\(\int \frac {(a+b \arctan (c x^3))^2}{x^7} \, dx\) [119]

Optimal result

Integrand size = 16, antiderivative size = 87 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )}{3 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+b^2 c^2 \log (x)-\frac {1}{6} b^2 c^2 \log \left (1+c^2 x^6\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4948, 4946, 5038, 272, 36, 29, 31, 5004} \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )}{3 x^3}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}-\frac {1}{6} b^2 c^2 \log \left (c^2 x^6+1\right )+b^2 c^2 \log (x) \]

[In]

Int[(a + b*ArcTan[c*x^3])^2/x^7,x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

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

Rule 272

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 4946

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

Rule 4948

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

Rule 5004

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

Rule 5038

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^3} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+\frac {1}{3} (b c) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x^2 \left (1+c^2 x^2\right )} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+\frac {1}{3} (b c) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x^2} \, dx,x,x^3\right )-\frac {1}{3} \left (b c^3\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx,x,x^3\right ) \\ & = -\frac {b c \left (a+b \arctan \left (c x^3\right )\right )}{3 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+\frac {1}{3} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x^2\right )} \, dx,x,x^3\right ) \\ & = -\frac {b c \left (a+b \arctan \left (c x^3\right )\right )}{3 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^6\right ) \\ & = -\frac {b c \left (a+b \arctan \left (c x^3\right )\right )}{3 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+\frac {1}{6} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^6\right )-\frac {1}{6} \left (b^2 c^4\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^6\right ) \\ & = -\frac {b c \left (a+b \arctan \left (c x^3\right )\right )}{3 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{6 x^6}+b^2 c^2 \log (x)-\frac {1}{6} b^2 c^2 \log \left (1+c^2 x^6\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {a^2+2 a b c x^3+2 b \left (a+b c x^3+a c^2 x^6\right ) \arctan \left (c x^3\right )+b^2 \left (1+c^2 x^6\right ) \arctan \left (c x^3\right )^2-6 b^2 c^2 x^6 \log (x)+b^2 c^2 x^6 \log \left (1+c^2 x^6\right )}{6 x^6} \]

[In]

Integrate[(a + b*ArcTan[c*x^3])^2/x^7,x]

[Out]

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

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {a^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2} c^{2}}{6}-\frac {b^{2} c \arctan \left (c \,x^{3}\right )}{3 x^{3}}+b^{2} c^{2} \ln \left (x \right )-\frac {b^{2} c^{2} \ln \left (c^{2} x^{6}+1\right )}{6}-\frac {a b \arctan \left (c \,x^{3}\right )}{3 x^{6}}-\frac {a b \arctan \left (c \,x^{3}\right ) c^{2}}{3}-\frac {a b c}{3 x^{3}}\)	\(118\)
parts	\(-\frac {a^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2} c^{2}}{6}-\frac {b^{2} c \arctan \left (c \,x^{3}\right )}{3 x^{3}}+b^{2} c^{2} \ln \left (x \right )-\frac {b^{2} c^{2} \ln \left (c^{2} x^{6}+1\right )}{6}-\frac {a b \arctan \left (c \,x^{3}\right )}{3 x^{6}}-\frac {a b \arctan \left (c \,x^{3}\right ) c^{2}}{3}-\frac {a b c}{3 x^{3}}\)	\(118\)
parallelrisch	\(\frac {-b^{2} \arctan \left (c \,x^{3}\right )^{2} x^{6} c^{2}+6 b^{2} c^{2} \ln \left (x \right ) x^{6}-b^{2} c^{2} \ln \left (c^{2} x^{6}+1\right ) x^{6}-2 a b \arctan \left (c \,x^{3}\right ) x^{6} c^{2}+a^{2} c^{2} x^{6}-2 b^{2} \arctan \left (c \,x^{3}\right ) x^{3} c -2 a b c \,x^{3}-b^{2} \arctan \left (c \,x^{3}\right )^{2}-2 a b \arctan \left (c \,x^{3}\right )-a^{2}}{6 x^{6}}\)	\(137\)
risch	\(\frac {b^{2} \left (c^{2} x^{6}+1\right ) \ln \left (i c \,x^{3}+1\right )^{2}}{24 x^{6}}+\frac {i b \left (i b \,c^{2} x^{6} \ln \left (-i c \,x^{3}+1\right )+2 b c \,x^{3}+2 a +i b \ln \left (-i c \,x^{3}+1\right )\right ) \ln \left (i c \,x^{3}+1\right )}{12 x^{6}}-\frac {4 i \ln \left (\left (-7 i b c +a c \right ) x^{3}+7 b +i a \right ) a b \,c^{2} x^{6}-4 i \ln \left (\left (7 i b c +a c \right ) x^{3}+7 b -i a \right ) a b \,c^{2} x^{6}-b^{2} c^{2} x^{6} \ln \left (-i c \,x^{3}+1\right )^{2}-24 b^{2} c^{2} \ln \left (x \right ) x^{6}+4 \ln \left (\left (-7 i b c +a c \right ) x^{3}+7 b +i a \right ) b^{2} c^{2} x^{6}+4 \ln \left (\left (7 i b c +a c \right ) x^{3}+7 b -i a \right ) b^{2} c^{2} x^{6}+4 i b^{2} c \,x^{3} \ln \left (-i c \,x^{3}+1\right )+8 a b c \,x^{3}+4 i b \ln \left (-i c \,x^{3}+1\right ) a -b^{2} \ln \left (-i c \,x^{3}+1\right )^{2}+4 a^{2}}{24 x^{6}}\)	\(332\)

