3.67 \(\int \frac{a+b \tan ^{-1}(c x^2)}{x^7} \, dx\)

Optimal. Leaf size=55 \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} b c^3 \log \left (c^2 x^4+1\right )-\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]

[Out]

-(b*c)/(12*x^4) - (a + b*ArcTan[c*x^2])/(6*x^6) - (b*c^3*Log[x])/3 + (b*c^3*Log[1 + c^2*x^4])/12

________________________________________________________________________________________

Rubi [A]  time = 0.0322249, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5033, 266, 44} \[ -\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} b c^3 \log \left (c^2 x^4+1\right )-\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x^2])/x^7,x]

[Out]

-(b*c)/(12*x^4) - (a + b*ArcTan[c*x^2])/(6*x^6) - (b*c^3*Log[x])/3 + (b*c^3*Log[1 + c^2*x^4])/12

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 266

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^2\right )}{x^7} \, dx &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{3} (b c) \int \frac{1}{x^5 \left (1+c^2 x^4\right )} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^4\right )\\ &=-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}+\frac{1}{12} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c^2}{x}+\frac{c^4}{1+c^2 x}\right ) \, dx,x,x^4\right )\\ &=-\frac{b c}{12 x^4}-\frac{a+b \tan ^{-1}\left (c x^2\right )}{6 x^6}-\frac{1}{3} b c^3 \log (x)+\frac{1}{12} b c^3 \log \left (1+c^2 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0137974, size = 60, normalized size = 1.09 \[ -\frac{a}{6 x^6}+\frac{1}{12} b c^3 \log \left (c^2 x^4+1\right )-\frac{1}{3} b c^3 \log (x)-\frac{b c}{12 x^4}-\frac{b \tan ^{-1}\left (c x^2\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a/(6*x^6) - (b*c)/(12*x^4) - (b*ArcTan[c*x^2])/(6*x^6) - (b*c^3*Log[x])/3 + (b*c^3*Log[1 + c^2*x^4])/12

________________________________________________________________________________________

Maple [A]  time = 0.029, size = 51, normalized size = 0.9 \begin{align*} -{\frac{a}{6\,{x}^{6}}}-{\frac{b\arctan \left ( c{x}^{2} \right ) }{6\,{x}^{6}}}+{\frac{b{c}^{3}\ln \left ({c}^{2}{x}^{4}+1 \right ) }{12}}-{\frac{bc}{12\,{x}^{4}}}-{\frac{b{c}^{3}\ln \left ( x \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00167, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{12} \,{\left ({\left (c^{2} \log \left (c^{2} x^{4} + 1\right ) - c^{2} \log \left (x^{4}\right ) - \frac{1}{x^{4}}\right )} c - \frac{2 \, \arctan \left (c x^{2}\right )}{x^{6}}\right )} b - \frac{a}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.82218, size = 130, normalized size = 2.36 \begin{align*} \frac{b c^{3} x^{6} \log \left (c^{2} x^{4} + 1\right ) - 4 \, b c^{3} x^{6} \log \left (x\right ) - b c x^{2} - 2 \, b \arctan \left (c x^{2}\right ) - 2 \, a}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 140.139, size = 784, normalized size = 14.25 \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**7,x)

[Out]

Piecewise((-(a - oo*I*b)/(6*x**6), Eq(c, -I/x**2)), (-(a + oo*I*b)/(6*x**6), Eq(c, I/x**2)), (-a/(6*x**6), Eq(
c, 0)), (2*I*a*c**6*x**4*(c**(-2))**(7/2)/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 12*I*c**4*x**6*(c**(-2))**(7/2)
) + 2*I*a*c**4*(c**(-2))**(7/2)/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 12*I*c**4*x**6*(c**(-2))**(7/2)) + 4*I*b*
c**9*x**10*(c**(-2))**(7/2)*log(x)/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 12*I*c**4*x**6*(c**(-2))**(7/2)) - 2*I
*b*c**9*x**10*(c**(-2))**(7/2)*log(x**2 + I*sqrt(c**(-2)))/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 12*I*c**4*x**6
*(c**(-2))**(7/2)) - I*b*c**9*x**10*(c**(-2))**(7/2)/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 12*I*c**4*x**6*(c**(
-2))**(7/2)) + 4*I*b*c**7*x**6*(c**(-2))**(7/2)*log(x)/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 12*I*c**4*x**6*(c*
*(-2))**(7/2)) - 2*I*b*c**7*x**6*(c**(-2))**(7/2)*log(x**2 + I*sqrt(c**(-2)))/(-12*I*c**6*x**10*(c**(-2))**(7/
2) - 12*I*c**4*x**6*(c**(-2))**(7/2)) + 2*I*b*c**6*x**4*(c**(-2))**(7/2)*atan(c*x**2)/(-12*I*c**6*x**10*(c**(-
2))**(7/2) - 12*I*c**4*x**6*(c**(-2))**(7/2)) + I*b*c**5*x**2*(c**(-2))**(7/2)/(-12*I*c**6*x**10*(c**(-2))**(7
/2) - 12*I*c**4*x**6*(c**(-2))**(7/2)) + 2*I*b*c**4*(c**(-2))**(7/2)*atan(c*x**2)/(-12*I*c**6*x**10*(c**(-2))*
*(7/2) - 12*I*c**4*x**6*(c**(-2))**(7/2)) + 2*b*c**2*x**10*atan(c*x**2)/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 1
2*I*c**4*x**6*(c**(-2))**(7/2)) + 2*b*x**6*atan(c*x**2)/(-12*I*c**6*x**10*(c**(-2))**(7/2) - 12*I*c**4*x**6*(c
**(-2))**(7/2)), True))

________________________________________________________________________________________

Giac [A]  time = 1.14157, size = 93, normalized size = 1.69 \begin{align*} \frac{b c^{7} x^{6} \log \left (c^{2} x^{4} + 1\right ) - 2 \, b c^{7} x^{6} \log \left (c x^{2}\right ) - b c^{5} x^{2} - 2 \, b c^{4} \arctan \left (c x^{2}\right ) - 2 \, a c^{4}}{12 \, c^{4} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b*c^7*x^6*log(c^2*x^4 + 1) - 2*b*c^7*x^6*log(c*x^2) - b*c^5*x^2 - 2*b*c^4*arctan(c*x^2) - 2*a*c^4)/(c^4*
x^6)