Integrand size = 22, antiderivative size = 50 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=-\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a^5 c^3}+\frac {\operatorname {CosIntegral}(4 \arctan (a x))}{8 a^5 c^3}+\frac {3 \log (\arctan (a x))}{8 a^5 c^3} \] Output:
-1/2*Ci(2*arctan(a*x))/a^5/c^3+1/8*Ci(4*arctan(a*x))/a^5/c^3+3/8*ln(arctan (a*x))/a^5/c^3
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {-4 \operatorname {CosIntegral}(2 \arctan (a x))+\operatorname {CosIntegral}(4 \arctan (a x))+3 \log (\arctan (a x))}{8 a^5 c^3} \] Input:
Integrate[x^4/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]
Output:
(-4*CosIntegral[2*ArcTan[a*x]] + CosIntegral[4*ArcTan[a*x]] + 3*Log[ArcTan [a*x]])/(8*a^5*c^3)
Time = 0.35 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5505, 3042, 3793, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^4}{\arctan (a x) \left (a^2 c x^2+c\right )^3} \, dx\) |
\(\Big \downarrow \) 5505 |
\(\displaystyle \frac {\int \frac {a^4 x^4}{\left (a^2 x^2+1\right )^2 \arctan (a x)}d\arctan (a x)}{a^5 c^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin (\arctan (a x))^4}{\arctan (a x)}d\arctan (a x)}{a^5 c^3}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {\int \left (-\frac {\cos (2 \arctan (a x))}{2 \arctan (a x)}+\frac {\cos (4 \arctan (a x))}{8 \arctan (a x)}+\frac {3}{8 \arctan (a x)}\right )d\arctan (a x)}{a^5 c^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^5 c^3}\) |
Input:
Int[x^4/((c + a^2*c*x^2)^3*ArcTan[a*x]),x]
Output:
(-1/2*CosIntegral[2*ArcTan[a*x]] + CosIntegral[4*ArcTan[a*x]]/8 + (3*Log[A rcTan[a*x]])/8)/(a^5*c^3)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1) Subst[Int[(a + b*x)^p*(Sin[x]^m/ Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p }, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q ] || GtQ[d, 0])
Time = 5.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {3 \ln \left (\arctan \left (a x \right )\right )-4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right )+\operatorname {Ci}\left (4 \arctan \left (a x \right )\right )}{8 a^{5} c^{3}}\) | \(33\) |
default | \(\frac {3 \ln \left (\arctan \left (a x \right )\right )-4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right )+\operatorname {Ci}\left (4 \arctan \left (a x \right )\right )}{8 a^{5} c^{3}}\) | \(33\) |
Input:
int(x^4/(a^2*c*x^2+c)^3/arctan(a*x),x,method=_RETURNVERBOSE)
Output:
1/8/a^5*(3*ln(arctan(a*x))-4*Ci(2*arctan(a*x))+Ci(4*arctan(a*x)))/c^3
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.48 \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {6 \, \log \left (\arctan \left (a x\right )\right ) + \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 4 \, \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right )}{16 \, a^{5} c^{3}} \] Input:
integrate(x^4/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="fricas")
Output:
1/16*(6*log(arctan(a*x)) + log_integral((a^4*x^4 + 4*I*a^3*x^3 - 6*a^2*x^2 - 4*I*a*x + 1)/(a^4*x^4 + 2*a^2*x^2 + 1)) + log_integral((a^4*x^4 - 4*I*a ^3*x^3 - 6*a^2*x^2 + 4*I*a*x + 1)/(a^4*x^4 + 2*a^2*x^2 + 1)) - 4*log_integ ral(-(a^2*x^2 + 2*I*a*x - 1)/(a^2*x^2 + 1)) - 4*log_integral(-(a^2*x^2 - 2 *I*a*x - 1)/(a^2*x^2 + 1)))/(a^5*c^3)
\[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {\int \frac {x^{4}}{a^{6} x^{6} \operatorname {atan}{\left (a x \right )} + 3 a^{4} x^{4} \operatorname {atan}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atan}{\left (a x \right )} + \operatorname {atan}{\left (a x \right )}}\, dx}{c^{3}} \] Input:
integrate(x**4/(a**2*c*x**2+c)**3/atan(a*x),x)
Output:
Integral(x**4/(a**6*x**6*atan(a*x) + 3*a**4*x**4*atan(a*x) + 3*a**2*x**2*a tan(a*x) + atan(a*x)), x)/c**3
\[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )} \,d x } \] Input:
integrate(x^4/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="maxima")
Output:
integrate(x^4/((a^2*c*x^2 + c)^3*arctan(a*x)), x)
\[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int { \frac {x^{4}}{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )} \,d x } \] Input:
integrate(x^4/(a^2*c*x^2+c)^3/arctan(a*x),x, algorithm="giac")
Output:
integrate(x^4/((a^2*c*x^2 + c)^3*arctan(a*x)), x)
Timed out. \[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\int \frac {x^4}{\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^3} \,d x \] Input:
int(x^4/(atan(a*x)*(c + a^2*c*x^2)^3),x)
Output:
int(x^4/(atan(a*x)*(c + a^2*c*x^2)^3), x)
\[ \int \frac {x^4}{\left (c+a^2 c x^2\right )^3 \arctan (a x)} \, dx=\frac {-2 \left (\int \frac {x^{2}}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{3}-\left (\int \frac {1}{\mathit {atan} \left (a x \right ) a^{6} x^{6}+3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a +\mathrm {log}\left (\mathit {atan} \left (a x \right )\right )}{a^{5} c^{3}} \] Input:
int(x^4/(a^2*c*x^2+c)^3/atan(a*x),x)
Output:
( - 2*int(x**2/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x)* a**2*x**2 + atan(a*x)),x)*a**3 - int(1/(atan(a*x)*a**6*x**6 + 3*atan(a*x)* a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a + log(atan(a*x)))/(a** 5*c**3)