\(\int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx\) [135]

Optimal result

Integrand size = 18, antiderivative size = 228 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 (d e+f-c f) (d e-(1+c) f) \arctan (c+d x)}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}+\frac {b d^2 (d e-c f) \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5154, 2007, 723, 814, 648, 632, 210, 642} \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 \arctan (c+d x) (-c f+d e+f) (d e-(c+1) f)}{2 f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )^2}+\frac {b d^2 (d e-c f) \log \left (c^2+2 c d x+d^2 x^2+1\right )}{2 \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )^2}+\frac {b d}{2 (e+f x) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}-\frac {b d^2 (d e-c f) \log (e+f x)}{\left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )^2} \]

[In]

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

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 723

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m
+ 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c
*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[m, -1]

Rule 814

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

Rule 2007

Int[(u_)^(m_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*ExpandToSum[v, x]^p, x] /; FreeQ[{m, p}, x] &&
LinearQ[u, x] && QuadraticQ[v, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[v, x])

Rule 5154

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

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {(b d) \int \frac {1}{(e+f x)^2 \left (1+(c+d x)^2\right )} \, dx}{2 f} \\ & = -\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {(b d) \int \frac {1}{(e+f x)^2 \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{2 f} \\ & = \frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {(b d) \int \frac {d (d e-2 c f)-d^2 f x}{(e+f x) \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ & = \frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {(b d) \int \left (\frac {2 d f^2 (d e-c f)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}+\frac {d^2 \left (d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2-2 d f (d e-c f) x\right )}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+c^2+2 c d x+d^2 x^2\right )}\right ) \, dx}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ & = \frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {\left (b d^3\right ) \int \frac {d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2-2 d f (d e-c f) x}{1+c^2+2 c d x+d^2 x^2} \, dx}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ & = \frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}+\frac {\left (b d^2 (d e-c f)\right ) \int \frac {2 c d+2 d^2 x}{1+c^2+2 c d x+d^2 x^2} \, dx}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {\left (b d \left (4 c d^2 f (d e-c f)+2 d^2 \left (d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2\right )\right )\right ) \int \frac {1}{1+c^2+2 c d x+d^2 x^2} \, dx}{4 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ & = \frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}+\frac {b d^2 (d e-c f) \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}+\frac {\left (b d \left (4 c d^2 f (d e-c f)+2 d^2 \left (d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 d^2-x^2} \, dx,x,2 c d+2 d^2 x\right )}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ & = \frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{2 f (e+f x)^2}-\frac {b d^2 (d e-f-c f) (d e+f-c f) \arctan (c+d x)}{2 f \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right )^2}-\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}+\frac {b d^2 (d e-c f) \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.47 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.79 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=\frac {\frac {b d f}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^2}+\frac {i b d^2 \log (i-c-d x)}{2 (d e-(-i+c) f)^2}-\frac {i b d^2 \log (i+c+d x)}{2 (d e-(i+c) f)^2}-\frac {2 b d^2 f (d e-c f) \log (d (e+f x))}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}}{2 f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.07

