+\frac {1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )-\frac {1}{8} b c^{4/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )-\frac {3 b c}{4 x} \]

[Out]

-3/4*b*c/x-1/4*b*c^(4/3)*arctan(c^(1/3)*x)+1/4*(-a-b*arctan(c*x^3))/x^4-1/8*b*c^(4/3)*arctan(2*c^(1/3)*x-3^(1/
2))-1/8*b*c^(4/3)*arctan(2*c^(1/3)*x+3^(1/2))-1/16*b*c^(4/3)*ln(1+c^(2/3)*x^2-c^(1/3)*x*3^(1/2))*3^(1/2)+1/16*
b*c^(4/3)*ln(1+c^(2/3)*x^2+c^(1/3)*x*3^(1/2))*3^(1/2)

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Rubi [A]  time = 0.41, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5033, 325, 295, 634, 618, 204, 628, 203} \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )-\frac {1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )+\frac {1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )-\frac {1}{8} b c^{4/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )-\frac {3 b c}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^3])/x^5,x]

[Out]

(-3*b*c)/(4*x) - (b*c^(4/3)*ArcTan[c^(1/3)*x])/4 - (a + b*ArcTan[c*x^3])/(4*x^4) + (b*c^(4/3)*ArcTan[Sqrt[3] -
2*c^(1/3)*x])/8 - (b*c^(4/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/8 - (Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^(1/3)*x
+ c^(2/3)*x^2])/16 + (Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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^3\right )}{x^5} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}+\frac {1}{4} (3 b c) \int \frac {1}{x^2 \left (1+c^2 x^6\right )} \, dx\\ &=-\frac {3 b c}{4 x}-\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{4} \left (3 b c^3\right ) \int \frac {x^4}{1+c^2 x^6} \, dx\\ &=-\frac {3 b c}{4 x}-\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{4} \left (b c^{5/3}\right ) \int \frac {1}{1+c^{2/3} x^2} \, dx-\frac {1}{4} \left (b c^{5/3}\right ) \int \frac {-\frac {1}{2}+\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx-\frac {1}{4} \left (b c^{5/3}\right ) \int \frac {-\frac {1}{2}-\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=-\frac {3 b c}{4 x}-\frac {1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} \left (\sqrt {3} b c^{4/3}\right ) \int \frac {-\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{16} \left (\sqrt {3} b c^{4/3}\right ) \int \frac {\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx-\frac {1}{16} \left (b c^{5/3}\right ) \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx-\frac {1}{16} \left (b c^{5/3}\right ) \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=-\frac {3 b c}{4 x}-\frac {1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )-\frac {\left (b c^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 \sqrt {3}}+\frac {\left (b c^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 \sqrt {3}}\\ &=-\frac {3 b c}{4 x}-\frac {1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tan ^{-1}\left (c x^3\right )}{4 x^4}+\frac {1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )-\frac {1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )-\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 179, normalized size = 1.03 \[ -\frac {a}{4 x^4}-\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{16} \sqrt {3} b c^{4/3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )-\frac {1}{4} b c^{4/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )+\frac {1}{8} b c^{4/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )-\frac {1}{8} b c^{4/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )-\frac {b \tan ^{-1}\left (c x^3\right )}{4 x^4}-\frac {3 b c}{4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x^3])/x^5,x]

[Out]

-1/4*a/x^4 - (3*b*c)/(4*x) - (b*c^(4/3)*ArcTan[c^(1/3)*x])/4 - (b*ArcTan[c*x^3])/(4*x^4) + (b*c^(4/3)*ArcTan[S
qrt[3] - 2*c^(1/3)*x])/8 - (b*c^(4/3)*ArcTan[Sqrt[3] + 2*c^(1/3)*x])/8 - (Sqrt[3]*b*c^(4/3)*Log[1 - Sqrt[3]*c^
(1/3)*x + c^(2/3)*x^2])/16 + (Sqrt[3]*b*c^(4/3)*Log[1 + Sqrt[3]*c^(1/3)*x + c^(2/3)*x^2])/16

