\(\int x^4 (a+b \arctan (c x))^2 \, dx\) [14]

Optimal result

Integrand size = 14, antiderivative size = 170 \[ \int x^4 (a+b \arctan (c x))^2 \, dx=-\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}+\frac {3 b^2 \arctan (c x)}{10 c^5}+\frac {b x^2 (a+b \arctan (c x))}{5 c^3}-\frac {b x^4 (a+b \arctan (c x))}{10 c}+\frac {i (a+b \arctan (c x))^2}{5 c^5}+\frac {1}{5} x^5 (a+b \arctan (c x))^2+\frac {2 b (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^5}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^5} \]

[Out]

-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*arctan(c*x))/c^3-1/10*b*x^4*(a+b*arct
an(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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4946, 5036, 308, 209, 327, 5040, 4964, 2449, 2352} \[ \int x^4 (a+b \arctan (c x))^2 \, dx=\frac {i (a+b \arctan (c x))^2}{5 c^5}+\frac {2 b \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{5 c^5}+\frac {b x^2 (a+b \arctan (c x))}{5 c^3}+\frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {b x^4 (a+b \arctan (c x))}{10 c}+\frac {3 b^2 \arctan (c x)}{10 c^5}+\frac {i b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{5 c^5}-\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2} \]

[In]

Int[x^4*(a + b*ArcTan[c*x])^2,x]

[Out]

(-3*b^2*x)/(10*c^4) + (b^2*x^3)/(30*c^2) + (3*b^2*ArcTan[c*x])/(10*c^5) + (b*x^2*(a + b*ArcTan[c*x]))/(5*c^3)
- (b*x^4*(a + b*ArcTan[c*x]))/(10*c) + ((I/5)*(a + b*ArcTan[c*x])^2)/c^5 + (x^5*(a + b*ArcTan[c*x])^2)/5 + (2*
b*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/(5*c^5) + ((I/5)*b^2*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^5

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-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]

Rule 4946

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] - Dist[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]

Rule 4964

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] + Dist[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]

Rule 5036

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x])^p, x], x] - Dist[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]

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {1}{5} (2 b c) \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx \\ & = \frac {1}{5} x^5 (a+b \arctan (c x))^2-\frac {(2 b) \int x^3 (a+b \arctan (c x)) \, dx}{5 c}+\frac {(2 b) \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c} \\ & = -\frac {b x^4 (a+b \arctan (c x))}{10 c}+\frac {1}{5} x^5 (a+b \arctan (c x))^2+\frac {1}{10} b^2 \int \frac {x^4}{1+c^2 x^2} \, dx+\frac {(2 b) \int x (a+b \arctan (c x)) \, dx}{5 c^3}-\frac {(2 b) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c^3} \\ & = \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 {1}{10} b^2 \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx+\frac {(2 b) \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{5 c^4}-\frac {b^2 \int \frac {x^2}{1+c^2 x^2} \, dx}{5 c^2} \\ & = -\frac {3 b^2 x}{10 c^4}+\frac {b^2 x^3}{30 c^2}+\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 {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{10 c^4}+\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{5 c^4}-\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^4} \\ & = -\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 {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^5} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int x^4 (a+b \arctan (c x))^2 \, dx=\frac {9 a b-9 b^2 c x+6 a b c^2 x^2+b^2 c^3 x^3-3 a b c^4 x^4+6 a^2 c^5 x^5+6 b^2 \left (-i+c^5 x^5\right ) \arctan (c x)^2-3 b \arctan (c x) \left (-4 a c^5 x^5+b \left (-3-2 c^2 x^2+c^4 x^4\right )-4 b \log \left (1+e^{2 i \arctan (c x)}\right )\right )-6 a b \log \left (1+c^2 x^2\right )-6 i b^2 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{30 c^5} \]

[In]

Integrate[x^4*(a + b*ArcTan[c*x])^2,x]

[Out]

(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)

Maple [A] (verified)