method	result	size
parts	\(-\frac {a}{2 \left (f x +e \right )^{2} f}+\frac {b \left (-\frac {d^{3} \operatorname {arccot}\left (d x +c \right )}{2 \left (f \left (d x +c \right )-c f +d e \right )^{2} f}-\frac {d^{3} \left (-\frac {f}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right ) \left (f \left (d x +c \right )-c f +d e \right )}-\frac {2 \left (c f -d e \right ) f \ln \left (f \left (d x +c \right )-c f +d e \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}+\frac {\frac {\left (2 c \,f^{2}-2 d e f \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-f^{2}\right ) \arctan \left (d x +c \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}\right )}{2 f}\right )}{d}\)	\(245\)
derivativedivides	\(\frac {-\frac {a \,d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-b \,d^{3} \left (\frac {\operatorname {arccot}\left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}+\frac {\frac {f}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right ) \left (c f -d e -f \left (d x +c \right )\right )}-\frac {2 f \left (c f -d e \right ) \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}+\frac {\frac {\left (2 c \,f^{2}-2 d e f \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-f^{2}\right ) \arctan \left (d x +c \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}}{2 f}\right )}{d}\)	\(260\)
default	\(\frac {-\frac {a \,d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-b \,d^{3} \left (\frac {\operatorname {arccot}\left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}+\frac {\frac {f}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right ) \left (c f -d e -f \left (d x +c \right )\right )}-\frac {2 f \left (c f -d e \right ) \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}+\frac {\frac {\left (2 c \,f^{2}-2 d e f \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-f^{2}\right ) \arctan \left (d x +c \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}}{2 f}\right )}{d}\)	\(260\)
parallelrisch	\(\frac {-a \,d^{2} f^{5}-2 x^{2} \operatorname {arccot}\left (d x +c \right ) b c \,d^{5} e \,f^{4}+2 x \,\operatorname {arccot}\left (d x +c \right ) b \,c^{2} d^{4} e \,f^{4}-4 x \,\operatorname {arccot}\left (d x +c \right ) b c \,d^{5} e^{2} f^{3}+4 \ln \left (f x +e \right ) x b c \,d^{4} e \,f^{4}-2 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x b c \,d^{4} e \,f^{4}+x^{2} \operatorname {arccot}\left (d x +c \right ) b \,c^{2} d^{4} f^{5}+x^{2} \operatorname {arccot}\left (d x +c \right ) b \,d^{6} e^{2} f^{3}+4 \,\operatorname {arccot}\left (d x +c \right ) b c \,d^{3} e \,f^{4}-2 x b c \,d^{4} e \,f^{4}+2 \ln \left (f x +e \right ) x^{2} b c \,d^{4} f^{5}-2 \ln \left (f x +e \right ) x^{2} b \,d^{5} e \,f^{4}-\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{2} b c \,d^{4} f^{5}+\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{2} b \,d^{5} e \,f^{4}-4 \ln \left (f x +e \right ) x b \,d^{5} e^{2} f^{3}+2 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x b \,d^{5} e^{2} f^{3}+2 \ln \left (f x +e \right ) b c \,d^{4} e^{2} f^{3}-\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b c \,d^{4} e^{2} f^{3}+2 x \,\operatorname {arccot}\left (d x +c \right ) b \,d^{6} e^{3} f^{2}+4 \,\operatorname {arccot}\left (d x +c \right ) b \,c^{3} d^{3} e \,f^{4}-5 \,\operatorname {arccot}\left (d x +c \right ) b \,c^{2} d^{4} e^{2} f^{3}+2 \,\operatorname {arccot}\left (d x +c \right ) b c \,d^{5} e^{3} f^{2}-2 x \,\operatorname {arccot}\left (d x +c \right ) b \,d^{4} e \,f^{4}+b \,d^{5} e^{3} f^{2}-2 a \,d^{4} e^{2} f^{3}+b \,d^{3} e \,f^{4}-a \,c^{4} d^{2} f^{5}-2 a \,c^{2} d^{2} f^{5}+x b \,d^{3} f^{5}-\operatorname {arccot}\left (d x +c \right ) b \,d^{2} f^{5}+4 a c \,d^{5} e^{3} f^{2}-a \,e^{4} f \,d^{6}+4 a \,c^{3} d^{3} e \,f^{4}-6 a \,c^{2} d^{4} e^{2} f^{3}+b \,c^{2} d^{3} e \,f^{4}-2 b c \,d^{4} e^{2} f^{3}+4 a c \,d^{3} e \,f^{4}-x^{2} \operatorname {arccot}\left (d x +c \right ) b \,d^{4} f^{5}-\operatorname {arccot}\left (d x +c \right ) b \,c^{4} d^{2} f^{5}+x b \,c^{2} d^{3} f^{5}+x b \,d^{5} e^{2} f^{3}-2 \,\operatorname {arccot}\left (d x +c \right ) b \,c^{2} d^{2} f^{5}-3 \,\operatorname {arccot}\left (d x +c \right ) b \,d^{4} e^{2} f^{3}-2 \ln \left (f x +e \right ) b \,d^{5} e^{3} f^{2}+\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,d^{5} e^{3} f^{2}}{2 \left (f x +e \right )^{2} \left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}+2 c^{2} f^{4}-4 c d e \,f^{3}+2 d^{2} e^{2} f^{2}+f^{4}\right ) d^{2} f^{2}}\)	\(888\)
risch	\(\text {Expression too large to display}\)	\(13315\)