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fricas [B]  time = 0.52, size = 595, normalized size = 3.42 \[ \frac {\sqrt {3} \left (b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \log \left (4 \, b^{10} c^{14} x^{2} + 4 \, \left (b^{6} c^{8}\right )^{\frac {2}{3}} b^{6} c^{8} + 4 \, \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {5}{6}} b^{5} c^{7} x\right ) - \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \log \left (4 \, b^{10} c^{14} x^{2} + 4 \, \left (b^{6} c^{8}\right )^{\frac {2}{3}} b^{6} c^{8} - 4 \, \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {5}{6}} b^{5} c^{7} x\right ) + \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \log \left (b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac {2}{3}} b^{6} c^{8} + \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {5}{6}} b^{5} c^{7} x\right ) - \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \log \left (b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac {2}{3}} b^{6} c^{8} - \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {5}{6}} b^{5} c^{7} x\right ) + 8 \, \left (b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \arctan \left (-\frac {\sqrt {3} b^{6} c^{8} + 2 \, \left (b^{6} c^{8}\right )^{\frac {1}{6}} b^{5} c^{7} x - 2 \, \sqrt {b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac {2}{3}} b^{6} c^{8} + \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {5}{6}} b^{5} c^{7} x} \left (b^{6} c^{8}\right )^{\frac {1}{6}}}{b^{6} c^{8}}\right ) + 8 \, \left (b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \arctan \left (\frac {\sqrt {3} b^{6} c^{8} - 2 \, \left (b^{6} c^{8}\right )^{\frac {1}{6}} b^{5} c^{7} x + 2 \, \sqrt {b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac {2}{3}} b^{6} c^{8} - \sqrt {3} \left (b^{6} c^{8}\right )^{\frac {5}{6}} b^{5} c^{7} x} \left (b^{6} c^{8}\right )^{\frac {1}{6}}}{b^{6} c^{8}}\right ) + 16 \, \left (b^{6} c^{8}\right )^{\frac {1}{6}} x^{4} \arctan \left (-\frac {\left (b^{6} c^{8}\right )^{\frac {1}{6}} b^{5} c^{7} x - \sqrt {b^{10} c^{14} x^{2} + \left (b^{6} c^{8}\right )^{\frac {2}{3}} b^{6} c^{8}} \left (b^{6} c^{8}\right )^{\frac {1}{6}}}{b^{6} c^{8}}\right ) - 24 \, b c x^{3} - 8 \, b \arctan \left (c x^{3}\right ) - 8 \, a}{32 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x^5,x, algorithm="fricas")

[Out]

1/32*(sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(4*b^10*c^14*x^2 + 4*(b^6*c^8)^(2/3)*b^6*c^8 + 4*sqrt(3)*(b^6*c^8)^(5/6)*
b^5*c^7*x) - sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(4*b^10*c^14*x^2 + 4*(b^6*c^8)^(2/3)*b^6*c^8 - 4*sqrt(3)*(b^6*c^8)
^(5/6)*b^5*c^7*x) + sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(b^10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8 + sqrt(3)*(b^6*c^8
)^(5/6)*b^5*c^7*x) - sqrt(3)*(b^6*c^8)^(1/6)*x^4*log(b^10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8 - sqrt(3)*(b^6*c^
8)^(5/6)*b^5*c^7*x) + 8*(b^6*c^8)^(1/6)*x^4*arctan(-(sqrt(3)*b^6*c^8 + 2*(b^6*c^8)^(1/6)*b^5*c^7*x - 2*sqrt(b^
10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8 + sqrt(3)*(b^6*c^8)^(5/6)*b^5*c^7*x)*(b^6*c^8)^(1/6))/(b^6*c^8)) + 8*(b^
6*c^8)^(1/6)*x^4*arctan((sqrt(3)*b^6*c^8 - 2*(b^6*c^8)^(1/6)*b^5*c^7*x + 2*sqrt(b^10*c^14*x^2 + (b^6*c^8)^(2/3
)*b^6*c^8 - sqrt(3)*(b^6*c^8)^(5/6)*b^5*c^7*x)*(b^6*c^8)^(1/6))/(b^6*c^8)) + 16*(b^6*c^8)^(1/6)*x^4*arctan(-((
b^6*c^8)^(1/6)*b^5*c^7*x - sqrt(b^10*c^14*x^2 + (b^6*c^8)^(2/3)*b^6*c^8)*(b^6*c^8)^(1/6))/(b^6*c^8)) - 24*b*c*
x^3 - 8*b*arctan(c*x^3) - 8*a)/x^4

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giac [A]  time = 2.98, size = 161, normalized size = 0.93 \[ \frac {1}{16} \, b c^{3} {\left (\frac {\sqrt {3} {\left | c \right |}^{\frac {1}{3}} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{2}} - \frac {\sqrt {3} {\left | c \right |}^{\frac {1}{3}} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{2}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}} - \frac {4 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}}\right )} - \frac {3 \, b c x^{3} + b \arctan \left (c x^{3}\right ) + a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x^5,x, algorithm="giac")

[Out]

1/16*b*c^3*(sqrt(3)*abs(c)^(1/3)*log(x^2 + sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - sqrt(3)*abs(c)^(1/3)
*log(x^2 - sqrt(3)*x/abs(c)^(1/3) + 1/abs(c)^(2/3))/c^2 - 2*abs(c)^(1/3)*arctan((2*x + sqrt(3)/abs(c)^(1/3))*a
bs(c)^(1/3))/c^2 - 2*abs(c)^(1/3)*arctan((2*x - sqrt(3)/abs(c)^(1/3))*abs(c)^(1/3))/c^2 - 4*abs(c)^(1/3)*arcta
n(x*abs(c)^(1/3))/c^2) - 1/4*(3*b*c*x^3 + b*arctan(c*x^3) + a)/x^4

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maple [A]  time = 0.10, size = 159, normalized size = 0.91 \[ -\frac {a}{4 x^{4}}-\frac {b \arctan \left (c \,x^{3}\right )}{4 x^{4}}-\frac {b c \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{4 \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {b \,c^{3} \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{16}-\frac {b c \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{8 \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {b \,c^{3} \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{16}-\frac {b c \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{8 \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {3 b c}{4 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^3))/x^5,x)

