\(\int (e+f x) (a+b \arctan (c+d x))^3 \, dx\) [37]

Optimal result

Integrand size = 18, antiderivative size = 337 \[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=-\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (c+d x))^3}{2 d^2 f}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b^2 f (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b (d e-c f) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2} \]

[Out]

-3/2*I*b*f*(a+b*arctan(d*x+c))^2/d^2-3/2*b*f*(d*x+c)*(a+b*arctan(d*x+c))^2/d^2+I*(-c*f+d*e)*(a+b*arctan(d*x+c)
)^3/d^2-1/2*(-c*f+d*e+f)*(d*e-(1+c)*f)*(a+b*arctan(d*x+c))^3/d^2/f+1/2*(f*x+e)^2*(a+b*arctan(d*x+c))^3/f-3*b^2
*f*(a+b*arctan(d*x+c))*ln(2/(1+I*(d*x+c)))/d^2+3*b*(-c*f+d*e)*(a+b*arctan(d*x+c))^2*ln(2/(1+I*(d*x+c)))/d^2-3/
2*I*b^3*f*polylog(2,1-2/(1+I*(d*x+c)))/d^2+3*I*b^2*(-c*f+d*e)*(a+b*arctan(d*x+c))*polylog(2,1-2/(1+I*(d*x+c)))
/d^2+3/2*b^3*(-c*f+d*e)*polylog(3,1-2/(1+I*(d*x+c)))/d^2

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5155, 4974, 4930, 5040, 4964, 2449, 2352, 5104, 5004, 5114, 6745} \[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\frac {3 i b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) (a+b \arctan (c+d x))}{d^2}-\frac {3 b^2 f \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))}{d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^3}{d^2}-\frac {(-c f+d e+f) (d e-(c+1) f) (a+b \arctan (c+d x))^3}{2 d^2 f}+\frac {3 b (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) (a+b \arctan (c+d x))^2}{d^2}-\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}+\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 d^2}-\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{2 d^2} \]

[In]

Int[(e + f*x)*(a + b*ArcTan[c + d*x])^3,x]

[Out]

(((-3*I)/2)*b*f*(a + b*ArcTan[c + d*x])^2)/d^2 - (3*b*f*(c + d*x)*(a + b*ArcTan[c + d*x])^2)/(2*d^2) + (I*(d*e
- c*f)*(a + b*ArcTan[c + d*x])^3)/d^2 - ((d*e + f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcTan[c + d*x])^3)/(2*d^2*
f) + ((e + f*x)^2*(a + b*ArcTan[c + d*x])^3)/(2*f) - (3*b^2*f*(a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x))]
)/d^2 + (3*b*(d*e - c*f)*(a + b*ArcTan[c + d*x])^2*Log[2/(1 + I*(c + d*x))])/d^2 - (((3*I)/2)*b^3*f*PolyLog[2,
1 - 2/(1 + I*(c + d*x))])/d^2 + ((3*I)*b^2*(d*e - c*f)*(a + b*ArcTan[c + d*x])*PolyLog[2, 1 - 2/(1 + I*(c + d
*x))])/d^2 + (3*b^3*(d*e - c*f)*PolyLog[3, 1 - 2/(1 + I*(c + d*x))])/(2*d^2)

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]

Rule 5114

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*(a + b*Ar
cTan[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x] + Dist[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]

Rule 5155

Int[((a_.) + ArcTan[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcTan[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

