\(\int (c+a^2 c x^2)^3 \arctan (a x) \, dx\) [168]

Optimal result

Integrand size = 17, antiderivative size = 161 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {4 c^3 \left (1+a^2 x^2\right )}{35 a}-\frac {3 c^3 \left (1+a^2 x^2\right )^2}{70 a}-\frac {c^3 \left (1+a^2 x^2\right )^3}{42 a}+\frac {16}{35} c^3 x \arctan (a x)+\frac {8}{35} c^3 x \left (1+a^2 x^2\right ) \arctan (a x)+\frac {6}{35} c^3 x \left (1+a^2 x^2\right )^2 \arctan (a x)+\frac {1}{7} c^3 x \left (1+a^2 x^2\right )^3 \arctan (a x)-\frac {8 c^3 \log \left (1+a^2 x^2\right )}{35 a} \] Output:

-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
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.52 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^3 \left (-a^2 x^2 \left (57+24 a^2 x^2+5 a^4 x^4\right )+6 a x \left (35+35 a^2 x^2+21 a^4 x^4+5 a^6 x^6\right ) \arctan (a x)-48 \log \left (1+a^2 x^2\right )\right )}{210 a} \] Input:

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

Output:

(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)
 

Rubi [A] (verified)

Time = 0.50 (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

\(\displaystyle \frac {1}{7} c^3 x \left (a^2 x^2+1\right )^3 \arctan (a x)+\frac {6}{7} c^3 \left (\frac {1}{5} x \left (a^2 x^2+1\right )^2 \arctan (a x)+\frac {4}{5} \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \arctan (a x)+\frac {2}{3} \left (x \arctan (a x)-\frac {\log \left (a^2 x^2+1\right )}{2 a}\right )-\frac {a^2 x^2+1}{6 a}\right )-\frac {\left (a^2 x^2+1\right )^2}{20 a}\right )-\frac {c^3 \left (a^2 x^2+1\right )^3}{42 a}\)

Input:

Int[(c + a^2*c*x^2)^3*ArcTan[a*x],x]
 

Output:

-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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 5345

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])
 

rule 5413

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]
 

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.61

method	result	size
parts	\(\frac {c^{3} \arctan \left (a x \right ) a^{6} x^{7}}{7}+\frac {3 c^{3} \arctan \left (a x \right ) a^{4} x^{5}}{5}+c^{3} \arctan \left (a x \right ) a^{2} x^{3}+c^{3} x \arctan \left (a x \right )-\frac {a \,c^{3} \left (\frac {5 a^{4} x^{6}}{6}+4 a^{2} x^{4}+\frac {19 x^{2}}{2}+\frac {8 \ln \left (a^{2} x^{2}+1\right )}{a^{2}}\right )}{35}\)	\(98\)
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	\(\frac {c^{3} \left (-\frac {a^{2} x^{2} \left (4 a^{4} x^{4}-6 a^{2} x^{2}+12\right )}{42}+\frac {4 a^{8} x^{8} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{7 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{7}\right )}{4 a}+\frac {3 c^{3} \left (\frac {a^{2} x^{2} \left (-3 a^{2} x^{2}+6\right )}{15}+\frac {4 a^{6} x^{6} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{5 \sqrt {a^{2} x^{2}}}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{5}\right )}{4 a}+\frac {3 c^{3} \left (-\frac {2 a^{2} x^{2}}{3}+\frac {4 a^{4} x^{4} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{3 \sqrt {a^{2} x^{2}}}+\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}\right )}{4 a}+\frac {c^{3} \left (\frac {4 a^{2} x^{2} \arctan \left (\sqrt {a^{2} x^{2}}\right )}{\sqrt {a^{2} x^{2}}}-2 \ln \left (a^{2} x^{2}+1\right )\right )}{4 a}\)	\(246\)

Input:

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

Output:

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*a*c^3*(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))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.63 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {5 \, a^{6} c^{3} x^{6} + 24 \, a^{4} c^{3} x^{4} + 57 \, a^{2} c^{3} x^{2} + 48 \, c^{3} \log \left (a^{2} x^{2} + 1\right ) - 6 \, {\left (5 \, a^{7} c^{3} x^{7} + 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} + 35 \, a c^{3} x\right )} \arctan \left (a x\right )}{210 \, a} \] Input:

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

Output:

-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
 

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.73 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\begin {cases} \frac {a^{6} c^{3} x^{7} \operatorname {atan}{\left (a x \right )}}{7} - \frac {a^{5} c^{3} x^{6}}{42} + \frac {3 a^{4} c^{3} x^{5} \operatorname {atan}{\left (a x \right )}}{5} - \frac {4 a^{3} c^{3} x^{4}}{35} + a^{2} c^{3} x^{3} \operatorname {atan}{\left (a x \right )} - \frac {19 a c^{3} x^{2}}{70} + c^{3} x \operatorname {atan}{\left (a x \right )} - \frac {8 c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{35 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

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))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.61 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {1}{210} \, {\left (5 \, a^{4} c^{3} x^{6} + 24 \, a^{2} c^{3} x^{4} + 57 \, c^{3} x^{2} + \frac {48 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{a^{2}}\right )} a + \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x\right )} \arctan \left (a x\right ) \] Input:

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

Output:

-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)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.65 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {8 \, c^{3} \log \left (a^{2} x^{2} + 1\right )}{35 \, a} + \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} + 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} + 35 \, c^{3} x\right )} \arctan \left (a x\right ) - \frac {5 \, a^{11} c^{3} x^{6} + 24 \, a^{9} c^{3} x^{4} + 57 \, a^{7} c^{3} x^{2}}{210 \, a^{6}} \] Input:

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

Output:

-8/35*c^3*log(a^2*x^2 + 1)/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) - 1/210*(5*a^11*c^3*x^6 + 24*a^9*c^3*x^ 
4 + 57*a^7*c^3*x^2)/a^6
 

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.55 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=-\frac {c^3\,\left (48\,\ln \left (a^2\,x^2+1\right )+57\,a^2\,x^2+24\,a^4\,x^4+5\,a^6\,x^6-210\,a^3\,x^3\,\mathrm {atan}\left (a\,x\right )-126\,a^5\,x^5\,\mathrm {atan}\left (a\,x\right )-30\,a^7\,x^7\,\mathrm {atan}\left (a\,x\right )-210\,a\,x\,\mathrm {atan}\left (a\,x\right )\right )}{210\,a} \] Input:

int(atan(a*x)*(c + a^2*c*x^2)^3,x)
 

Output:

-(c^3*(48*log(a^2*x^2 + 1) + 57*a^2*x^2 + 24*a^4*x^4 + 5*a^6*x^6 - 210*a^3 
*x^3*atan(a*x) - 126*a^5*x^5*atan(a*x) - 30*a^7*x^7*atan(a*x) - 210*a*x*at 
an(a*x)))/(210*a)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.55 \[ \int \left (c+a^2 c x^2\right )^3 \arctan (a x) \, dx=\frac {c^{3} \left (30 \mathit {atan} \left (a x \right ) a^{7} x^{7}+126 \mathit {atan} \left (a x \right ) a^{5} x^{5}+210 \mathit {atan} \left (a x \right ) a^{3} x^{3}+210 \mathit {atan} \left (a x \right ) a x -48 \,\mathrm {log}\left (a^{2} x^{2}+1\right )-5 a^{6} x^{6}-24 a^{4} x^{4}-57 a^{2} x^{2}\right )}{210 a} \] Input:

int((a^2*c*x^2+c)^3*atan(a*x),x)
 

Output:

(c**3*(30*atan(a*x)*a**7*x**7 + 126*atan(a*x)*a**5*x**5 + 210*atan(a*x)*a* 
*3*x**3 + 210*atan(a*x)*a*x - 48*log(a**2*x**2 + 1) - 5*a**6*x**6 - 24*a** 
4*x**4 - 57*a**2*x**2))/(210*a)