[Out]

-1/4*a/x^4-1/4*b/x^4*arctan(c*x^3)-1/4*b*c/(1/c^2)^(1/6)*arctan(x/(1/c^2)^(1/6))-1/16*b*c^3*3^(1/2)*(1/c^2)^(5
/6)*ln(x^2-3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))-1/8*b*c/(1/c^2)^(1/6)*arctan(2*x/(1/c^2)^(1/6)-3^(1/2))+1/16
*b*c^3*3^(1/2)*(1/c^2)^(5/6)*ln(x^2+3^(1/2)*(1/c^2)^(1/6)*x+(1/c^2)^(1/3))-1/8*b*c/(1/c^2)^(1/6)*arctan(2*x/(1
/c^2)^(1/6)+3^(1/2))-3/4*b*c/x

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maxima [A]  time = 0.42, size = 147, normalized size = 0.84 \[ \frac {1}{16} \, {\left ({\left (c^{2} {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )} - \frac {12}{x}\right )} c - \frac {4 \, \arctan \left (c x^{3}\right )}{x^{4}}\right )} b - \frac {a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))/x^5,x, algorithm="maxima")

[Out]

1/16*((c^2*(sqrt(3)*log(c^(2/3)*x^2 + sqrt(3)*c^(1/3)*x + 1)/c^(5/3) - sqrt(3)*log(c^(2/3)*x^2 - sqrt(3)*c^(1/
3)*x + 1)/c^(5/3) - 4*arctan(c^(1/3)*x)/c^(5/3) - 2*arctan((2*c^(2/3)*x + sqrt(3)*c^(1/3))/c^(1/3))/c^(5/3) -
2*arctan((2*c^(2/3)*x - sqrt(3)*c^(1/3))/c^(1/3))/c^(5/3)) - 12/x)*c - 4*arctan(c*x^3)/x^4)*b - 1/4*a/x^4

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mupad [B]  time = 0.71, size = 120, normalized size = 0.69 \[ -\frac {a}{4\,x^4}+\frac {b\,c^{4/3}\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )+\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{8}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{4\,x^4}-\frac {3\,b\,c}{4\,x}-\frac {\sqrt {3}\,b\,c^{4/3}\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )-\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )\,1{}\mathrm {i}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x^3))/x^5,x)

[Out]

(b*c^(4/3)*(atan((-1)^(2/3)*c^(1/3)*x) + atan(((-1)^(2/3)*c^(1/3)*x*(3^(1/2)*1i - 1))/2) + 2*atan(((-1)^(2/3)*
c^(1/3)*x*(3^(1/2)*1i + 1))/2)))/8 - a/(4*x^4) - (b*atan(c*x^3))/(4*x^4) - (3*b*c)/(4*x) - (3^(1/2)*b*c^(4/3)*
(atan((-1)^(2/3)*c^(1/3)*x) - atan(((-1)^(2/3)*c^(1/3)*x*(3^(1/2)*1i - 1))/2))*1i)/8

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sympy [A]  time = 109.84, size = 320, normalized size = 1.84 \[ \begin {cases} - \frac {a}{4 x^{4}} + \frac {3 \left (-1\right )^{\frac {5}{6}} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \log {\left (4 x^{2} - 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{16} - \frac {3 \left (-1\right )^{\frac {5}{6}} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \log {\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{16} - \frac {\left (-1\right )^{\frac {5}{6}} \sqrt {3} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{8} - \frac {\left (-1\right )^{\frac {5}{6}} \sqrt {3} b c^{3} \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{8} - \frac {\sqrt [3]{-1} b c^{2} \sqrt [3]{\frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{4} - \frac {3 b c}{4 x} - \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{4 x^{4}} & \text {for}\: c \neq 0 \\- \frac {a}{4 x^{4}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x**3))/x**5,x)

[Out]

Piecewise((-a/(4*x**4) + 3*(-1)**(5/6)*b*c**3*(c**(-2))**(5/6)*log(4*x**2 - 4*(-1)**(1/6)*x*(c**(-2))**(1/6) +
4*(-1)**(1/3)*(c**(-2))**(1/3))/16 - 3*(-1)**(5/6)*b*c**3*(c**(-2))**(5/6)*log(4*x**2 + 4*(-1)**(1/6)*x*(c**(
-2))**(1/6) + 4*(-1)**(1/3)*(c**(-2))**(1/3))/16 - (-1)**(5/6)*sqrt(3)*b*c**3*(c**(-2))**(5/6)*atan(2*(-1)**(5
/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) - sqrt(3)/3)/8 - (-1)**(5/6)*sqrt(3)*b*c**3*(c**(-2))**(5/6)*atan(2*(-1)**(
5/6)*sqrt(3)*x/(3*(c**(-2))**(1/6)) + sqrt(3)/3)/8 - (-1)**(1/3)*b*c**2*(c**(-2))**(1/3)*atan(c*x**3)/4 - 3*b*
c/(4*x) - b*atan(c*x**3)/(4*x**4), Ne(c, 0)), (-a/(4*x**4), True))

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