\(\int (d+e x) (a+b \arctan (c x))^2 \, dx\) [11]

Optimal result

Integrand size = 16, antiderivative size = 171 \[ \int (d+e x) (a+b \arctan (c x))^2 \, dx=-\frac {a b e x}{c}-\frac {b^2 e x \arctan (c x)}{c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \arctan (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c} \]

[Out]

-a*b*e*x/c-b^2*e*x*arctan(c*x)/c+I*d*(a+b*arctan(c*x))^2/c-1/2*(d^2-e^2/c^2)*(a+b*arctan(c*x))^2/e+1/2*(e*x+d)
^2*(a+b*arctan(c*x))^2/e+2*b*d*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/c+1/2*b^2*e*ln(c^2*x^2+1)/c^2+I*b^2*d*polylog
(2,1-2/(1+I*c*x))/c

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {4974, 4930, 266, 5104, 5004, 5040, 4964, 2449, 2352} \[ \int (d+e x) (a+b \arctan (c x))^2 \, dx=-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \arctan (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}+\frac {i d (a+b \arctan (c x))^2}{c}+\frac {2 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{c}-\frac {a b e x}{c}-\frac {b^2 e x \arctan (c x)}{c}+\frac {b^2 e \log \left (c^2 x^2+1\right )}{2 c^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{c} \]

[In]

Int[(d + e*x)*(a + b*ArcTan[c*x])^2,x]

[Out]

-((a*b*e*x)/c) - (b^2*e*x*ArcTan[c*x])/c + (I*d*(a + b*ArcTan[c*x])^2)/c - ((d^2 - e^2/c^2)*(a + b*ArcTan[c*x]
)^2)/(2*e) + ((d + e*x)^2*(a + b*ArcTan[c*x])^2)/(2*e) + (2*b*d*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c + (b
^2*e*Log[1 + c^2*x^2])/(2*c^2) + (I*b^2*d*PolyLog[2, 1 - 2/(1 + I*c*x)])/c

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 4930

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTan[c*x^n])^p, x] - Dist[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 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 4974

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a
+ b*ArcTan[c*x])^p/(e*(q + 1))), x] - Dist[b*c*(p/(e*(q + 1))), Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p -
1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && N
eQ[q, -1]

Rule 5004

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -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]

Rule 5104

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}-\frac {(b c) \int \left (\frac {e^2 (a+b \arctan (c x))}{c^2}+\frac {\left (c^2 d^2-e^2+2 c^2 d e x\right ) (a+b \arctan (c x))}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{e} \\ & = \frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}-\frac {b \int \frac {\left (c^2 d^2-e^2+2 c^2 d e x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c e}-\frac {(b e) \int (a+b \arctan (c x)) \, dx}{c} \\ & = -\frac {a b e x}{c}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}-\frac {b \int \left (\frac {c^2 d^2 \left (1-\frac {e^2}{c^2 d^2}\right ) (a+b \arctan (c x))}{1+c^2 x^2}+\frac {2 c^2 d e x (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{c e}-\frac {\left (b^2 e\right ) \int \arctan (c x) \, dx}{c} \\ & = -\frac {a b e x}{c}-\frac {b^2 e x \arctan (c x)}{c}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}-(2 b c d) \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\left (b^2 e\right ) \int \frac {x}{1+c^2 x^2} \, dx-\frac {(b (c d-e) (c d+e)) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{c e} \\ & = -\frac {a b e x}{c}-\frac {b^2 e x \arctan (c x)}{c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \arctan (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+(2 b d) \int \frac {a+b \arctan (c x)}{i-c x} \, dx \\ & = -\frac {a b e x}{c}-\frac {b^2 e x \arctan (c x)}{c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \arctan (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}-\left (2 b^2 d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx \\ & = -\frac {a b e x}{c}-\frac {b^2 e x \arctan (c x)}{c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \arctan (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {\left (2 i b^2 d\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c} \\ & = -\frac {a b e x}{c}-\frac {b^2 e x \arctan (c x)}{c}+\frac {i d (a+b \arctan (c x))^2}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \arctan (c x))^2}{2 e}+\frac {(d+e x)^2 (a+b \arctan (c x))^2}{2 e}+\frac {2 b d (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {b^2 e \log \left (1+c^2 x^2\right )}{2 c^2}+\frac {i b^2 d \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.01 \[ \int (d+e x) (a+b \arctan (c x))^2 \, dx=\frac {2 a^2 c^2 d x-2 a b c e x+a^2 c^2 e x^2+b^2 (-i+c x) (2 c d+i e+c e x) \arctan (c x)^2+2 b \arctan (c x) \left (-b c e x+a \left (e+2 c^2 d x+c^2 e x^2\right )+2 b c d \log \left (1+e^{2 i \arctan (c x)}\right )\right )-2 a b c d \log \left (1+c^2 x^2\right )+b^2 e \log \left (1+c^2 x^2\right )-2 i b^2 c d \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{2 c^2} \]

