Integrand size = 20, antiderivative size = 169 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=-\frac {c \arctan (a x)^3}{x}+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,1-\frac {2}{1+i a x}\right ) \] Output:
-c*arctan(a*x)^3/x+a^2*c*x*arctan(a*x)^3+3*a*c*arctan(a*x)^2*ln(2/(1+I*a*x ))+3*a*c*arctan(a*x)^2*ln(2-2/(1-I*a*x))-3*I*a*c*arctan(a*x)*polylog(2,-1+ 2/(1-I*a*x))+3*I*a*c*arctan(a*x)*polylog(2,1-2/(1+I*a*x))+3/2*a*c*polylog( 3,-1+2/(1-I*a*x))+3/2*a*c*polylog(3,1-2/(1+I*a*x))
Time = 0.13 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.11 \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=-i a c \arctan (a x)^3+a^2 c x \arctan (a x)^3+3 a c \arctan (a x)^2 \log \left (1+e^{2 i \arctan (a x)}\right )-3 i a c \arctan (a x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )+a c \left (-\frac {i \pi ^3}{8}+i \arctan (a x)^3-\frac {\arctan (a x)^3}{a x}+3 \arctan (a x)^2 \log \left (1-e^{-2 i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arctan (a x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arctan (a x)}\right )\right )+\frac {3}{2} a c \operatorname {PolyLog}\left (3,-e^{2 i \arctan (a x)}\right ) \] Input:
Integrate[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]
Output:
(-I)*a*c*ArcTan[a*x]^3 + a^2*c*x*ArcTan[a*x]^3 + 3*a*c*ArcTan[a*x]^2*Log[1 + E^((2*I)*ArcTan[a*x])] - (3*I)*a*c*ArcTan[a*x]*PolyLog[2, -E^((2*I)*Arc Tan[a*x])] + a*c*((-1/8*I)*Pi^3 + I*ArcTan[a*x]^3 - ArcTan[a*x]^3/(a*x) + 3*ArcTan[a*x]^2*Log[1 - E^((-2*I)*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLo g[2, E^((-2*I)*ArcTan[a*x])] + (3*PolyLog[3, E^((-2*I)*ArcTan[a*x])])/2) + (3*a*c*PolyLog[3, -E^((2*I)*ArcTan[a*x])])/2
Time = 1.71 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.40, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5485, 5345, 5361, 5455, 5379, 5459, 5403, 5527, 5529, 7164}
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.
\(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle a^2 c \int \arctan (a x)^3dx+c \int \frac {\arctan (a x)^3}{x^2}dx\) |
\(\Big \downarrow \) 5345 |
\(\displaystyle a^2 c \left (x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx\right )+c \int \frac {\arctan (a x)^3}{x^2}dx\) |
\(\Big \downarrow \) 5361 |
\(\displaystyle a^2 c \left (x \arctan (a x)^3-3 a \int \frac {x \arctan (a x)^2}{a^2 x^2+1}dx\right )+c \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{x}\right )\) |
\(\Big \downarrow \) 5455 |
\(\displaystyle c \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{x}\right )+a^2 c \left (x \arctan (a x)^3-3 a \left (-\frac {\int \frac {\arctan (a x)^2}{i-a x}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )\) |
\(\Big \downarrow \) 5379 |
\(\displaystyle c \left (3 a \int \frac {\arctan (a x)^2}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^3}{x}\right )+a^2 c \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )\) |
\(\Big \downarrow \) 5459 |
\(\displaystyle a^2 c \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+c \left (-\frac {\arctan (a x)^3}{x}+3 a \left (i \int \frac {\arctan (a x)^2}{x (a x+i)}dx-\frac {1}{3} i \arctan (a x)^3\right )\right )\) |
\(\Big \downarrow \) 5403 |
\(\displaystyle a^2 c \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+c \left (-\frac {\arctan (a x)^3}{x}+3 a \left (i \left (2 i a \int \frac {\arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )\) |
\(\Big \downarrow \) 5527 |
\(\displaystyle c \left (-\frac {\arctan (a x)^3}{x}+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )+a^2 c \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \int \frac {\arctan (a x) \log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )\) |
\(\Big \downarrow \) 5529 |
\(\displaystyle a^2 c \left (x \arctan (a x)^3-3 a \left (-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}\right )}{a}-\frac {i \arctan (a x)^3}{3 a^2}\right )\right )+c \left (-\frac {\arctan (a x)^3}{x}+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{a^2 x^2+1}dx\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )\) |
\(\Big \downarrow \) 7164 |