[In]

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

[Out]

-1/2*a/(f*x+e)^2/f+b/d*(-1/2*d^3/(f*(d*x+c)-c*f+d*e)^2/f*arccot(d*x+c)-1/2*d^3/f*(-f/(c^2*f^2-2*c*d*e*f+d^2*e^
2+f^2)/(f*(d*x+c)-c*f+d*e)-2*(c*f-d*e)*f/(c^2*f^2-2*c*d*e*f+d^2*e^2+f^2)^2*ln(f*(d*x+c)-c*f+d*e)+1/(c^2*f^2-2*
c*d*e*f+d^2*e^2+f^2)^2*(1/2*(2*c*f^2-2*d*e*f)*ln(1+(d*x+c)^2)+(c^2*f^2-2*c*d*e*f+d^2*e^2-f^2)*arctan(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (220) = 440\).

Time = 1.21 (sec) , antiderivative size = 728, normalized size of antiderivative = 3.19 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=-\frac {a d^{4} e^{4} - {\left (4 \, a c + b\right )} d^{3} e^{3} f + 2 \, {\left (3 \, a c^{2} + b c + a\right )} d^{2} e^{2} f^{2} - {\left (4 \, a c^{3} + b c^{2} + 4 \, a c + b\right )} d e f^{3} + {\left (a c^{4} + 2 \, a c^{2} + a\right )} f^{4} - {\left (b d^{3} e^{2} f^{2} - 2 \, b c d^{2} e f^{3} + {\left (b c^{2} + b\right )} d f^{4}\right )} x + {\left (b d^{4} e^{4} - 4 \, b c d^{3} e^{3} f + 2 \, {\left (3 \, b c^{2} + b\right )} d^{2} e^{2} f^{2} - 4 \, {\left (b c^{3} + b c\right )} d e f^{3} + {\left (b c^{4} + 2 \, b c^{2} + b\right )} f^{4}\right )} \operatorname {arccot}\left (d x + c\right ) + {\left (b d^{4} e^{4} - 2 \, b c d^{3} e^{3} f + {\left (b c^{2} - b\right )} d^{2} e^{2} f^{2} + {\left (b d^{4} e^{2} f^{2} - 2 \, b c d^{3} e f^{3} + {\left (b c^{2} - b\right )} d^{2} f^{4}\right )} x^{2} + 2 \, {\left (b d^{4} e^{3} f - 2 \, b c d^{3} e^{2} f^{2} + {\left (b c^{2} - b\right )} d^{2} e f^{3}\right )} x\right )} \arctan \left (d x + c\right ) - {\left (b d^{3} e^{3} f - b c d^{2} e^{2} f^{2} + {\left (b d^{3} e f^{3} - b c d^{2} f^{4}\right )} x^{2} + 2 \, {\left (b d^{3} e^{2} f^{2} - b c d^{2} e f^{3}\right )} x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + 2 \, {\left (b d^{3} e^{3} f - b c d^{2} e^{2} f^{2} + {\left (b d^{3} e f^{3} - b c d^{2} f^{4}\right )} x^{2} + 2 \, {\left (b d^{3} e^{2} f^{2} - b c d^{2} e f^{3}\right )} x\right )} \log \left (f x + e\right )}{2 \, {\left (d^{4} e^{6} f - 4 \, c d^{3} e^{5} f^{2} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{4} f^{3} - 4 \, {\left (c^{3} + c\right )} d e^{3} f^{4} + {\left (c^{4} + 2 \, c^{2} + 1\right )} e^{2} f^{5} + {\left (d^{4} e^{4} f^{3} - 4 \, c d^{3} e^{3} f^{4} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{5} - 4 \, {\left (c^{3} + c\right )} d e f^{6} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{7}\right )} x^{2} + 2 \, {\left (d^{4} e^{5} f^{2} - 4 \, c d^{3} e^{4} f^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{3} f^{4} - 4 \, {\left (c^{3} + c\right )} d e^{2} f^{5} + {\left (c^{4} + 2 \, c^{2} + 1\right )} e f^{6}\right )} x\right )}} \]

