Integrand size = 17, antiderivative size = 161 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {16}{35} c^3 x \arctan (a x)+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)-\frac {8 c^3 \log \left (1+a^2 x^2\right )}{35 a} \]
-4/35*c^3*(a^2*x^2+1)/a-3/70*c^3*(a^2*x^2+1)^2/a-1/42*c^3*(a^2*x^2+1)^3/a+ 16/35*c^3*x*arctan(a*x)+8/35*c^3*x*(a^2*x^2+1)*arctan(a*x)+6/35*c^3*x*(a^2 *x^2+1)^2*arctan(a*x)+1/7*c^3*x*(a^2*x^2+1)^3*arctan(a*x)-8/35*c^3*ln(a^2* x^2+1)/a
Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^3 \left (-a^2 x^2 \left (57+24 a^2 x^2+5 a^4 x^4\right )+6 a x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right ) \arctan (a x)-48 \log \left (1+a^2 x^2\right )\right )}{210 a} \]
(c^3*(-(a^2*x^2*(57 + 24*a^2*x^2 + 5*a^4*x^4)) + 6*a*x*(35 + 35*a^2*x^2 + 21*a^4*x^4 + 5*a^6*x^6)*ArcTan[a*x] - 48*Log[1 + a^2*x^2]))/(210*a)
Time = 0.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5413, 27, 5413, 5413, 5345, 240}
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.
\(\displaystyle \int \arctan (a x) \left (a^2 c x^2+c\right )^3 \, dx\) |
\(\Big \downarrow \) 5413 |
\(\displaystyle \frac {6}{7} c \int c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {6}{7} c^3 \int \left (a^2 x^2+1\right )^2 \arctan (a x)dx+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}\) |
\(\Big \downarrow \) 5413 |
\(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \int \left (a^2 x^2+1\right ) \arctan (a x)dx+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}\) |
\(\Big \downarrow \) 5413 |
\(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \int \arctan (a x)dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}\) |
\(\Big \downarrow \) 5345 |
\(\displaystyle \frac {6}{7} c^3 \left (\frac {4}{5} \left (\frac {2}{3} \left (x \arctan (a x)-a \int \frac {x}{a^2 x^2+1}dx\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)-\frac {a^2 x^2+1}{6 a}\right )+\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )+\frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)+\frac {6}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}\) |
-1/42*(c^3*(1 + a^2*x^2)^3)/a + (c^3*x*(1 + a^2*x^2)^3*ArcTan[a*x])/7 + (6 *c^3*(-1/20*(1 + a^2*x^2)^2/a + (x*(1 + a^2*x^2)^2*ArcTan[a*x])/5 + (4*(-1 /6*(1 + a^2*x^2)/a + (x*(1 + a^2*x^2)*ArcTan[a*x])/3 + (2*(x*ArcTan[a*x] - Log[1 + a^2*x^2]/(2*a)))/3))/5))/7
3.2.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0]
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61
method | result | size |
parts | \(\frac {c^{3} \arctan \left (a x \right ) a^{6} x^{7}}{7}+\frac {3 c^{3} \arctan \left (a x \right ) a^{4} x^{5}}{5}+c^{3} \arctan \left (a x \right ) a^{2} x^{3}+c^{3} x \arctan \left (a x \right )-\frac {c^{3} a \left (\frac {5 a^{4} x^{6}}{6}+4 a^{2} x^{4}+\frac {19 x^{2}}{2}+\frac {8 \ln \left (a^{2} x^{2}+1\right )}{a^{2}}\right )}{35}\) | \(98\) |
derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )+a \,c^{3} x \arctan \left (a x \right )-\frac {c^{3} \left (\frac {5 a^{6} x^{6}}{6}+4 a^{4} x^{4}+\frac {19 a^{2} x^{2}}{2}+8 \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\) | \(102\) |
default | \(\frac {\frac {c^{3} \arctan \left (a x \right ) a^{7} x^{7}}{7}+\frac {3 a^{5} c^{3} x^{5} \arctan \left (a x \right )}{5}+a^{3} c^{3} x^{3} \arctan \left (a x \right )+a \,c^{3} x \arctan \left (a x \right )-\frac {c^{3} \left (\frac {5 a^{6} x^{6}}{6}+4 a^{4} x^{4}+\frac {19 a^{2} x^{2}}{2}+8 \ln \left (a^{2} x^{2}+1\right )\right )}{35}}{a}\) | \(102\) |
parallelrisch | \(-\frac {-30 c^{3} \arctan \left (a x \right ) a^{7} x^{7}+5 a^{6} c^{3} x^{6}-126 a^{5} c^{3} x^{5} \arctan \left (a x \right )+24 a^{4} c^{3} x^{4}-210 a^{3} c^{3} x^{3} \arctan \left (a x \right )+57 a^{2} c^{3} x^{2}-210 a \,c^{3} x \arctan \left (a x \right )+48 c^{3} \ln \left (a^{2} x^{2}+1\right )}{210 a}\) | \(111\) |