\(\displaystyle a^2 c \left (x \arctan (a x)^3-3 a \left (-\frac {i \arctan (a x)^3}{3 a^2}-\frac {\frac {\arctan (a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a}-2 \left (-\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{i a x+1}\right )}{4 a}\right )}{a}\right )\right )+c \left (-\frac {\arctan (a x)^3}{x}+3 a \left (i \left (2 i a \left (\frac {i \arctan (a x) \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )}{2 a}-\frac {\operatorname {PolyLog}\left (3,\frac {2}{1-i a x}-1\right )}{4 a}\right )-i \arctan (a x)^2 \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{3} i \arctan (a x)^3\right )\right )\) |
Input:
Int[((c + a^2*c*x^2)*ArcTan[a*x]^3)/x^2,x]
Output:
c*(-(ArcTan[a*x]^3/x) + 3*a*((-1/3*I)*ArcTan[a*x]^3 + I*((-I)*ArcTan[a*x]^ 2*Log[2 - 2/(1 - I*a*x)] + (2*I)*a*(((I/2)*ArcTan[a*x]*PolyLog[2, -1 + 2/( 1 - I*a*x)])/a - PolyLog[3, -1 + 2/(1 - I*a*x)]/(4*a))))) + a^2*c*(x*ArcTa n[a*x]^3 - 3*a*(((-1/3*I)*ArcTan[a*x]^3)/a^2 - ((ArcTan[a*x]^2*Log[2/(1 + I*a*x)])/a - 2*(((-1/2*I)*ArcTan[a*x]*PolyLog[2, 1 - 2/(1 + I*a*x)])/a - P olyLog[3, 1 - 2/(1 + I*a*x)]/(4*a)))/a))
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_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( p/e) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 + c^2*x^2)) , x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0 ]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si mp[b*c*(p/d) Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2* d^2 + e^2, 0]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*e*(p + 1))), x] - Si mp[1/(c*d) Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcTan[c*x])^(p + 1)/(b*d*(p + 1))), x] + Si mp[I/d Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[I*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] - Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2* d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I + c*x)))^2, 0]
Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[(-I)*(a + b*ArcTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)) , x] + Simp[b*p*(I/2) Int[(a + b*ArcTan[c*x])^(p - 1)*(PolyLog[2, 1 - u]/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c ^2*d] && EqQ[(1 - u)^2 - (1 - 2*(I/(I - c*x)))^2, 0]
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /; !FalseQ[w]] /; FreeQ[n, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 152.19 (sec) , antiderivative size = 1653, normalized size of antiderivative = 9.78
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1653\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1654\) |
default | \(\text {Expression too large to display}\) | \(1654\) |
Input:
int((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x,method=_RETURNVERBOSE)
Output:
a^2*c*x*arctan(a*x)^3-c*arctan(a*x)^3/x-3*c*(-a*arctan(a*x)^2*ln(a*x)+a*ar ctan(a*x)^2*ln(a^2*x^2+1)-2*a*(arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2 ))-1/3*I*arctan(a*x)^3-1/2*arctan(a*x)^2*ln((1+I*a*x)^2/(a^2*x^2+1)-1)+1/2 *arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*arctan(a*x)*polylog(2,( 1+I*a*x)/(a^2*x^2+1)^(1/2))+polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/2*arc tan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*arctan(a*x)*polylog(2,-(1+I *a*x)/(a^2*x^2+1)^(1/2))+polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+1/4*(-I*P i*csgn(I*(1+I*a*x)/(a^2*x^2+1)^(1/2))^2*csgn(I*(1+I*a*x)^2/(a^2*x^2+1))-I* Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn((( 1+I*a*x)^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2-I*Pi*csgn(I*(1+I* a*x)^2/(a^2*x^2+1))^3-I*Pi*csgn(I*(1+I*a*x)^2/(a^2*x^2+1)/((1+I*a*x)^2/(a^ 2*x^2+1)+1)^2)^3-I*Pi*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x )^2/(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))^2+I*Pi*csgn(I*((1+I*a*x)^2 /(a^2*x^2+1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(((1+I*a*x)^2/(a^2*x^2+1) -1)/((1+I*a*x)^2/(a^2*x^2+1)+1))-2*I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1) )*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^ 2+1)-1))*csgn(I/((1+I*a*x)^2/(a^2*x^2+1)+1))*csgn(I*((1+I*a*x)^2/(a^2*x^2+ 1)-1)/((1+I*a*x)^2/(a^2*x^2+1)+1))-I*Pi*csgn(((1+I*a*x)^2/(a^2*x^2+1)-1)/( (1+I*a*x)^2/(a^2*x^2+1)+1))^2+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1))^2*c sgn(I*((1+I*a*x)^2/(a^2*x^2+1)+1)^2)+I*Pi*csgn(I*((1+I*a*x)^2/(a^2*x^2+...