[In]

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

[Out]

-1/2*(a*d^4*e^4 - (4*a*c + b)*d^3*e^3*f + 2*(3*a*c^2 + b*c + a)*d^2*e^2*f^2 - (4*a*c^3 + b*c^2 + 4*a*c + b)*d*
e*f^3 + (a*c^4 + 2*a*c^2 + a)*f^4 - (b*d^3*e^2*f^2 - 2*b*c*d^2*e*f^3 + (b*c^2 + b)*d*f^4)*x + (b*d^4*e^4 - 4*b
*c*d^3*e^3*f + 2*(3*b*c^2 + b)*d^2*e^2*f^2 - 4*(b*c^3 + b*c)*d*e*f^3 + (b*c^4 + 2*b*c^2 + b)*f^4)*arccot(d*x +
c) + (b*d^4*e^4 - 2*b*c*d^3*e^3*f + (b*c^2 - b)*d^2*e^2*f^2 + (b*d^4*e^2*f^2 - 2*b*c*d^3*e*f^3 + (b*c^2 - b)*
d^2*f^4)*x^2 + 2*(b*d^4*e^3*f - 2*b*c*d^3*e^2*f^2 + (b*c^2 - b)*d^2*e*f^3)*x)*arctan(d*x + c) - (b*d^3*e^3*f -
b*c*d^2*e^2*f^2 + (b*d^3*e*f^3 - b*c*d^2*f^4)*x^2 + 2*(b*d^3*e^2*f^2 - b*c*d^2*e*f^3)*x)*log(d^2*x^2 + 2*c*d*
x + c^2 + 1) + 2*(b*d^3*e^3*f - b*c*d^2*e^2*f^2 + (b*d^3*e*f^3 - b*c*d^2*f^4)*x^2 + 2*(b*d^3*e^2*f^2 - b*c*d^2
*e*f^3)*x)*log(f*x + e))/(d^4*e^6*f - 4*c*d^3*e^5*f^2 + 2*(3*c^2 + 1)*d^2*e^4*f^3 - 4*(c^3 + c)*d*e^3*f^4 + (c
^4 + 2*c^2 + 1)*e^2*f^5 + (d^4*e^4*f^3 - 4*c*d^3*e^3*f^4 + 2*(3*c^2 + 1)*d^2*e^2*f^5 - 4*(c^3 + c)*d*e*f^6 + (
c^4 + 2*c^2 + 1)*f^7)*x^2 + 2*(d^4*e^5*f^2 - 4*c*d^3*e^4*f^3 + 2*(3*c^2 + 1)*d^2*e^3*f^4 - 4*(c^3 + c)*d*e^2*f
^5 + (c^4 + 2*c^2 + 1)*e*f^6)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=\text {Timed out} \]

[In]

integrate((a+b*acot(d*x+c))/(f*x+e)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.80 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=\frac {1}{2} \, {\left (d {\left (\frac {{\left (d^{2} e - c d f\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{2} - 4 \, {\left (c^{3} + c\right )} d e f^{3} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{4}} - \frac {2 \, {\left (d^{2} e - c d f\right )} \log \left (f x + e\right )}{d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{2} - 4 \, {\left (c^{3} + c\right )} d e f^{3} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{4}} - \frac {{\left (d^{4} e^{2} - 2 \, c d^{3} e f + {\left (c^{2} - 1\right )} d^{2} f^{2}\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{{\left (d^{4} e^{4} f - 4 \, c d^{3} e^{3} f^{2} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{3} - 4 \, {\left (c^{3} + c\right )} d e f^{4} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{5}\right )} d} + \frac {1}{d^{2} e^{3} - 2 \, c d e^{2} f + {\left (c^{2} + 1\right )} e f^{2} + {\left (d^{2} e^{2} f - 2 \, c d e f^{2} + {\left (c^{2} + 1\right )} f^{3}\right )} x}\right )} - \frac {\operatorname {arccot}\left (d x + c\right )}{f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f}\right )} b - \frac {a}{2 \, {\left (f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f\right )}} \]