Rule 6745

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) (a+b \arctan (x))^3 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {(3 b) \text {Subst}\left (\int \left (\frac {f^2 (a+b \arctan (x))^2}{d^2}+\frac {((d e-f-c f) (d e+f-c f)+2 f (d e-c f) x) (a+b \arctan (x))^2}{d^2 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{2 f} \\ & = \frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {(3 b) \text {Subst}\left (\int \frac {((d e-f-c f) (d e+f-c f)+2 f (d e-c f) x) (a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{2 d^2 f}-\frac {(3 b f) \text {Subst}\left (\int (a+b \arctan (x))^2 \, dx,x,c+d x\right )}{2 d^2} \\ & = -\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {(3 b) \text {Subst}\left (\int \left (\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (x))^2}{1+x^2}-\frac {2 f (-d e+c f) x (a+b \arctan (x))^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{2 d^2 f}+\frac {\left (3 b^2 f\right ) \text {Subst}\left (\int \frac {x (a+b \arctan (x))}{1+x^2} \, dx,x,c+d x\right )}{d^2} \\ & = -\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {\left (3 b^2 f\right ) \text {Subst}\left (\int \frac {a+b \arctan (x)}{i-x} \, dx,x,c+d x\right )}{d^2}-\frac {(3 b (d e-c f)) \text {Subst}\left (\int \frac {x (a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{d^2}-\frac {(3 b (d e+f-c f) (d e-(1+c) f)) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{1+x^2} \, dx,x,c+d x\right )}{2 d^2 f} \\ & = -\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (c+d x))^3}{2 d^2 f}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b^2 f (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {\left (3 b^3 f\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}+\frac {(3 b (d e-c f)) \text {Subst}\left (\int \frac {(a+b \arctan (x))^2}{i-x} \, dx,x,c+d x\right )}{d^2} \\ & = -\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (c+d x))^3}{2 d^2 f}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b^2 f (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b (d e-c f) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {\left (3 i b^3 f\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d^2}-\frac {\left (6 b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {(a+b \arctan (x)) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2} \\ & = -\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (c+d x))^3}{2 d^2 f}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b^2 f (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b (d e-c f) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {\left (3 i b^3 (d e-c f)\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2} \\ & = -\frac {3 i b f (a+b \arctan (c+d x))^2}{2 d^2}-\frac {3 b f (c+d x) (a+b \arctan (c+d x))^2}{2 d^2}+\frac {i (d e-c f) (a+b \arctan (c+d x))^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) (a+b \arctan (c+d x))^3}{2 d^2 f}+\frac {(e+f x)^2 (a+b \arctan (c+d x))^3}{2 f}-\frac {3 b^2 f (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b (d e-c f) (a+b \arctan (c+d x))^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) (a+b \arctan (c+d x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.07 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.76 \[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\frac {a^2 (2 a d e-3 b f-2 a c f) (c+d x)+a^3 f (c+d x)^2+3 a^2 b f \arctan (c+d x)-3 a^2 b (c+d x) (c f-d (2 e+f x)) \arctan (c+d x)+6 a b^2 f \left (-((c+d x) \arctan (c+d x))+\frac {1}{2} \left (1+(c+d x)^2\right ) \arctan (c+d x)^2-\log \left (\frac {1}{\sqrt {1+(c+d x)^2}}\right )\right )-3 a^2 b (d e-c f) \log \left (1+(c+d x)^2\right )+6 a b^2 d e \left (\arctan (c+d x) \left ((-i+c+d x) \arctan (c+d x)+2 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )-6 a b^2 c f \left (\arctan (c+d x) \left ((-i+c+d x) \arctan (c+d x)+2 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )+b^3 f \left (\arctan (c+d x) \left (3 i \arctan (c+d x)-3 (c+d x) \arctan (c+d x)+\left (1+(c+d x)^2\right ) \arctan (c+d x)^2-6 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )+3 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )+2 b^3 d e \left (\arctan (c+d x)^2 \left ((-i+c+d x) \arctan (c+d x)+3 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-3 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )-2 b^3 c f \left (\arctan (c+d x)^2 \left ((-i+c+d x) \arctan (c+d x)+3 \log \left (1+e^{2 i \arctan (c+d x)}\right )\right )-3 i \arctan (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{2 i \arctan (c+d x)}\right )\right )}{2 d^2} \]

[In]

Integrate[(e + f*x)*(a + b*ArcTan[c + d*x])^3,x]

[Out]

(a^2*(2*a*d*e - 3*b*f - 2*a*c*f)*(c + d*x) + a^3*f*(c + d*x)^2 + 3*a^2*b*f*ArcTan[c + d*x] - 3*a^2*b*(c + d*x)
*(c*f - d*(2*e + f*x))*ArcTan[c + d*x] + 6*a*b^2*f*(-((c + d*x)*ArcTan[c + d*x]) + ((1 + (c + d*x)^2)*ArcTan[c
+ d*x]^2)/2 - Log[1/Sqrt[1 + (c + d*x)^2]]) - 3*a^2*b*(d*e - c*f)*Log[1 + (c + d*x)^2] + 6*a*b^2*d*e*(ArcTan[
c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*Log[1 + E^((2*I)*ArcTan[c + d*x])]) - I*PolyLog[2, -E^((2*I)*ArcT
an[c + d*x])]) - 6*a*b^2*c*f*(ArcTan[c + d*x]*((-I + c + d*x)*ArcTan[c + d*x] + 2*Log[1 + E^((2*I)*ArcTan[c +
d*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]) + b^3*f*(ArcTan[c + d*x]*((3*I)*ArcTan[c + d*x] - 3*(c + d
*x)*ArcTan[c + d*x] + (1 + (c + d*x)^2)*ArcTan[c + d*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + (3*I)*Poly
Log[2, -E^((2*I)*ArcTan[c + d*x])]) + 2*b^3*d*e*(ArcTan[c + d*x]^2*((-I + c + d*x)*ArcTan[c + d*x] + 3*Log[1 +
E^((2*I)*ArcTan[c + d*x])]) - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E
^((2*I)*ArcTan[c + d*x])])/2) - 2*b^3*c*f*(ArcTan[c + d*x]^2*((-I + c + d*x)*ArcTan[c + d*x] + 3*Log[1 + E^((2
*I)*ArcTan[c + d*x])]) - (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I
)*ArcTan[c + d*x])])/2))/(2*d^2)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.33 (sec) , antiderivative size = 8267, normalized size of antiderivative = 24.53

method	result	size
parts	\(\text {Expression too large to display}\)	\(8267\)
derivativedivides	\(\text {Expression too large to display}\)	\(8269\)
default	\(\text {Expression too large to display}\)	\(8269\)