[In]

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

[Out]

(2*a^2*c^2*d*x - 2*a*b*c*e*x + a^2*c^2*e*x^2 + b^2*(-I + c*x)*(2*c*d + I*e + c*e*x)*ArcTan[c*x]^2 + 2*b*ArcTan
[c*x]*(-(b*c*e*x) + a*(e + 2*c^2*d*x + c^2*e*x^2) + 2*b*c*d*Log[1 + E^((2*I)*ArcTan[c*x])]) - 2*a*b*c*d*Log[1
+ c^2*x^2] + b^2*e*Log[1 + c^2*x^2] - (2*I)*b^2*c*d*PolyLog[2, -E^((2*I)*ArcTan[c*x])])/(2*c^2)

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.71

method	result	size
parts	\(a^{2} \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b^{2} \left (\frac {\arctan \left (c x \right )^{2} c \,x^{2} e}{2}+\arctan \left (c x \right )^{2} c x d -\frac {\ln \left (c^{2} x^{2}+1\right ) \arctan \left (c x \right ) c d -\frac {\arctan \left (c x \right )^{2} e}{2}+\arctan \left (c x \right ) e c x -\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}-d c \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )}{c}\right )}{c}+a b \arctan \left (c x \right ) x^{2} e +2 a b \arctan \left (c x \right ) x d -\frac {a b d \ln \left (c^{2} x^{2}+1\right )}{c}-\frac {a b e x}{c}+\frac {e b a \arctan \left (c x \right )}{c^{2}}\)	\(292\)
derivativedivides	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} d \,c^{2} x +\frac {\arctan \left (c x \right )^{2} e \,c^{2} x^{2}}{2}-\ln \left (c^{2} x^{2}+1\right ) \arctan \left (c x \right ) c d +\frac {\arctan \left (c x \right )^{2} e}{2}-\arctan \left (c x \right ) e c x +\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )\right )}{c}+\frac {2 a b \left (\arctan \left (c x \right ) d \,c^{2} x +\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {c e x}{2}-\frac {d c \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {e \arctan \left (c x \right )}{2}\right )}{c}}{c}\)	\(302\)
default	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (\arctan \left (c x \right )^{2} d \,c^{2} x +\frac {\arctan \left (c x \right )^{2} e \,c^{2} x^{2}}{2}-\ln \left (c^{2} x^{2}+1\right ) \arctan \left (c x \right ) c d +\frac {\arctan \left (c x \right )^{2} e}{2}-\arctan \left (c x \right ) e c x +\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \left (-\frac {i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{2}\right )\right )}{c}+\frac {2 a b \left (\arctan \left (c x \right ) d \,c^{2} x +\frac {\arctan \left (c x \right ) e \,c^{2} x^{2}}{2}-\frac {c e x}{2}-\frac {d c \ln \left (c^{2} x^{2}+1\right )}{2}+\frac {e \arctan \left (c x \right )}{2}\right )}{c}}{c}\)	\(302\)
risch	\(-\frac {a b e x}{c}+\frac {b^{2} e \ln \left (c^{2} x^{2}+1\right )}{4 c^{2}}+a^{2} d x +\frac {a^{2} e \,x^{2}}{2}-\frac {e \,b^{2} \ln \left (-i c x +1\right )^{2}}{8 c^{2}}+\frac {e \,b^{2} \ln \left (-i c x +1\right )}{2 c^{2}}-\frac {e \,b^{2} \ln \left (-i c x +1\right )^{2} x^{2}}{8}-\frac {\ln \left (-i c x +1\right )^{2} b^{2} d x}{4}+\frac {i a^{2} d}{c}-\frac {b^{2} \left (c^{2} e \,x^{2}+2 d \,c^{2} x -2 i d c +e \right ) \ln \left (i c x +1\right )^{2}}{8 c^{2}}-\frac {a b d \ln \left (c^{2} x^{2}+1\right )}{2 c}+\frac {e b a \arctan \left (c x \right )}{2 c^{2}}-\frac {i e b a \ln \left (c^{2} x^{2}+1\right )}{4 c^{2}}-\frac {i a b d \arctan \left (c x \right )}{c}-\frac {i e \,b^{2} \ln \left (-i c x +1\right ) x}{2 c}+i \ln \left (-i c x +1\right ) a b d x +\frac {i e b a \ln \left (-i c x +1\right )}{2 c^{2}}+\frac {i e b a \ln \left (-i c x +1\right ) x^{2}}{2}+\frac {i b^{2} \ln \left (\frac {1}{2}-\frac {i c x}{2}\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d}{c}-\frac {i b^{2} \ln \left (-i c x +1\right ) \ln \left (\frac {1}{2}+\frac {i c x}{2}\right ) d}{c}+\frac {a^{2} e}{2 c^{2}}+\left (\frac {b^{2} x \left (e x +2 d \right ) \ln \left (-i c x +1\right )}{4}-\frac {i b \left (2 a \,c^{2} e \,x^{2}+4 a \,c^{2} d x -2 \ln \left (-i c x +1\right ) b c d -2 b c e x +i \ln \left (-i c x +1\right ) b e \right )}{4 c^{2}}\right ) \ln \left (i c x +1\right )+\frac {i e \,b^{2} \arctan \left (c x \right )}{2 c^{2}}-\frac {i b a e}{c^{2}}-\frac {\ln \left (-i c x +1\right ) a b d}{c}-\frac {i \ln \left (-i c x +1\right )^{2} b^{2} d}{4 c}+\frac {i b^{2} \operatorname {dilog}\left (\frac {1}{2}-\frac {i c x}{2}\right ) d}{c}\)	\(524\)