[In]

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

[Out]

1/2*(d*((d^2*e - c*d*f)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^4*e^4 - 4*c*d^3*e^3*f + 2*(3*c^2 + 1)*d^2*e^2*f^2
- 4*(c^3 + c)*d*e*f^3 + (c^4 + 2*c^2 + 1)*f^4) - 2*(d^2*e - c*d*f)*log(f*x + e)/(d^4*e^4 - 4*c*d^3*e^3*f + 2*(
3*c^2 + 1)*d^2*e^2*f^2 - 4*(c^3 + c)*d*e*f^3 + (c^4 + 2*c^2 + 1)*f^4) - (d^4*e^2 - 2*c*d^3*e*f + (c^2 - 1)*d^2
*f^2)*arctan((d^2*x + c*d)/d)/((d^4*e^4*f - 4*c*d^3*e^3*f^2 + 2*(3*c^2 + 1)*d^2*e^2*f^3 - 4*(c^3 + c)*d*e*f^4
+ (c^4 + 2*c^2 + 1)*f^5)*d) + 1/(d^2*e^3 - 2*c*d*e^2*f + (c^2 + 1)*e*f^2 + (d^2*e^2*f - 2*c*d*e*f^2 + (c^2 + 1
)*f^3)*x)) - arccot(d*x + c)/(f^3*x^2 + 2*e*f^2*x + e^2*f))*b - 1/2*a/(f^3*x^2 + 2*e*f^2*x + e^2*f)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6173 vs. \(2 (220) = 440\).

