\(\int \frac {x (a+b \arctan (c x))}{d+e x} \, dx\) [136]

Optimal result

Integrand size = 17, antiderivative size = 179 \[ \int \frac {x (a+b \arctan (c x))}{d+e x} \, dx=\frac {a x}{e}+\frac {b x \arctan (c x)}{e}+\frac {d (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {d (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{e^2}-\frac {b \log \left (1+c^2 x^2\right )}{2 c e}-\frac {i b d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4996, 4930, 266, 4966, 2449, 2352, 2497} \[ \int \frac {x (a+b \arctan (c x))}{d+e x} \, dx=\frac {d \log \left (\frac {2}{1-i c x}\right ) (a+b \arctan (c x))}{e^2}-\frac {d (a+b \arctan (c x)) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e^2}+\frac {a x}{e}+\frac {b x \arctan (c x)}{e}-\frac {b \log \left (c^2 x^2+1\right )}{2 c e}-\frac {i b d \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {i b d \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e^2} \]

[In]

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

[Out]

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

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 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((
d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcTan[c*x])*(Log[2*c*((d + e*x)/((c*
d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4996

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

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

Mathematica [A] (verified)

Time = 1.17 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.84 \[ \int \frac {x (a+b \arctan (c x))}{d+e x} \, dx=\frac {2 a e x-2 a d \log (d+e x)+\frac {b \left (-i c d \pi \arctan (c x)+2 c e x \arctan (c x)+2 i c d \arctan \left (\frac {c d}{e}\right ) \arctan (c x)-i c d \arctan (c x)^2-e \arctan (c x)^2+\sqrt {1+\frac {c^2 d^2}{e^2}} e e^{i \arctan \left (\frac {c d}{e}\right )} \arctan (c x)^2-c d \pi \log \left (1+e^{-2 i \arctan (c x)}\right )+2 c d \arctan (c x) \log \left (1+e^{2 i \arctan (c x)}\right )-2 c d \arctan \left (\frac {c d}{e}\right ) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )-2 c d \arctan (c x) \log \left (1-e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )-e \log \left (1+c^2 x^2\right )-\frac {1}{2} c d \pi \log \left (1+c^2 x^2\right )+2 c d \arctan \left (\frac {c d}{e}\right ) \log \left (\sin \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )\right )-i c d \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )+i c d \operatorname {PolyLog}\left (2,e^{2 i \left (\arctan \left (\frac {c d}{e}\right )+\arctan (c x)\right )}\right )\right )}{c}}{2 e^2} \]

[In]

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

[Out]

(2*a*e*x - 2*a*d*Log[d + e*x] + (b*((-I)*c*d*Pi*ArcTan[c*x] + 2*c*e*x*ArcTan[c*x] + (2*I)*c*d*ArcTan[(c*d)/e]*
ArcTan[c*x] - I*c*d*ArcTan[c*x]^2 - e*ArcTan[c*x]^2 + Sqrt[1 + (c^2*d^2)/e^2]*e*E^(I*ArcTan[(c*d)/e])*ArcTan[c
*x]^2 - c*d*Pi*Log[1 + E^((-2*I)*ArcTan[c*x])] + 2*c*d*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] - 2*c*d*ArcT
an[(c*d)/e]*Log[1 - E^((2*I)*(ArcTan[(c*d)/e] + ArcTan[c*x]))] - 2*c*d*ArcTan[c*x]*Log[1 - E^((2*I)*(ArcTan[(c
*d)/e] + ArcTan[c*x]))] - e*Log[1 + c^2*x^2] - (c*d*Pi*Log[1 + c^2*x^2])/2 + 2*c*d*ArcTan[(c*d)/e]*Log[Sin[Arc
Tan[(c*d)/e] + ArcTan[c*x]]] - I*c*d*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + I*c*d*PolyLog[2, E^((2*I)*(ArcTan[(c
*d)/e] + ArcTan[c*x]))]))/c)/(2*e^2)

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.25

method	result	size
parts	\(\frac {a x}{e}-\frac {a d \ln \left (e x +d \right )}{e^{2}}+\frac {b \left (\frac {c^{2} \arctan \left (c x \right ) x}{e}-\frac {c^{2} \arctan \left (c x \right ) d \ln \left (e c x +c d \right )}{e^{2}}-\frac {c \left (\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{2}-c d \left (-\frac {i \ln \left (e c x +c d \right ) \left (\ln \left (\frac {-e c x +i e}{c d +i e}\right )-\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )\right )}{e}\right )}{c^{2}}\)	\(224\)
derivativedivides	\(\frac {\frac {a \,c^{2} x}{e}-\frac {a \,c^{2} d \ln \left (e c x +c d \right )}{e^{2}}+b c \left (\frac {\arctan \left (c x \right ) c x}{e}-\frac {\arctan \left (c x \right ) d c \ln \left (e c x +c d \right )}{e^{2}}-\frac {\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{2}-c d \left (\frac {i \ln \left (e c x +c d \right ) \left (-\ln \left (\frac {-e c x +i e}{c d +i e}\right )+\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e}\right )}{c^{2}}\)	\(230\)
default	\(\frac {\frac {a \,c^{2} x}{e}-\frac {a \,c^{2} d \ln \left (e c x +c d \right )}{e^{2}}+b c \left (\frac {\arctan \left (c x \right ) c x}{e}-\frac {\arctan \left (c x \right ) d c \ln \left (e c x +c d \right )}{e^{2}}-\frac {\frac {\ln \left (c^{2} d^{2}-2 c d \left (e c x +c d \right )+e^{2}+\left (e c x +c d \right )^{2}\right )}{2}-c d \left (\frac {i \ln \left (e c x +c d \right ) \left (-\ln \left (\frac {-e c x +i e}{c d +i e}\right )+\ln \left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}-\frac {i \left (\operatorname {dilog}\left (\frac {-e c x +i e}{c d +i e}\right )-\operatorname {dilog}\left (\frac {e c x +i e}{-c d +i e}\right )\right )}{2 e}\right )}{e}\right )}{c^{2}}\)	\(230\)
risch	\(\frac {i a}{c e}+\frac {i b d \operatorname {dilog}\left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{2}}-\frac {b \ln \left (c^{2} x^{2}+1\right )}{4 c e}+\frac {i b \ln \left (-i c x +1\right ) x}{2 e}+\frac {b}{c e}-\frac {i b d \ln \left (-i c x +1\right ) \ln \left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{2}}-\frac {i b d \operatorname {dilog}\left (\frac {-i c d +\left (-i c x +1\right ) e -e}{-i c d -e}\right )}{2 e^{2}}+\frac {a x}{e}+\frac {i b \arctan \left (c x \right )}{2 c e}-\frac {a d \ln \left (i c d -\left (-i c x +1\right ) e +e \right )}{e^{2}}-\frac {i b \ln \left (i c x +1\right ) x}{2 e}+\frac {i b d \ln \left (i c x +1\right ) \ln \left (\frac {i c d +\left (i c x +1\right ) e -e}{i c d -e}\right )}{2 e^{2}}-\frac {b \ln \left (i c x +1\right )}{2 c e}\)	\(305\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

a*(x/e - d*log(e*x + d)/e^2) + 2*b*integrate(1/2*x*arctan(c*x)/(e*x + d), x)

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

int((x*(a + b*atan(c*x)))/(d + e*x),x)

[Out]

int((x*(a + b*atan(c*x)))/(d + e*x), x)