[In]

int((a+b*arctan(c*x^3))^2/x^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {b^{2} c^{2} x^{6} \log \left (c^{2} x^{6} + 1\right ) - 6 \, b^{2} c^{2} x^{6} \log \left (x\right ) + 2 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} + b^{2}\right )} \arctan \left (c x^{3}\right )^{2} + a^{2} + 2 \, {\left (a b c^{2} x^{6} + b^{2} c x^{3} + a b\right )} \arctan \left (c x^{3}\right )}{6 \, x^{6}} \]

[In]

integrate((a+b*arctan(c*x^3))^2/x^7,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (80) = 160\).

Time = 71.42 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=\begin {cases} - \frac {a^{2}}{6 x^{6}} - \frac {a b c^{2} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {a b c}{3 x^{3}} - \frac {a b \operatorname {atan}{\left (c x^{3} \right )}}{3 x^{6}} + \frac {b^{2} c^{3} \sqrt {- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{3} + b^{2} c^{2} \log {\left (x \right )} - \frac {b^{2} c^{2} \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{3} - \frac {b^{2} c^{2} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{3} - \frac {b^{2} c^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6} - \frac {b^{2} c \operatorname {atan}{\left (c x^{3} \right )}}{3 x^{3}} - \frac {b^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6 x^{6}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{6 x^{6}} & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*atan(c*x**3))**2/x**7,x)

[Out]

Piecewise((-a**2/(6*x**6) - a*b*c**2*atan(c*x**3)/3 - a*b*c/(3*x**3) - a*b*atan(c*x**3)/(3*x**6) + b**2*c**3*s
qrt(-1/c**2)*atan(c*x**3)/3 + b**2*c**2*log(x) - b**2*c**2*log(x - (-1/c**2)**(1/6))/3 - b**2*c**2*log(4*x**2
+ 4*x*(-1/c**2)**(1/6) + 4*(-1/c**2)**(1/3))/3 - b**2*c**2*atan(c*x**3)**2/6 - b**2*c*atan(c*x**3)/(3*x**3) -
b**2*atan(c*x**3)**2/(6*x**6), Ne(c, 0)), (-a**2/(6*x**6), True))

Maxima [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {1}{3} \, {\left ({\left (c \arctan \left (c x^{3}\right ) + \frac {1}{x^{3}}\right )} c + \frac {\arctan \left (c x^{3}\right )}{x^{6}}\right )} a b + \frac {1}{6} \, {\left ({\left (\arctan \left (c x^{3}\right )^{2} - \log \left (c^{2} x^{6} + 1\right ) + 6 \, \log \left (x\right )\right )} c^{2} - 2 \, {\left (c \arctan \left (c x^{3}\right ) + \frac {1}{x^{3}}\right )} c \arctan \left (c x^{3}\right )\right )} b^{2} - \frac {b^{2} \arctan \left (c x^{3}\right )^{2}}{6 \, x^{6}} - \frac {a^{2}}{6 \, x^{6}} \]

[In]

integrate((a+b*arctan(c*x^3))^2/x^7,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{2}}{x^{7}} \,d x } \]

[In]

integrate((a+b*arctan(c*x^3))^2/x^7,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x^3) + a)^2/x^7, x)

Mupad [B] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.75 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{x^7} \, dx=b^2\,c^2\,\ln \left (x\right )-\frac {b^2\,c^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6}-\frac {b^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6\,x^6}-\frac {b^2\,c^2\,\ln \left (c^2\,x^6+1\right )}{6}-\frac {a^2}{6\,x^6}-\frac {b^2\,c\,\mathrm {atan}\left (c\,x^3\right )}{3\,x^3}-\frac {a\,b\,c}{3\,x^3}-\frac {a\,b\,c^2\,\mathrm {atan}\left (\frac {a^2\,c\,x^3}{a^2+49\,b^2}+\frac {49\,b^2\,c\,x^3}{a^2+49\,b^2}\right )}{3}-\frac {a\,b\,\mathrm {atan}\left (c\,x^3\right )}{3\,x^6} \]

[In]

int((a + b*atan(c*x^3))^2/x^7,x)

[Out]

b^2*c^2*log(x) - (b^2*c^2*atan(c*x^3)^2)/6 - (b^2*atan(c*x^3)^2)/(6*x^6) - (b^2*c^2*log(c^2*x^6 + 1))/6 - a^2/
(6*x^6) - (b^2*c*atan(c*x^3))/(3*x^3) - (a*b*c)/(3*x^3) - (a*b*c^2*atan((a^2*c*x^3)/(a^2 + 49*b^2) + (49*b^2*c
*x^3)/(a^2 + 49*b^2)))/3 - (a*b*atan(c*x^3))/(3*x^6)