Time = 2.13 (sec) , antiderivative size = 6173, normalized size of antiderivative = 27.07 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*(4*b*d^4*e^3*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c)))^3 - 12*b*c*d^3*e^2*f*arctan(1/(d*x + c))*ta
n(1/2*arctan(1/(d*x + c)))^3 + 12*b*c^2*d^2*e*f^2*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c)))^3 - 4*b*c^3
*d*f^3*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c)))^3 - b*d^3*e^2*f*arctan(1/(d*x + c))*tan(1/2*arctan(1/(
d*x + c)))^4 + 2*b*c*d^2*e*f^2*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c)))^4 - b*c^2*d*f^3*arctan(1/(d*x
+ c))*tan(1/2*arctan(1/(d*x + c)))^4 + 4*b*d^4*e^3*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f
*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c
)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1
/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arc
tan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^2 - 12*b*c*d^3*e^2*f
*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/
2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f
^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))
) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)
))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^2 + 12*b*c^2*d^2*e*f^2*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2
- 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arcta
n(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(
1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/
(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^2 - 4*b*
c^3*d*f^3*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f
^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c
)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(
d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/
(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^2 + 4*a*d^4*e^3*tan(1/2*arctan(1/(d*x + c)))^3 - 12*a*c*d^3*e
^2*f*tan(1/2*arctan(1/(d*x + c)))^3 + 12*a*c^2*d^2*e*f^2*tan(1/2*arctan(1/(d*x + c)))^3 - 4*a*c^3*d*f^3*tan(1/
2*arctan(1/(d*x + c)))^3 - 4*b*d^3*e^2*f*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*a
rctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*
c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c)
)) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x
+ c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^3 + 8*b*c*d^2*e*f^2*log(4*(4*d
^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/
(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*
arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*t
an(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*
tan(1/2*arctan(1/(d*x + c)))^3 - 4*b*c^2*d*f^3*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan
(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^
3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*
x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(
1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^3 - a*d^3*e^2*f*tan(1/2*
arctan(1/(d*x + c)))^4 + 2*a*c*d^2*e*f^2*tan(1/2*arctan(1/(d*x + c)))^4 - a*c^2*d*f^3*tan(1/2*arctan(1/(d*x +
c)))^4 + b*d^2*e*f^2*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^
2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan
(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2
*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/
2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^4 - b*c*d*f^3*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d
*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*ta
n(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4
*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c))
)^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)
))^4 - 4*b*d^4*e^3*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c))) + 12*b*c*d^3*e^2*f*arctan(1/(d*x + c))*tan
(1/2*arctan(1/(d*x + c))) - 12*b*c^2*d^2*e*f^2*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c))) + 4*b*c^3*d*f^
3*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c))) + 14*b*d^3*e^2*f*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x
+ c)))^2 - 28*b*c*d^2*e*f^2*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c)))^2 + 14*b*c^2*d*f^3*arctan(1/(d*x
+ c))*tan(1/2*arctan(1/(d*x + c)))^2 - 2*b*d^3*e^2*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*b*c*d^2*e*f^2*tan(1/2*
arctan(1/(d*x + c)))^3 - 2*b*c^2*d*f^3*tan(1/2*arctan(1/(d*x + c)))^3 - 4*b*d^2*e*f^2*arctan(1/(d*x + c))*tan(
1/2*arctan(1/(d*x + c)))^3 + 4*b*c*d*f^3*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c)))^3 + b*d^2*e*f^2*tan(
1/2*arctan(1/(d*x + c)))^4 - b*c*d*f^3*tan(1/2*arctan(1/(d*x + c)))^4 + b*d*f^3*arctan(1/(d*x + c))*tan(1/2*ar
ctan(1/(d*x + c)))^4 - 4*a*d^4*e^3*tan(1/2*arctan(1/(d*x + c))) + 12*a*c*d^3*e^2*f*tan(1/2*arctan(1/(d*x + c))
) - 12*a*c^2*d^2*e*f^2*tan(1/2*arctan(1/(d*x + c))) + 4*a*c^3*d*f^3*tan(1/2*arctan(1/(d*x + c))) + 4*b*d^3*e^2
*f*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(
1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 +
f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c
))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x +
c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c))) - 8*b*c*d^2*e*f^2*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 -
8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1
/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2
