3.13.92 \(\int \frac {(a+b \arctan (c x)) (d+e \log (1+c^2 x^2))}{x^2} \, dx\) [1292]

3.13.92.1 Optimal result

Integrand size = 26, antiderivative size = 100 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx=\frac {c e (a+b \arctan (c x))^2}{b}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}+\frac {1}{2} b c \left (d+e \log \left (1+c^2 x^2\right )\right ) \log \left (1-\frac {1}{1+c^2 x^2}\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {1}{1+c^2 x^2}\right ) \]

output

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

3.13.92.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx=\frac {c e (a+b \arctan (c x))^2}{b}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x}+b c \left (-\frac {\left (d+e \log \left (1+c^2 x^2\right )\right ) \left (d-2 e \log \left (-c^2 x^2\right )+e \log \left (1+c^2 x^2\right )\right )}{4 e}+\frac {1}{2} e \operatorname {PolyLog}\left (2,1+c^2 x^2\right )\right ) \]

input

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

output

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

3.13.92.3 Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5552, 2925, 2858, 25, 27, 2779, 2838, 5419}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x^2} \, dx\)

\(\Big \downarrow \) 5552

\(\displaystyle 2 c^2 e \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx+b c \int \frac {d+e \log \left (c^2 x^2+1\right )}{x \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}\)

\(\Big \downarrow \) 2925

\(\displaystyle 2 c^2 e \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx+\frac {1}{2} b c \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^2 \left (c^2 x^2+1\right )}dx^2-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}\)

\(\Big \downarrow \) 2858

\(\displaystyle 2 c^2 e \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx+\frac {b \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^4}d\left (c^2 x^2+1\right )}{2 c}-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}\)

\(\Big \downarrow \) 25

\(\displaystyle 2 c^2 e \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx-\frac {b \int -\frac {d+e \log \left (c^2 x^2+1\right )}{x^4}d\left (c^2 x^2+1\right )}{2 c}-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 c^2 e \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx-\frac {1}{2} b c \int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^2 x^4}d\left (c^2 x^2+1\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}\)

\(\Big \downarrow \) 2779

\(\displaystyle 2 c^2 e \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx-\frac {1}{2} b c \left (e \int \frac {\log \left (1-\frac {1}{x^2}\right )}{x^2}d\left (c^2 x^2+1\right )-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}\)

\(\Big \downarrow \) 2838

\(\displaystyle 2 c^2 e \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}-\frac {1}{2} b c \left (e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )\right )\)

\(\Big \downarrow \) 5419

\(\displaystyle -\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x}+\frac {c e (a+b \arctan (c x))^2}{b}-\frac {1}{2} b c \left (e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )\right )\)

input

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

output

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

3.13.92.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2838

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 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + 
e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - 
d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
 

rule 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], 
x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

rule 5419

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> 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 5552

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*( 
e_.))*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(d + e*Log[f + g*x^2])*((a + 
b*ArcTan[c*x])/(m + 1)), x] + (-Simp[b*(c/(m + 1))   Int[x^(m + 1)*((d + e* 
Log[f + g*x^2])/(1 + c^2*x^2)), x], x] - Simp[2*e*(g/(m + 1))   Int[x^(m + 
2)*((a + b*ArcTan[c*x])/(f + g*x^2)), x], x]) /; FreeQ[{a, b, c, d, e, f, g 
}, x] && ILtQ[m/2, 0]
 

3.13.92.4 Maple [F]

\[\int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (c^{2} x^{2}+1\right )\right )}{x^{2}}d x\]

input

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

output

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

3.13.92.5 Fricas [F]

\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{2}} \,d x } \]

input

integrate((a+b*arctan(c*x))*(d+e*log(c^2*x^2+1))/x^2,x, algorithm="fricas" 
)
 

output

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

3.13.92.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*atan(c*x))*(d+e*ln(c**2*x**2+1))/x**2,x)
 

output

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

3.13.92.7 Maxima [F]

\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{2}} \,d x } \]

input

integrate((a+b*arctan(c*x))*(d+e*log(c^2*x^2+1))/x^2,x, algorithm="maxima" 
)
 

output

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

3.13.92.8 Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx=\text {Timed out} \]

input

integrate((a+b*arctan(c*x))*(d+e*log(c^2*x^2+1))/x^2,x, algorithm="giac")
 

output

Timed out
 

3.13.92.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (c^2\,x^2+1\right )\right )}{x^2} \,d x \]

input

int(((a + b*atan(c*x))*(d + e*log(c^2*x^2 + 1)))/x^2,x)
 

output

int(((a + b*atan(c*x))*(d + e*log(c^2*x^2 + 1)))/x^2, x)