3.102 \(\int \frac {a+b \tan ^{-1}(c x^3)}{x^7} \, dx\)

Optimal. Leaf size=41 \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{6 x^6}-\frac {1}{6} b c^2 \tan ^{-1}\left (c x^3\right )-\frac {b c}{6 x^3} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5033, 275, 325, 203} \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{6 x^6}-\frac {1}{6} b c^2 \tan ^{-1}\left (c x^3\right )-\frac {b c}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^3])/x^7,x]

[Out]

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

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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^7} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{6 x^6}+\frac {1}{2} (b c) \int \frac {1}{x^4 \left (1+c^2 x^6\right )} \, dx\\ &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{6 x^6}+\frac {1}{6} (b c) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx,x,x^3\right )\\ &=-\frac {b c}{6 x^3}-\frac {a+b \tan ^{-1}\left (c x^3\right )}{6 x^6}-\frac {1}{6} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,x^3\right )\\ &=-\frac {b c}{6 x^3}-\frac {1}{6} b c^2 \tan ^{-1}\left (c x^3\right )-\frac {a+b \tan ^{-1}\left (c x^3\right )}{6 x^6}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 48, normalized size = 1.17 \[ -\frac {a}{6 x^6}-\frac {b c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^6\right )}{6 x^3}-\frac {b \tan ^{-1}\left (c x^3\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*a/x^6 - (b*ArcTan[c*x^3])/(6*x^6) - (b*c*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^6)])/(6*x^3)

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fricas [A]  time = 0.41, size = 30, normalized size = 0.73 \[ -\frac {b c x^{3} + {\left (b c^{2} x^{6} + b\right )} \arctan \left (c x^{3}\right ) + a}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.17, size = 74, normalized size = 1.80 \[ \frac {b c^{5} i x^{6} \log \left (c i x^{3} + 1\right ) - b c^{5} i x^{6} \log \left (-c i x^{3} + 1\right ) - 2 \, b c^{4} x^{3} - 2 \, b c^{3} \arctan \left (c x^{3}\right ) - 2 \, a c^{3}}{12 \, c^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 39, normalized size = 0.95 \[ -\frac {a}{6 x^{6}}-\frac {b \arctan \left (c \,x^{3}\right )}{6 x^{6}}-\frac {b \,c^{2} \arctan \left (c \,x^{3}\right )}{6}-\frac {b c}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 35, normalized size = 0.85 \[ -\frac {1}{6} \, {\left ({\left (c \arctan \left (c x^{3}\right ) + \frac {1}{x^{3}}\right )} c + \frac {\arctan \left (c x^{3}\right )}{x^{6}}\right )} b - \frac {a}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.39, size = 41, normalized size = 1.00 \[ -\frac {\frac {b\,c\,x^3}{3}+\frac {a}{3}}{2\,x^6}-\frac {b\,c^2\,\mathrm {atan}\left (c\,x^3\right )}{6}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{6\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x^3))/x^7,x)

[Out]

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

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sympy [A]  time = 88.00, size = 42, normalized size = 1.02 \[ - \frac {a}{6 x^{6}} - \frac {b c^{2} \operatorname {atan}{\left (c x^{3} \right )}}{6} - \frac {b c}{6 x^{3}} - \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/(6*x**6) - b*c**2*atan(c*x**3)/6 - b*c/(6*x**3) - b*atan(c*x**3)/(6*x**6)

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