*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(ta
n(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c))) + 4*b*c^2*d
*f^3*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*ta
n(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3
+ f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x +
c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x
+ c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c))) + 14*a*d^3*e^2*f*tan(1/2*arctan(1/(d*x + c)))^2 - 28*a*c*d^2*e*f^2
*tan(1/2*arctan(1/(d*x + c)))^2 + 14*a*c^2*d*f^3*tan(1/2*arctan(1/(d*x + c)))^2 - 2*b*d^2*e*f^2*log(4*(4*d^2*e
^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x
+ c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arct
an(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1
/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(
1/2*arctan(1/(d*x + c)))^2 + 2*b*c*d*f^3*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*a
rctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*
c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c)
)) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x
+ c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1))*tan(1/2*arctan(1/(d*x + c)))^2 - 4*a*d^2*e*f^2*tan(1/2*arct
an(1/(d*x + c)))^3 + 4*a*c*d*f^3*tan(1/2*arctan(1/(d*x + c)))^3 + a*d*f^3*tan(1/2*arctan(1/(d*x + c)))^4 - b*d
^3*e^2*f*arctan(1/(d*x + c)) + 2*b*c*d^2*e*f^2*arctan(1/(d*x + c)) - b*c^2*d*f^3*arctan(1/(d*x + c)) + 2*b*d^3
*e^2*f*tan(1/2*arctan(1/(d*x + c))) - 4*b*c*d^2*e*f^2*tan(1/2*arctan(1/(d*x + c))) + 2*b*c^2*d*f^3*tan(1/2*arc
tan(1/(d*x + c))) + 4*b*d^2*e*f^2*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c))) - 4*b*c*d*f^3*arctan(1/(d*x
+ c))*tan(1/2*arctan(1/(d*x + c))) - 6*b*d^2*e*f^2*tan(1/2*arctan(1/(d*x + c)))^2 + 6*b*c*d*f^3*tan(1/2*arcta
n(1/(d*x + c)))^2 + 2*b*d*f^3*arctan(1/(d*x + c))*tan(1/2*arctan(1/(d*x + c)))^2 + 2*b*d*f^3*tan(1/2*arctan(1/
(d*x + c)))^3 - a*d^3*e^2*f + 2*a*c*d^2*e*f^2 - a*c^2*d*f^3 + b*d^2*e*f^2*log(4*(4*d^2*e^2*tan(1/2*arctan(1/(d
*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 4*d*e*f*ta
n(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arctan(1/(d*x + c)))^4 + 4
*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1/2*arctan(1/(d*x + c))
)^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1)) - b*c*d*f^3*log(4*(4*d^2*e
^2*tan(1/2*arctan(1/(d*x + c)))^2 - 8*c*d*e*f*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^2*f^2*tan(1/2*arctan(1/(d*x
+ c)))^2 - 4*d*e*f*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^2*tan(1/2*arctan(1/(d*x + c)))^3 + f^2*tan(1/2*arct
an(1/(d*x + c)))^4 + 4*d*e*f*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^2*tan(1/2*arctan(1/(d*x + c))) - 2*f^2*tan(1
/2*arctan(1/(d*x + c)))^2 + f^2)/(tan(1/2*arctan(1/(d*x + c)))^4 + 2*tan(1/2*arctan(1/(d*x + c)))^2 + 1)) + 4*
a*d^2*e*f^2*tan(1/2*arctan(1/(d*x + c))) - 4*a*c*d*f^3*tan(1/2*arctan(1/(d*x + c))) + 2*a*d*f^3*tan(1/2*arctan
(1/(d*x + c)))^2 + b*d^2*e*f^2 - b*c*d*f^3 + b*d*f^3*arctan(1/(d*x + c)) - 2*b*d*f^3*tan(1/2*arctan(1/(d*x + c
))) + a*d*f^3)*d/(4*d^6*e^6*tan(1/2*arctan(1/(d*x + c)))^2 - 24*c*d^5*e^5*f*tan(1/2*arctan(1/(d*x + c)))^2 + 6
0*c^2*d^4*e^4*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 80*c^3*d^3*e^3*f^3*tan(1/2*arctan(1/(d*x + c)))^2 + 60*c^4*
d^2*e^2*f^4*tan(1/2*arctan(1/(d*x + c)))^2 - 24*c^5*d*e*f^5*tan(1/2*arctan(1/(d*x + c)))^2 + 4*c^6*f^6*tan(1/2
*arctan(1/(d*x + c)))^2 - 4*d^5*e^5*f*tan(1/2*arctan(1/(d*x + c)))^3 + 20*c*d^4*e^4*f^2*tan(1/2*arctan(1/(d*x
+ c)))^3 - 40*c^2*d^3*e^3*f^3*tan(1/2*arctan(1/(d*x + c)))^3 + 40*c^3*d^2*e^2*f^4*tan(1/2*arctan(1/(d*x + c)))
^3 - 20*c^4*d*e*f^5*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c^5*f^6*tan(1/2*arctan(1/(d*x + c)))^3 + d^4*e^4*f^2*ta
n(1/2*arctan(1/(d*x + c)))^4 - 4*c*d^3*e^3*f^3*tan(1/2*arctan(1/(d*x + c)))^4 + 6*c^2*d^2*e^2*f^4*tan(1/2*arct
an(1/(d*x + c)))^4 - 4*c^3*d*e*f^5*tan(1/2*arctan(1/(d*x + c)))^4 + c^4*f^6*tan(1/2*arctan(1/(d*x + c)))^4 + 4
*d^5*e^5*f*tan(1/2*arctan(1/(d*x + c))) - 20*c*d^4*e^4*f^2*tan(1/2*arctan(1/(d*x + c))) + 40*c^2*d^3*e^3*f^3*t
an(1/2*arctan(1/(d*x + c))) - 40*c^3*d^2*e^2*f^4*tan(1/2*arctan(1/(d*x + c))) + 20*c^4*d*e*f^5*tan(1/2*arctan(
1/(d*x + c))) - 4*c^5*f^6*tan(1/2*arctan(1/(d*x + c))) + 6*d^4*e^4*f^2*tan(1/2*arctan(1/(d*x + c)))^2 - 24*c*d
^3*e^3*f^3*tan(1/2*arctan(1/(d*x + c)))^2 + 36*c^2*d^2*e^2*f^4*tan(1/2*arctan(1/(d*x + c)))^2 - 24*c^3*d*e*f^5
*tan(1/2*arctan(1/(d*x + c)))^2 + 6*c^4*f^6*tan(1/2*arctan(1/(d*x + c)))^2 - 8*d^3*e^3*f^3*tan(1/2*arctan(1/(d
*x + c)))^3 + 24*c*d^2*e^2*f^4*tan(1/2*arctan(1/(d*x + c)))^3 - 24*c^2*d*e*f^5*tan(1/2*arctan(1/(d*x + c)))^3
+ 8*c^3*f^6*tan(1/2*arctan(1/(d*x + c)))^3 + 2*d^2*e^2*f^4*tan(1/2*arctan(1/(d*x + c)))^4 - 4*c*d*e*f^5*tan(1/
2*arctan(1/(d*x + c)))^4 + 2*c^2*f^6*tan(1/2*arctan(1/(d*x + c)))^4 + d^4*e^4*f^2 - 4*c*d^3*e^3*f^3 + 6*c^2*d^
2*e^2*f^4 - 4*c^3*d*e*f^5 + c^4*f^6 + 8*d^3*e^3*f^3*tan(1/2*arctan(1/(d*x + c))) - 24*c*d^2*e^2*f^4*tan(1/2*ar
ctan(1/(d*x + c))) + 24*c^2*d*e*f^5*tan(1/2*arctan(1/(d*x + c))) - 8*c^3*f^6*tan(1/2*arctan(1/(d*x + c))) - 4*
d*e*f^5*tan(1/2*arctan(1/(d*x + c)))^3 + 4*c*f^6*tan(1/2*arctan(1/(d*x + c)))^3 + f^6*tan(1/2*arctan(1/(d*x +
c)))^4 + 2*d^2*e^2*f^4 - 4*c*d*e*f^5 + 2*c^2*f^6 + 4*d*e*f^5*tan(1/2*arctan(1/(d*x + c))) - 4*c*f^6*tan(1/2*ar
ctan(1/(d*x + c))) - 2*f^6*tan(1/2*arctan(1/(d*x + c)))^2 + f^6)

