\(\int (e+f x) (a+b \cot ^{-1}(c+d x))^2 \, dx\) [137]

Optimal result

Integrand size = 18, antiderivative size = 220 \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {b^2 f \log \left (1+(c+d x)^2\right )}{2 d^2}+\frac {i b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5156, 4975, 4931, 266, 5105, 5005, 5041, 4965, 2449, 2352} \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}-\frac {(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2 f}-\frac {2 b (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 f}+\frac {a b f x}{d}+\frac {i b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{d^2}+\frac {b^2 f \log \left ((c+d x)^2+1\right )}{2 d^2}+\frac {b^2 f (c+d x) \cot ^{-1}(c+d x)}{d^2} \]

[In]

Int[(e + f*x)*(a + b*ArcCot[c + d*x])^2,x]

[Out]

(a*b*f*x)/d + (b^2*f*(c + d*x)*ArcCot[c + d*x])/d^2 + (I*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^2 - ((d*e +
f - c*f)*(d*e - (1 + c)*f)*(a + b*ArcCot[c + d*x])^2)/(2*d^2*f) + ((e + f*x)^2*(a + b*ArcCot[c + d*x])^2)/(2*f
) - (2*b*(d*e - c*f)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/d^2 + (b^2*f*Log[1 + (c + d*x)^2])/(2*d
^2) + (I*b^2*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/d^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 4931

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Dist[b*c
*n*p, Int[x^n*((a + b*ArcCot[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 4965

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] - Dist[b*c*(p/e), Int[(a + b*ArcCot[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 4975

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a
+ b*ArcCot[c*x])^p/(e*(q + 1))), x] + Dist[b*c*(p/(e*(q + 1))), Int[ExpandIntegrand[(a + b*ArcCot[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 5005

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-(a + b*ArcCot[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 5041

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

Rule 5105

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcCot[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 5156

Int[((a_.) + ArcCot[(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*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.30 \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {2 a^2 c d e+2 a b c f-a^2 c^2 f+2 a^2 d^2 e x+2 a b d f x+a^2 d^2 f x^2+b^2 (i+c+d x) (-((i+c) f)+d (2 e+f x)) \cot ^{-1}(c+d x)^2-2 a b f \arctan (c+d x)+2 b \cot ^{-1}(c+d x) \left (-((c+d x) (-b f+a c f-a d (2 e+f x)))-2 b (d e-c f) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )-4 a b d e \log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )-2 b^2 f \log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+4 a b c f \log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+2 i b^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )}{2 d^2} \]

[In]

Integrate[(e + f*x)*(a + b*ArcCot[c + d*x])^2,x]

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (210 ) = 420\).

Time = 0.88 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.94

method	result	size
parts	\(a^{2} \left (\frac {1}{2} f \,x^{2}+e x \right )+\frac {b^{2} \left (\frac {\operatorname {arccot}\left (d x +c \right )^{2} \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arccot}\left (d x +c \right )^{2} c f \left (d x +c \right )}{d}+\operatorname {arccot}\left (d x +c \right )^{2} e \left (d x +c \right )+\frac {-\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right ) c f +\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right ) d e -\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) f +\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right ) f +\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\arctan \left (d x +c \right )^{2} f}{2}+\frac {\left (-2 c f +2 d e \right ) \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2}}{d}\right )}{d}+\frac {2 a b \left (\frac {\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arccot}\left (d x +c \right ) c f \left (d x +c \right )}{d}+\operatorname {arccot}\left (d x +c \right ) e \left (d x +c \right )+\frac {f \left (d x +c \right )+\frac {\left (-2 c f +2 d e \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}-f \arctan \left (d x +c \right )}{2 d}\right )}{d}\)	\(427\)
derivativedivides	\(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arccot}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arccot}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}+\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right ) c f -\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right ) d e +\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) f -\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right ) f -\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {\arctan \left (d x +c \right )^{2} f}{2}+\frac {\left (2 c f -2 d e \right ) \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arccot}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccot}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccot}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}+\frac {f \arctan \left (d x +c \right )}{2}\right )}{d}}{d}\)	\(434\)
default	\(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arccot}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arccot}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arccot}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}+\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right ) c f -\operatorname {arccot}\left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right ) d e +\operatorname {arccot}\left (d x +c \right ) \arctan \left (d x +c \right ) f -\operatorname {arccot}\left (d x +c \right ) \left (d x +c \right ) f -\frac {f \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\frac {\arctan \left (d x +c \right )^{2} f}{2}+\frac {\left (2 c f -2 d e \right ) \left (-\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{2}\right )}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arccot}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccot}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccot}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{4}+\frac {f \arctan \left (d x +c \right )}{2}\right )}{d}}{d}\)	\(434\)
risch	\(\text {Expression too large to display}\)	\(1781\)

[In]

int((f*x+e)*(a+b*arccot(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((f*x+e)*(a+b*arccot(d*x+c))^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \]

[In]

integrate((f*x+e)*(a+b*acot(d*x+c))**2,x)

[Out]

Integral((a + b*acot(c + d*x))**2*(e + f*x), x)

Maxima [F]

\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((f*x+e)*(a+b*arccot(d*x+c))^2,x, algorithm="maxima")

[Out]

1/8*b^2*f*x^2*arctan2(1, d*x + c)^2 + 1/4*b^2*e*x*arctan2(1, d*x + c)^2 + 1/2*a^2*f*x^2 + (x^2*arccot(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*b*f + a^2*e*x
+ (2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a*b*e/d - 1/32*(b^2*f*x^2 + 2*b^2*e*x)*log(d^2*x^2 + 2
*c*d*x + c^2 + 1)^2 + integrate(1/16*(12*b^2*d^2*f*x^3*arctan2(1, d*x + c)^2 + 4*(3*b^2*d^2*e*arctan2(1, d*x +
c)^2 + (6*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*f)*x^2 + (b^2*d^2*f*x^3 + (b^2*d^2*e + 2*b
^2*c*d*f)*x^2 + (b^2*c^2 + b^2)*e + (2*b^2*c*d*e + (b^2*c^2 + b^2)*f)*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 +
12*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*e + 4*(2*(3*b^2*c*arctan2(1, d*x + c)^2 + b^2*a
rctan2(1, d*x + c))*d*e + 3*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*f)*x + 2*(b^2*d^2*f*x^
3 + 2*b^2*c*d*e*x + (2*b^2*d^2*e + b^2*c*d*f)*x^2)*log(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^2*x^2 + 2*c*d*x + c^2
+ 1), x)

Giac [F]

\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((f*x+e)*(a+b*arccot(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((f*x + e)*(b*arccot(d*x + c) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^2 \,d x \]

[In]

int((e + f*x)*(a + b*acot(c + d*x))^2,x)

[Out]

int((e + f*x)*(a + b*acot(c + d*x))^2, x)