[In]

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

[Out]

a^2*(1/2*e*x^2+d*x)+b^2/c*(1/2*arctan(c*x)^2*c*x^2*e+arctan(c*x)^2*c*x*d-1/c*(ln(c^2*x^2+1)*arctan(c*x)*c*d-1/
2*arctan(c*x)^2*e+arctan(c*x)*e*c*x-1/2*e*ln(c^2*x^2+1)-d*c*(-1/2*I*(ln(c*x-I)*ln(c^2*x^2+1)-dilog(-1/2*I*(c*x
+I))-ln(c*x-I)*ln(-1/2*I*(c*x+I))-1/2*ln(c*x-I)^2)+1/2*I*(ln(c*x+I)*ln(c^2*x^2+1)-dilog(1/2*I*(c*x-I))-ln(c*x+
I)*ln(1/2*I*(c*x-I))-1/2*ln(c*x+I)^2))))+a*b*arctan(c*x)*x^2*e+2*a*b*arctan(c*x)*x*d-1/c*a*b*d*ln(c^2*x^2+1)-a
*b*e*x/c+1/c^2*e*b*a*arctan(c*x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

12*b^2*c^2*e*integrate(1/16*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + b^2*c^2*e*integrate(1/16*x^3*log(c^2*x^2 + 1
)^2/(c^2*x^2 + 1), x) + 12*b^2*c^2*d*integrate(1/16*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 2*b^2*c^2*e*integrat
e(1/16*x^3*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + b^2*c^2*d*integrate(1/16*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1),
x) + 4*b^2*c^2*d*integrate(1/16*x^2*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 1/2*a^2*e*x^2 + 1/4*b^2*d*arctan(c*x
)^3/c - 4*b^2*c*e*integrate(1/16*x^2*arctan(c*x)/(c^2*x^2 + 1), x) - 8*b^2*c*d*integrate(1/16*x*arctan(c*x)/(c
^2*x^2 + 1), x) + (x^2*arctan(c*x) - c*(x/c^2 - arctan(c*x)/c^3))*a*b*e + a^2*d*x + 12*b^2*e*integrate(1/16*x*
arctan(c*x)^2/(c^2*x^2 + 1), x) + b^2*e*integrate(1/16*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + b^2*d*integrat
e(1/16*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + (2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a*b*d/c + 1/8*(b^2*e*x^2
+ 2*b^2*d*x)*arctan(c*x)^2 - 1/32*(b^2*e*x^2 + 2*b^2*d*x)*log(c^2*x^2 + 1)^2

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int((a + b*atan(c*x))^2*(d + e*x),x)

[Out]

int((a + b*atan(c*x))^2*(d + e*x), x)