Integrand size = 14, antiderivative size = 170 \[ \int x^4 (a+b \arctan (c x))^2 \, dx=-\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}+\frac {3 b^2 \arctan (c x)}{10 c^5}+\frac {b x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b x^4 (a+b \arctan (c x))}{10 c}+\frac {i (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{5} x^5 (a+b \arctan (c x))^2+\frac {2 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5} \] Output:
-3/10*b^2*x/c^4+1/30*b^2*x^3/c^2+3/10*b^2*arctan(c*x)/c^5+1/5*b*x^2*(a+b*a rctan(c*x))/c^3-1/10*b*x^4*(a+b*arctan(c*x))/c+1/5*I*(a+b*arctan(c*x))^2/c ^5+1/5*x^5*(a+b*arctan(c*x))^2+2/5*b*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c^5 +1/5*I*b^2*polylog(2,1-2/(1+I*c*x))/c^5
Time = 0.35 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int x^4 (a+b \arctan (c x))^2 \, dx=\frac {9 a b-9 b^2 c x+6 a b c^2 x^2+b^2 c^3 x^3-3 a b c^4 x^4+6 a^2 c^5 x^5+6 b^2 \left (-i+c^5 x^5\right ) \arctan (c x)^2-3 b \arctan (c x) \left (-4 a c^5 x^5+b \left (-3-2 c^2 x^2+c^4 x^4\right )-4 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-6 a b \log \left (1+c^2 x^2\right )-6 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{30 c^5} \] Input:
Integrate[x^4*(a + b*ArcTan[c*x])^2,x]
Output:
(9*a*b - 9*b^2*c*x + 6*a*b*c^2*x^2 + b^2*c^3*x^3 - 3*a*b*c^4*x^4 + 6*a^2*c ^5*x^5 + 6*b^2*(-I + c^5*x^5)*ArcTan[c*x]^2 - 3*b*ArcTan[c*x]*(-4*a*c^5*x^ 5 + b*(-3 - 2*c^2*x^2 + c^4*x^4) - 4*b*Log[1 + E^((2*I)*ArcTan[c*x])]) - 6 *a*b*Log[1 + c^2*x^2] - (6*I)*b^2*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(30* c^5)
Time = 1.25 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.22, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {5361, 5451, 5361, 254, 2009, 5451, 5361, 262, 216, 5455, 5379, 2849, 2752}
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.
| \(\displaystyle \int x^4 (a+b \arctan (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arctan (c x))}{c^2 x^2+1}dx\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\int x^3 (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \int \frac {x^4}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \int \left (\frac {x^2}{c^2}+\frac {1}{c^4 \left (c^2 x^2+1\right )}-\frac {1}{c^4}\right )dx}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\int \frac {x^3 (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}\right )\) |
| \(\Big \downarrow \) 5451 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\int x (a+b \arctan (c x))dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \int \frac {x^2}{c^2 x^2+1}dx}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\int \frac {1}{c^2 x^2+1}dx}{c^2}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {\int \frac {x (a+b \arctan (c x))}{c^2 x^2+1}dx}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5455 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\int \frac {a+b \arctan (c x)}{i-c x}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 5379 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{c^2 x^2+1}dx}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {\frac {i b \int \frac {\log \left (\frac {2}{i c x+1}\right )}{1-\frac {2}{i c x+1}}d\frac {1}{i c x+1}}{c}+\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}}{c}-\frac {i (a+b \arctan (c x))^2}{2 b c^2}}{c^2}}{c^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {2}{5} b c \left (\frac {\frac {1}{4} x^4 (a+b \arctan (c x))-\frac {1}{4} b c \left (\frac {\arctan (c x)}{c^5}-\frac {x}{c^4}+\frac {x^3}{3 c^2}\right )}{c^2}-\frac {\frac {\frac {1}{2} x^2 (a+b \arctan (c x))-\frac {1}{2} b c \left (\frac {x}{c^2}-\frac {\arctan (c x)}{c^3}\right )}{c^2}-\frac {-\frac {i (a+b \arctan (c x))^2}{2 b c^2}-\frac {\frac {\log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}+\frac {i b \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{2 c}}{c}}{c^2}}{c^2}\right )\) |
