\(\int \frac {(d+e x^2)^2 (a+b \arctan (c x))}{x^3} \, dx\) [1131]

Optimal result

Integrand size = 21, antiderivative size = 128 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=-\frac {b c d^2}{2 x}-\frac {b e^2 x}{2 c}-\frac {1}{2} b c^2 d^2 \arctan (c x)+\frac {b e^2 \arctan (c x)}{2 c^2}-\frac {d^2 (a+b \arctan (c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))+2 a d e \log (x)+i b d e \operatorname {PolyLog}(2,-i c x)-i b d e \operatorname {PolyLog}(2,i c x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5100, 4946, 331, 209, 4940, 2438, 327} \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=-\frac {d^2 (a+b \arctan (c x))}{2 x^2}+\frac {1}{2} e^2 x^2 (a+b \arctan (c x))+2 a d e \log (x)-\frac {1}{2} b c^2 d^2 \arctan (c x)+\frac {b e^2 \arctan (c x)}{2 c^2}-\frac {b c d^2}{2 x}+i b d e \operatorname {PolyLog}(2,-i c x)-i b d e \operatorname {PolyLog}(2,i c x)-\frac {b e^2 x}{2 c} \]

[In]

Int[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^3,x]

[Out]

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

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 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 331

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

Rule 2438

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

Rule 4940

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

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 5100

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With
[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b,
c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=\frac {1}{2} \left (-\frac {b e^2 (c x-\arctan (c x))}{c^2}-\frac {d^2 (a+b \arctan (c x))}{x^2}+e^2 x^2 (a+b \arctan (c x))-\frac {b c d^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-c^2 x^2\right )}{x}+4 a d e \log (x)+2 i b d e \operatorname {PolyLog}(2,-i c x)-2 i b d e \operatorname {PolyLog}(2,i c x)\right ) \]

[In]

Integrate[((d + e*x^2)^2*(a + b*ArcTan[c*x]))/x^3,x]

[Out]

(-((b*e^2*(c*x - ArcTan[c*x]))/c^2) - (d^2*(a + b*ArcTan[c*x]))/x^2 + e^2*x^2*(a + b*ArcTan[c*x]) - (b*c*d^2*H
ypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)])/x + 4*a*d*e*Log[x] + (2*I)*b*d*e*PolyLog[2, (-I)*c*x] - (2*I)*b*d*
e*PolyLog[2, I*c*x])/2

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.39

method	result	size
parts	\(a \left (\frac {e^{2} x^{2}}{2}+2 e d \ln \left (x \right )-\frac {d^{2}}{2 x^{2}}\right )+b \,c^{2} \left (\frac {\arctan \left (c x \right ) e^{2} x^{2}}{2 c^{2}}+\frac {2 \arctan \left (c x \right ) d e \ln \left (c x \right )}{c^{2}}-\frac {\arctan \left (c x \right ) d^{2}}{2 c^{2} x^{2}}-\frac {c x \,e^{2}+\frac {c^{3} d^{2}}{x}+\left (c^{4} d^{2}-e^{2}\right ) \arctan \left (c x \right )+4 c^{2} d e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )}{2 c^{4}}\right )\)	\(178\)
derivativedivides	\(c^{2} \left (\frac {a \,e^{2} x^{2}}{2 c^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {b \left (\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{2}+2 \arctan \left (c x \right ) c^{2} d e \ln \left (c x \right )-\frac {\arctan \left (c x \right ) c^{2} d^{2}}{2 x^{2}}-\frac {c x \,e^{2}}{2}-\frac {c^{3} d^{2}}{2 x}+\frac {\left (-c^{4} d^{2}+e^{2}\right ) \arctan \left (c x \right )}{2}-2 c^{2} d e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )\right )}{c^{4}}\right )\)	\(189\)
default	\(c^{2} \left (\frac {a \,e^{2} x^{2}}{2 c^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {b \left (\frac {\arctan \left (c x \right ) e^{2} c^{2} x^{2}}{2}+2 \arctan \left (c x \right ) c^{2} d e \ln \left (c x \right )-\frac {\arctan \left (c x \right ) c^{2} d^{2}}{2 x^{2}}-\frac {c x \,e^{2}}{2}-\frac {c^{3} d^{2}}{2 x}+\frac {\left (-c^{4} d^{2}+e^{2}\right ) \arctan \left (c x \right )}{2}-2 c^{2} d e \left (-\frac {i \ln \left (c x \right ) \ln \left (i c x +1\right )}{2}+\frac {i \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i c x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i c x +1\right )}{2}\right )\right )}{c^{4}}\right )\)	\(189\)
risch	\(-\frac {i b \,d^{2} \ln \left (-i c x +1\right )}{4 x^{2}}+\frac {i b \,c^{2} d^{2} \ln \left (i c x +1\right )}{4}+\frac {i b \,d^{2} \ln \left (i c x +1\right )}{4 x^{2}}+\frac {i b \,e^{2} \ln \left (-i c x +1\right ) x^{2}}{4}+i b e d \operatorname {dilog}\left (i c x +1\right )+\frac {i c^{2} b \,d^{2} \ln \left (-i c x \right )}{4}-\frac {i b \,c^{2} d^{2} \ln \left (i c x \right )}{4}-i b d e \operatorname {dilog}\left (-i c x +1\right )-\frac {i b \,e^{2} \ln \left (i c x +1\right )}{4 c^{2}}-\frac {b \,e^{2} x}{2 c}+\frac {b \,e^{2} \arctan \left (c x \right )}{4 c^{2}}-\frac {b c \,d^{2}}{2 x}-\frac {b \,c^{2} d^{2} \arctan \left (c x \right )}{4}+2 a d e \ln \left (-i c x \right )+\frac {a \,e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 x^{2}}+\frac {a \,e^{2} x^{2}}{2}+\frac {i b \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{8 c^{2}}-\frac {i c^{2} b \,d^{2} \ln \left (c^{2} x^{2}+1\right )}{8}-\frac {i b \,e^{2} \ln \left (i c x +1\right ) x^{2}}{4}\)	\(294\)