[In]

int((f*x+e)*(a+b*arctan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [F]

\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((f*x+e)*(a+b*arctan(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(a^3*f*x + a^3*e + (b^3*f*x + b^3*e)*arctan(d*x + c)^3 + 3*(a*b^2*f*x + a*b^2*e)*arctan(d*x + c)^2 + 3
*(a^2*b*f*x + a^2*b*e)*arctan(d*x + c), x)

Sympy [F]

\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int \left (a + b \operatorname {atan}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )\, dx \]

[In]

integrate((f*x+e)*(a+b*atan(d*x+c))**3,x)

[Out]

Integral((a + b*atan(c + d*x))**3*(e + f*x), x)

Maxima [F]

\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((f*x+e)*(a+b*arctan(d*x+c))^3,x, algorithm="maxima")

[Out]

7/8*b^3*c^2*e*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 3*a*b^2*c^2*e*arctan(d*x + c)^2*arctan((d^2*x + c*
d)/d)/d - (3*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*c^2*e - 7/32*(6*
arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^3/d + arctan((d^2*x
+ c*d)/d)^4/d)*b^3*c^2*e + 7/8*b^3*e*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 56*b^3*d^2*f*integrate(1/64
*x^3*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*d^2*f*integrate(1/64*x^3*arctan(d*x + c)*log(
d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*d^2*f*integrate(1/64*x^3*arctan(d
*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*d^2*e*integrate(1/64*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c
*d*x + c^2 + 1), x) + 112*b^3*c*d*f*integrate(1/64*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 1
2*b^3*d^2*f*integrate(1/64*x^3*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1),
x) + 6*b^3*d^2*e*integrate(1/64*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c
^2 + 1), x) + 12*b^3*c*d*f*integrate(1/64*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*
c*d*x + c^2 + 1), x) + 192*a*b^2*d^2*e*integrate(1/64*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)
+ 384*a*b^2*c*d*f*integrate(1/64*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 112*b^3*c*d*e*integ
rate(1/64*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c^2*f*integrate(1/64*x*arctan(d*x + c
)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 24*b^3*d^2*e*integrate(1/64*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x
+ c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*c*d*f*integrate(1/64*x^2*arctan(d*x + c)*log(d^2*x^2 + 2
*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*c*d*e*integrate(1/64*x*arctan(d*x + c)*log(d^2*x^
2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*c^2*f*integrate(1/64*x*arctan(d*x + c)*log(
d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 384*a*b^2*c*d*e*integrate(1/64*x*arctan(d*x
+ c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*c^2*f*integrate(1/64*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*
d*x + c^2 + 1), x) + 24*b^3*c*d*e*integrate(1/64*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 +
2*c*d*x + c^2 + 1), x) + 6*b^3*c^2*e*integrate(1/64*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x
^2 + 2*c*d*x + c^2 + 1), x) + 1/2*a^3*f*x^2 + 3*a*b^2*e*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 12*b^3*d
*f*integrate(1/64*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d*f*integrate(1/64*x^2*log(d
^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 24*b^3*d*e*integrate(1/64*x*arctan(d*x + c)^
2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*d*e*integrate(1/64*x*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 +
2*c*d*x + c^2 + 1), x) - (3*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*
e - 7/32*(6*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^3/d + ar
ctan((d^2*x + c*d)/d)^4/d)*b^3*e + 3/2*(x^2*arctan(d*x + c) - d*(x/d^2 + (c^2 - 1)*arctan((d^2*x + c*d)/d)/d^3
- c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^3))*a^2*b*f + a^3*e*x + 56*b^3*f*integrate(1/64*x*arctan(d*x + c)^3/(d
^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*f*integrate(1/64*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/
(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*f*integrate(1/64*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 +
1), x) + 6*b^3*e*integrate(1/64*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 +
1), x) + 3/2*(2*(d*x + c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*a^2*b*e/d + 1/16*(b^3*f*x^2 + 2*b^3*e*x)*arc
tan(d*x + c)^3 - 3/64*(b^3*f*x^2 + 2*b^3*e*x)*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2

Giac [F]

\[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int { {\left (f x + e\right )} {\left (b \arctan \left (d x + c\right ) + a\right )}^{3} \,d x } \]

[In]

integrate((f*x+e)*(a+b*arctan(d*x+c))^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (e+f x) (a+b \arctan (c+d x))^3 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {atan}\left (c+d\,x\right )\right )}^3 \,d x \]

[In]

int((e + f*x)*(a + b*atan(c + d*x))^3,x)

[Out]

int((e + f*x)*(a + b*atan(c + d*x))^3, x)