Time = 1.99 (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	\(\frac {\frac {a^{2} c^{5} x^{5}}{5}+b^{2} \left (\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{5}-\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{5}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3}}{30}-\frac {3 c x}{10}+\frac {3 \arctan \left (c x \right )}{10}-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{10}\right )+2 a b \left (\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}-\frac {\ln \left (c^{2} x^{2}+1\right )}{10}\right )}{c^{5}}\)	\(267\)
default	\(\frac {\frac {a^{2} c^{5} x^{5}}{5}+b^{2} \left (\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{5}-\frac {c^{4} x^{4} \arctan \left (c x \right )}{10}+\frac {c^{2} x^{2} \arctan \left (c x \right )}{5}-\frac {\arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{5}+\frac {c^{3} x^{3}}{30}-\frac {3 c x}{10}+\frac {3 \arctan \left (c x \right )}{10}-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{10}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{10}\right )+2 a b \left (\frac {c^{5} x^{5} \arctan \left (c x \right )}{5}-\frac {c^{4} x^{4}}{20}+\frac {c^{2} x^{2}}{10}-\frac {\ln \left (c^{2} x^{2}+1\right )}{10}\right )}{c^{5}}\)	\(267\)
risch	\(\frac {i b^{2} \ln \left (-i c x +1\right ) x^{2}}{10 c^{3}}+\frac {i b^{2} \ln \left (i c x +1\right ) x^{4}}{20 c}+\frac {137 a b}{150 c^{5}}-\frac {a b \,x^{4}}{10 c}+\frac {413 i b^{2}}{2250 c^{5}}+\frac {i a^{2}}{5 c^{5}}-\frac {b^{2} \ln \left (i c x +1\right )^{2} x^{5}}{20}-\frac {b^{2} \ln \left (-i c x +1\right )^{2} x^{5}}{20}-\frac {3 b^{2} x}{10 c^{4}}+\frac {b^{2} x^{3}}{30 c^{2}}+\frac {a b \,x^{2}}{5 c^{3}}+\frac {3 b^{2} \arctan \left (c x \right )}{20 c^{5}}-\frac {a b \ln \left (c^{2} x^{2}+1\right )}{5 c^{5}}+\frac {b^{2} \ln \left (i c x +1\right ) \ln \left (-i c x +1\right ) x^{5}}{10}-\frac {i b^{2} \ln \left (-i c x +1\right )^{2}}{20 c^{5}}-\frac {137 i b^{2} \ln \left (i c x +1\right )}{600 c^{5}}+\frac {i b^{2} \ln \left (i c x +1\right )^{2}}{20 c^{5}}+\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{5}}-\frac {47 i b^{2} \ln \left (-i c x +1\right )}{600 c^{5}}+\frac {23 i b^{2} \ln \left (c^{2} x^{2}+1\right )}{150 c^{5}}+\frac {a^{2} x^{5}}{5}-\frac {i b^{2} \ln \left (i c x +1\right ) x^{2}}{10 c^{3}}-\frac {i b a \ln \left (i c x +1\right ) x^{5}}{5}+\frac {i b^{2} \ln \left (i c x +1\right ) \ln \left (-i c x +1\right )}{10 c^{5}}-\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (-i c x +1\right )}{5 c^{5}}+\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c x}{2}\right )}{5 c^{5}}-\frac {i b^{2} \ln \left (-i c x +1\right ) x^{4}}{20 c}+\frac {i a b \ln \left (-i c x +1\right ) x^{5}}{5}\)	\(459\)

[In]

int(x^4*(a+b*arctan(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

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))

Fricas [F]

\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b*arctan(c*x))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^4*arctan(c*x)^2 + 2*a*b*x^4*arctan(c*x) + a^2*x^4, x)

Sympy [F]

\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int x^{4} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}\, dx \]

[In]

integrate(x**4*(a+b*atan(c*x))**2,x)

[Out]

Integral(x**4*(a + b*atan(c*x))**2, x)

Maxima [F]

\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b*arctan(c*x))^2,x, algorithm="maxima")

[Out]

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) +
60*(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

Giac [F]

\[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{4} \,d x } \]

[In]

integrate(x^4*(a+b*arctan(c*x))^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^4 (a+b \arctan (c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2 \,d x \]

[In]

int(x^4*(a + b*atan(c*x))^2,x)

[Out]

int(x^4*(a + b*atan(c*x))^2, x)