[In]

int((e*x^2+d)^2*(a+b*arctan(c*x))/x^3,x,method=_RETURNVERBOSE)

[Out]

a*(1/2*e^2*x^2+2*e*d*ln(x)-1/2*d^2/x^2)+b*c^2*(1/2*arctan(c*x)/c^2*e^2*x^2+2*arctan(c*x)/c^2*d*e*ln(c*x)-1/2*a
rctan(c*x)*d^2/c^2/x^2-1/2/c^4*(c*x*e^2+c^3*d^2/x+(c^4*d^2-e^2)*arctan(c*x)+4*c^2*d*e*(-1/2*I*ln(c*x)*ln(1+I*c
*x)+1/2*I*ln(c*x)*ln(1-I*c*x)-1/2*I*dilog(1+I*c*x)+1/2*I*dilog(1-I*c*x))))

Fricas [F]

\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

integrate((e*x^2+d)^2*(a+b*arctan(c*x))/x^3,x, algorithm="fricas")

[Out]

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*arctan(c*x))/x^3, x)

Sympy [F]

\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \]

[In]

integrate((e*x**2+d)**2*(a+b*atan(c*x))/x**3,x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**2/x**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.20 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=\frac {1}{2} \, a e^{2} x^{2} - \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b d^{2} + 2 \, a d e \log \left (x\right ) - \frac {a d^{2}}{2 \, x^{2}} - \frac {\pi b c^{2} d e \log \left (c^{2} x^{2} + 1\right ) - 4 \, b c^{2} d e \arctan \left (c x\right ) \log \left (c x\right ) + 2 i \, b c^{2} d e {\rm Li}_2\left (i \, c x + 1\right ) - 2 i \, b c^{2} d e {\rm Li}_2\left (-i \, c x + 1\right ) + b c e^{2} x - {\left (b c^{2} e^{2} x^{2} + b e^{2}\right )} \arctan \left (c x\right )}{2 \, c^{2}} \]

[In]

integrate((e*x^2+d)^2*(a+b*arctan(c*x))/x^3,x, algorithm="maxima")

[Out]

1/2*a*e^2*x^2 - 1/2*((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*b*d^2 + 2*a*d*e*log(x) - 1/2*a*d^2/x^2 - 1/2*(
pi*b*c^2*d*e*log(c^2*x^2 + 1) - 4*b*c^2*d*e*arctan(c*x)*log(c*x) + 2*I*b*c^2*d*e*dilog(I*c*x + 1) - 2*I*b*c^2*
d*e*dilog(-I*c*x + 1) + b*c*e^2*x - (b*c^2*e^2*x^2 + b*e^2)*arctan(c*x))/c^2

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

integrate((e*x^2+d)^2*(a+b*arctan(c*x))/x^3,x, algorithm="giac")

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d+e x^2\right )^2 (a+b \arctan (c x))}{x^3} \, dx=\left \{\begin {array}{cl} \frac {a\,\left (e^2\,x^4-d^2+4\,d\,e\,x^2\,\ln \left (x\right )\right )}{2\,x^2} & \text {\ if\ \ }c=0\\ \frac {a\,\left (e^2\,x^4-d^2+4\,d\,e\,x^2\,\ln \left (x\right )\right )}{2\,x^2}-b\,e^2\,\left (\frac {x}{2\,c}-\mathrm {atan}\left (c\,x\right )\,\left (\frac {1}{2\,c^2}+\frac {x^2}{2}\right )\right )-\frac {b\,d^2\,\left (c^3\,\mathrm {atan}\left (c\,x\right )+\frac {c^2}{x}\right )}{2\,c}-\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{2\,x^2}-b\,d\,e\,\left ({\mathrm {Li}}_{\mathrm {2}}\left (1-c\,x\,1{}\mathrm {i}\right )-{\mathrm {Li}}_{\mathrm {2}}\left (1+c\,x\,1{}\mathrm {i}\right )\right )\,1{}\mathrm {i} & \text {\ if\ \ }c\neq 0 \end {array}\right . \]

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^2)/x^3,x)

[Out]

piecewise(c == 0, (a*(- d^2 + e^2*x^4 + 4*d*e*x^2*log(x)))/(2*x^2), c ~= 0, - b*e^2*(x/(2*c) - atan(c*x)*(1/(2
*c^2) + x^2/2)) + (a*(- d^2 + e^2*x^4 + 4*d*e*x^2*log(x)))/(2*x^2) - b*d*e*(dilog(- c*x*1i + 1) - dilog(c*x*1i
+ 1))*1i - (b*d^2*(c^3*atan(c*x) + c^2/x))/(2*c) - (b*d^2*atan(c*x))/(2*x^2))