Input:
Int[x^4*(a + b*ArcTan[c*x])^2,x]
Output:
(x^5*(a + b*ArcTan[c*x])^2)/5 - (2*b*c*(((x^4*(a + b*ArcTan[c*x]))/4 - (b* c*(-(x/c^4) + x^3/(3*c^2) + ArcTan[c*x]/c^5))/4)/c^2 - (((x^2*(a + b*ArcTa n[c*x]))/2 - (b*c*(x/c^2 - ArcTan[c*x]/c^3))/2)/c^2 - (((-1/2*I)*(a + b*Ar cTan[c*x])^2)/(b*c^2) - (((a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + ((I/ 2)*b*PolyLog[2, 1 - 2/(1 + I*c*x)])/c)/c)/c^2)/c^2))/5
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] )^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
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]
Time = 0.42 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.56
| method | result | size |
| parts | \(\frac {a^{2} x^{5}}{5}+\frac {b^{2} \left (\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{5}-\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{5}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3}}{30}-\frac {3 c x}{10}+\frac {3 \arctan \left (c x \right )}{10}-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{10}\right )}{c^{5}}+\frac {2 a b \left (\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}-\frac {\ln \left (c^{2} x^{2}+1\right )}{10}\right )}{c^{5}}\) | \(266\) |
| derivativedivides | \(\frac {\frac {c^{5} x^{5} a^{2}}{5}+b^{2} \left (\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{5}-\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{5}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3}}{30}-\frac {3 c x}{10}+\frac {3 \arctan \left (c x \right )}{10}-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{10}\right )+2 a b \left (\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}-\frac {\ln \left (c^{2} x^{2}+1\right )}{10}\right )}{c^{5}}\) | \(267\) |
| default | \(\frac {\frac {c^{5} x^{5} a^{2}}{5}+b^{2} \left (\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{5}-\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{5}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3}}{30}-\frac {3 c x}{10}+\frac {3 \arctan \left (c x \right )}{10}-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{10}\right )+2 a b \left (\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}-\frac {\ln \left (c^{2} x^{2}+1\right )}{10}\right )}{c^{5}}\) | \(267\) |
| risch | \(\frac {i b^{2} \ln \left (i c x +1\right ) x^{4}}{20 c}-\frac {i b^{2} \ln \left (i c x +1\right ) x^{2}}{10 c^{3}}-\frac {i b^{2} \ln \left (-i c x +1\right ) x^{4}}{20 c}+\frac {i b^{2} \ln \left (-i c x +1\right ) x^{2}}{10 c^{3}}+\frac {i b^{2} \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{10 c^{5}}-\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{5 c^{5}}+\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{5}}-\frac {i b a \ln \left (i c x +1\right ) x^{5}}{5}+\frac {i a b \ln \left (-i c x +1\right ) x^{5}}{5}+\frac {a b \,x^{2}}{5 c^{3}}-\frac {3 b^{2} x}{10 c^{4}}+\frac {b^{2} x^{3}}{30 c^{2}}+\frac {3 b^{2} \arctan \left (c x \right )}{20 c^{5}}-\frac {a b \,x^{4}}{10 c}+\frac {i b^{2} \ln \left (i c x +1\right )^{2}}{20 c^{5}}-\frac {137 i b^{2} \ln \left (i c x +1\right )}{600 c^{5}}-\frac {i b^{2} \ln \left (-i c x +1\right )^{2}}{20 c^{5}}-\frac {47 i b^{2} \ln \left (-i c x +1\right )}{600 c^{5}}+\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{5}}+\frac {23 i b^{2} \ln \left (c^{2} x^{2}+1\right )}{150 c^{5}}+\frac {b^{2} \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x^{5}}{10}+\frac {413 i b^{2}}{2250 c^{5}}+\frac {i a^{2}}{5 c^{5}}-\frac {b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}-\frac {b^{2} \ln \left (i c x +1\right )^{2} x^{5}}{20}+\frac {a^{2} x^{5}}{5}-\frac {a b \ln \left (c^{2} x^{2}+1\right )}{5 c^{5}}+\frac {137 a b}{150 c^{5}}\) | \(459\) |
Input:
int(x^4*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)
Output:
1/5*a^2*x^5+b^2/c^5*(1/5*c^5*x^5*arctan(c*x)^2-1/10*c^4*x^4*arctan(c*x)+1/ 5*c^2*x^2*arctan(c*x)-1/5*arctan(c*x)*ln(c^2*x^2+1)+1/30*c^3*x^3-3/10*c*x+ 3/10*arctan(c*x)-1/10*I*(ln(c*x-I)*ln(c^2*x^2+1)-1/2*ln(c*x-I)^2-dilog(-1/ 2*I*(c*x+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I)))+1/10*I*(ln(c*x+I)*ln(c^2*x^2+1) -1/2*ln(c*x+I)^2-dilog(1/2*I*(c*x-I))-ln(c*x+I)*ln(1/2*I*(c*x-I))))+2*a*b/ c^5*(1/5*c^5*x^5*arctan(c*x)-1/20*c^4*x^4+1/10*c^2*x^2-1/10*ln(c^2*x^2+1))
\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4} \,d x } \] Input:
integrate(x^4*(a+b*arctan(c*x))^2,x, algorithm="fricas")
Output:
integral(b^2*x^4*arctan(c*x)^2 + 2*a*b*x^4*arctan(c*x) + a^2*x^4, x)
\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int x^{4} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}\, dx \] Input:
integrate(x**4*(a+b*atan(c*x))**2,x)
Output:
Integral(x**4*(a + b*atan(c*x))**2, x)
\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4} \,d x } \] Input:
integrate(x^4*(a+b*arctan(c*x))^2,x, algorithm="maxima")
Output:
1/5*a^2*x^5 + 1/10*(4*x^5*arctan(c*x) - c*((c^2*x^4 - 2*x^2)/c^4 + 2*log(c ^2*x^2 + 1)/c^6))*a*b + 1/80*(4*x^5*arctan(c*x)^2 - x^5*log(c^2*x^2 + 1)^2 + 80*integrate(1/80*(4*c^2*x^6*log(c^2*x^2 + 1) - 8*c*x^5*arctan(c*x) + 6 0*(c^2*x^6 + x^4)*arctan(c*x)^2 + 5*(c^2*x^6 + x^4)*log(c^2*x^2 + 1)^2)/(c ^2*x^2 + 1), x))*b^2
\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4} \,d x } \] Input:
integrate(x^4*(a+b*arctan(c*x))^2,x, algorithm="giac")
Output:
integrate((b*arctan(c*x) + a)^2*x^4, x)
Timed out. \[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2 \,d x \] Input:
int(x^4*(a + b*atan(c*x))^2,x)
Output:
int(x^4*(a + b*atan(c*x))^2, x)
\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\frac {6 \mathit {atan} \left (c x \right )^{2} b^{2} c^{5} x^{5}-6 \mathit {atan} \left (c x \right )^{2} b^{2} c x +12 \mathit {atan} \left (c x \right ) a b \,c^{5} x^{5}-3 \mathit {atan} \left (c x \right ) b^{2} c^{4} x^{4}+6 \mathit {atan} \left (c x \right ) b^{2} c^{2} x^{2}+9 \mathit {atan} \left (c x \right ) b^{2}+6 \left (\int \mathit {atan} \left (c x \right )^{2}d x \right ) b^{2} c -6 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a b +6 a^{2} c^{5} x^{5}-3 a b \,c^{4} x^{4}+6 a b \,c^{2} x^{2}+b^{2} c^{3} x^{3}-9 b^{2} c x}{30 c^{5}} \] Input:
int(x^4*(a+b*atan(c*x))^2,x)
Output:
(6*atan(c*x)**2*b**2*c**5*x**5 - 6*atan(c*x)**2*b**2*c*x + 12*atan(c*x)*a* b*c**5*x**5 - 3*atan(c*x)*b**2*c**4*x**4 + 6*atan(c*x)*b**2*c**2*x**2 + 9* atan(c*x)*b**2 + 6*int(atan(c*x)**2,x)*b**2*c - 6*log(c**2*x**2 + 1)*a*b + 6*a**2*c**5*x**5 - 3*a*b*c**4*x**4 + 6*a*b*c**2*x**2 + b**2*c**3*x**3 - 9 *b**2*c*x)/(30*c**5)