risch | \(-\frac {i c^{3} x \left (5 a^{6} x^{6}+21 a^{4} x^{4}+35 a^{2} x^{2}+35\right ) \ln \left (i a x +1\right )}{70}+\frac {i c^{3} a^{6} x^{7} \ln \left (-i a x +1\right )}{14}-\frac {a^{5} c^{3} x^{6}}{42}+\frac {3 i c^{3} a^{4} x^{5} \ln \left (-i a x +1\right )}{10}-\frac {4 a^{3} c^{3} x^{4}}{35}+\frac {i c^{3} a^{2} x^{3} \ln \left (-i a x +1\right )}{2}-\frac {19 a \,c^{3} x^{2}}{70}+\frac {i c^{3} x \ln \left (-i a x +1\right )}{2}-\frac {8 c^{3} \ln \left (-a^{2} x^{2}-1\right )}{35 a}\) | \(168\) |
meijerg | \(\frac {c^{3} \left (-\frac {a^{2} x^{2} \left (4 a^{4} x^{4}-6 a^{2} x^{2}+12\right )}{42}+\frac {4 a^{8} x^{8} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{7}\right )}{4 a}+\frac {3 c^{3} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{4 a}+\frac {3 c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4 a}+\frac {c^{3} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4 a}\) | \(246\) |
1/7*c^3*arctan(a*x)*a^6*x^7+3/5*c^3*arctan(a*x)*a^4*x^5+c^3*arctan(a*x)*a^ 2*x^3+c^3*x*arctan(a*x)-1/35*c^3*a*(5/6*a^4*x^6+4*a^2*x^4+19/2*x^2+8/a^2*l n(a^2*x^2+1))
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.63 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {5 \, a^{6} c^{3} x^{6} + 24 \, a^{4} c^{3} x^{4} + 57 \, a^{2} c^{3} x^{2} + 48 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 6 \, {\left (5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x\right )} \arctan \left (a x\right )}{210 \, a} \]
-1/210*(5*a^6*c^3*x^6 + 24*a^4*c^3*x^4 + 57*a^2*c^3*x^2 + 48*c^3*log(a^2*x ^2 + 1) - 6*(5*a^7*c^3*x^7 + 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 + 35*a*c^3*x) *arctan(a*x))/a
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.73 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\begin {cases} \frac {a^{6} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{7} - \frac {a^{5} c^{3} x^{6}}{42} + \frac {3 a^{4} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {4 a^{3} c^{3} x^{4}}{35} + a^{2} c^{3} x^{3} \operatorname {atan}{\left (a x \right )} - \frac {19 a c^{3} x^{2}}{70} + c^{3} x \operatorname {atan}{\left (a x \right )} - \frac {8 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{35 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**6*c**3*x**7*atan(a*x)/7 - a**5*c**3*x**6/42 + 3*a**4*c**3*x* *5*atan(a*x)/5 - 4*a**3*c**3*x**4/35 + a**2*c**3*x**3*atan(a*x) - 19*a*c** 3*x**2/70 + c**3*x*atan(a*x) - 8*c**3*log(x**2 + a**(-2))/(35*a), Ne(a, 0) ), (0, True))
Time = 0.19 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.61 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {1}{210} \, {\left (5 \, a^{4} c^{3} x^{6} + 24 \, a^{2} c^{3} x^{4} + 57 \, c^{3} x^{2} + \frac {48 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a + \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x\right )} \arctan \left (a x\right ) \]
-1/210*(5*a^4*c^3*x^6 + 24*a^2*c^3*x^4 + 57*c^3*x^2 + 48*c^3*log(a^2*x^2 + 1)/a^2)*a + 1/35*(5*a^6*c^3*x^7 + 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 + 35*c^ 3*x)*arctan(a*x)
\[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right ) \,d x } \]
Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.55 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {c^3\,\left (48\,\ln \left (a^2\,x^2+1\right )+57\,a^2\,x^2+24\,a^4\,x^4+5\,a^6\,x^6-210\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )-126\,a^5\,x^5\,\mathrm {atan}\left (a\,x\right )-30\,a^7\,x^7\,\mathrm {atan}\left (a\,x\right )-210\,a\,x\,\mathrm {atan}\left (a\,x\right )\right )}{210\,a} \]