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \] Input:
integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x, algorithm="fricas")
Output:
integral((a^2*c*x^2 + c)*arctan(a*x)^3/x^2, x)
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=c \left (\int a^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx\right ) \] Input:
integrate((a**2*c*x**2+c)*atan(a*x)**3/x**2,x)
Output:
c*(Integral(a**2*atan(a*x)**3, x) + Integral(atan(a*x)**3/x**2, x))
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \] Input:
integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x, algorithm="maxima")
Output:
1/64*(8*(a^2*c*x^2 - c)*arctan(a*x)^3 - 6*(a^2*c*x^2 - c)*arctan(a*x)*log( a^2*x^2 + 1)^2 + (28*a*c*arctan(a*x)^4 + 1792*a^4*c*integrate(1/32*x^4*arc tan(a*x)^3/(a^2*x^4 + x^2), x) + 192*a^4*c*integrate(1/32*x^4*arctan(a*x)* log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 768*a^4*c*integrate(1/32*x^4*arct an(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 768*a^3*c*integrate(1/32*x^ 3*arctan(a*x)^2/(a^2*x^4 + x^2), x) + a*c*log(a^2*x^2 + 1)^3 + 384*a^2*c*i ntegrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) - 768 *a^2*c*integrate(1/32*x^2*arctan(a*x)*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) + 768*a*c*integrate(1/32*x*arctan(a*x)^2/(a^2*x^4 + x^2), x) - 192*a*c*in tegrate(1/32*x*log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x) + 1792*c*integrate(1 /32*arctan(a*x)^3/(a^2*x^4 + x^2), x) + 192*c*integrate(1/32*arctan(a*x)*l og(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x)/x
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{2}} \,d x } \] Input:
integrate((a^2*c*x^2+c)*arctan(a*x)^3/x^2,x, algorithm="giac")
Output:
integrate((a^2*c*x^2 + c)*arctan(a*x)^3/x^2, x)
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right )}{x^2} \,d x \] Input:
int((atan(a*x)^3*(c + a^2*c*x^2))/x^2,x)
Output:
int((atan(a*x)^3*(c + a^2*c*x^2))/x^2, x)
\[ \int \frac {\left (c+a^2 c x^2\right ) \arctan (a x)^3}{x^2} \, dx=\frac {c \left (-\mathit {atan} \left (a x \right )^{3}+\left (\int \mathit {atan} \left (a x \right )^{3}d x \right ) a^{2} x +3 \left (\int \frac {\mathit {atan} \left (a x \right )^{2}}{a^{2} x^{3}+x}d x \right ) a x \right )}{x} \] Input:
int((a^2*c*x^2+c)*atan(a*x)^3/x^2,x)
Output:
(c*( - atan(a*x)**3 + int(atan(a*x)**3,x)*a**2*x + 3*int(atan(a*x)**2/(a** 2*x**3 + x),x)*a*x))/x