3.13.94 \(\int \frac {(a+b \arctan (c x)) (d+e \log (1+c^2 x^2))}{x^4} \, dx\) [1294]

3.13.94.1 Optimal result

Integrand size = 26, antiderivative size = 189 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=-\frac {2 c^2 e (a+b \arctan (c x))}{3 x}-\frac {c^3 e (a+b \arctan (c x))^2}{3 b}+b c^3 e \log (x)-\frac {1}{3} b c^3 e \log \left (1+c^2 x^2\right )-\frac {b c \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{6 x^2}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{3 x^3}-\frac {1}{6} b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right ) \log \left (1-\frac {1}{1+c^2 x^2}\right )+\frac {1}{6} b c^3 e \operatorname {PolyLog}\left (2,\frac {1}{1+c^2 x^2}\right ) \]

output

-2/3*c^2*e*(a+b*arctan(c*x))/x-1/3*c^3*e*(a+b*arctan(c*x))^2/b+b*c^3*e*ln( 
x)-1/3*b*c^3*e*ln(c^2*x^2+1)-1/6*b*c*(c^2*x^2+1)*(d+e*ln(c^2*x^2+1))/x^2-1 
/3*(a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1))/x^3-1/6*b*c^3*(d+e*ln(c^2*x^2+1)) 
*ln(1-1/(c^2*x^2+1))+1/6*b*c^3*e*polylog(2,1/(c^2*x^2+1))
 

3.13.94.2 Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\frac {1}{12} \left (-\frac {8 c^2 e (a+b \arctan (c x))}{x}-\frac {4 c^3 e (a+b \arctan (c x))^2}{b}+6 b c^3 e \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )-\frac {2 b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2}-\frac {4 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^3}+\frac {b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{e}-2 b c^3 \left (\log \left (-c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )+e \operatorname {PolyLog}\left (2,1+c^2 x^2\right )\right )\right ) \]

input

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

output

((-8*c^2*e*(a + b*ArcTan[c*x]))/x - (4*c^3*e*(a + b*ArcTan[c*x])^2)/b + 6* 
b*c^3*e*(2*Log[x] - Log[1 + c^2*x^2]) - (2*b*c*(d + e*Log[1 + c^2*x^2]))/x 
^2 - (4*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^3 + (b*c^3*(d + e* 
Log[1 + c^2*x^2])^2)/e - 2*b*c^3*(Log[-(c^2*x^2)]*(d + e*Log[1 + c^2*x^2]) 
 + e*PolyLog[2, 1 + c^2*x^2]))/12
 

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

Time = 1.33 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {5552, 2925, 2858, 27, 2789, 2751, 16, 2779, 2838, 5453, 5361, 243, 47, 14, 16, 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.

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

\(\Big \downarrow \) 5552

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{3} b c \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^3 \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2925

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{6} b c \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^4 \left (c^2 x^2+1\right )}dx^2-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2858

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {b \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^6}d\left (c^2 x^2+1\right )}{6 c}-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{6} b c^3 \int \frac {d+e \log \left (c^2 x^2+1\right )}{c^4 x^6}d\left (c^2 x^2+1\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2789

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{6} b c^3 \left (\int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^2 x^4}d\left (c^2 x^2+1\right )+\int \frac {d+e \log \left (c^2 x^2+1\right )}{c^4 x^4}d\left (c^2 x^2+1\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2751

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{6} b c^3 \left (\int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^2 x^4}d\left (c^2 x^2+1\right )-e \int -\frac {1}{c^2 x^2}d\left (c^2 x^2+1\right )-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{6} b c^3 \left (\int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^2 x^4}d\left (c^2 x^2+1\right )-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}+e \log \left (-c^2 x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2779

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{6} b c^3 \left (e \int \frac {\log \left (1-\frac {1}{x^2}\right )}{x^2}d\left (c^2 x^2+1\right )-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2}{3} c^2 e \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {2}{3} c^2 e \left (\int \frac {a+b \arctan (c x)}{x^2}dx-c^2 \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {2}{3} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+b c \int \frac {1}{x \left (c^2 x^2+1\right )}dx-\frac {a+b \arctan (c x)}{x}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {2}{3} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx^2-\frac {a+b \arctan (c x)}{x}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {2}{3} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+\frac {1}{2} b c \left (\int \frac {1}{x^2}dx^2-c^2 \int \frac {1}{c^2 x^2+1}dx^2\right )-\frac {a+b \arctan (c x)}{x}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {2}{3} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+\frac {1}{2} b c \left (\log \left (x^2\right )-c^2 \int \frac {1}{c^2 x^2+1}dx^2\right )-\frac {a+b \arctan (c x)}{x}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {2}{3} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )-\frac {a+b \arctan (c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

\(\Big \downarrow \) 5419

\(\displaystyle -\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{3 x^3}+\frac {2}{3} c^2 e \left (-\frac {c (a+b \arctan (c x))^2}{2 b}-\frac {a+b \arctan (c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )\right )+\frac {1}{6} b c^3 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\)

input

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

output

(2*c^2*e*(-((a + b*ArcTan[c*x])/x) - (c*(a + b*ArcTan[c*x])^2)/(2*b) + (b* 
c*(Log[x^2] - Log[1 + c^2*x^2]))/2))/3 - ((a + b*ArcTan[c*x])*(d + e*Log[1 
 + c^2*x^2]))/(3*x^3) + (b*c^3*(e*Log[-(c^2*x^2)] - ((1 + c^2*x^2)*(d + e* 
Log[1 + c^2*x^2]))/(c^2*x^2) - Log[1 - x^(-2)]*(d + e*Log[1 + c^2*x^2]) + 
e*PolyLog[2, x^(-2)]))/6
 

3.13.94.3.1 Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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 47

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

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 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

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 5361

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] - Simp[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 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 5453

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

3.13.94.4 Maple [F]

\[\int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (c^{2} x^{2}+1\right )\right )}{x^{4}}d x\]

input

int((a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1))/x^4,x)
 

output

int((a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1))/x^4,x)
 

3.13.94.5 Fricas [F]

\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{4}} \,d x } \]

input

integrate((a+b*arctan(c*x))*(d+e*log(c^2*x^2+1))/x^4,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^4, x)
 

3.13.94.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a+b*atan(c*x))*(d+e*ln(c**2*x**2+1))/x**4,x)
 

output

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

3.13.94.7 Maxima [F]

\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{4}} \,d x } \]

input

integrate((a+b*arctan(c*x))*(d+e*log(c^2*x^2+1))/x^4,x, algorithm="maxima" 
)
 

output

1/6*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x^2)*c - 2*arctan(c*x)/x^3)* 
b*d - 1/3*(2*(c*arctan(c*x) + 1/x)*c^2 + log(c^2*x^2 + 1)/x^3)*a*e + b*e*i 
ntegrate(arctan(c*x)*log(c^2*x^2 + 1)/x^4, x) - 1/3*a*d/x^3
 

3.13.94.8 Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\text {Timed out} \]

input

integrate((a+b*arctan(c*x))*(d+e*log(c^2*x^2+1))/x^4,x, algorithm="giac")
 

output

Timed out
 

3.13.94.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (c^2\,x^2+1\right )\right )}{x^4} \,d x \]

input

int(((a + b*atan(c*x))*(d + e*log(c^2*x^2 + 1)))/x^4,x)
 

output

int(((a + b*atan(c*x))*(d + e*log(c^2*x^2 + 1)))/x^4, x)