Mupad [B] (verification not implemented)

Time = 8.33 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.75 \[ \int \frac {a+b \cot ^{-1}(c+d x)}{(e+f x)^3} \, dx=\frac {b\,d\,e}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {a\,f}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {b\,\mathrm {acot}\left (c+d\,x\right )}{2\,f\,{\left (e+f\,x\right )}^2}-\frac {a\,c^2\,f}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {b\,d^3\,e\,\ln \left (e+f\,x\right )}{{\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}^2}+\frac {b\,c\,d^2\,f\,\ln \left (e+f\,x\right )}{{\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}^2}+\frac {a\,c\,d\,e}{{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}+\frac {b\,d\,f\,x}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {a\,d^2\,e^2}{2\,f\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}+\frac {b\,d^2\,\ln \left (c+d\,x-\mathrm {i}\right )\,1{}\mathrm {i}}{4\,f\,{\left (d\,e-c\,f+f\,1{}\mathrm {i}\right )}^2}-\frac {b\,d^2\,\ln \left (c+d\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,f\,{\left (c\,f-d\,e+f\,1{}\mathrm {i}\right )}^2} \]

[In]

int((a + b*acot(c + d*x))/(e + f*x)^3,x)

[Out]

(b*d*e)/(2*(e + f*x)^2*(f^2 + c^2*f^2 + d^2*e^2 - 2*c*d*e*f)) - (a*f)/(2*(e + f*x)^2*(f^2 + c^2*f^2 + d^2*e^2
- 2*c*d*e*f)) - (b*acot(c + d*x))/(2*f*(e + f*x)^2) + (b*d^2*log(c + d*x - 1i)*1i)/(4*f*(f*1i - c*f + d*e)^2)
- (b*d^2*log(c + d*x + 1i)*1i)/(4*f*(f*1i + c*f - d*e)^2) - (a*c^2*f)/(2*(e + f*x)^2*(f^2 + c^2*f^2 + d^2*e^2
- 2*c*d*e*f)) - (b*d^3*e*log(e + f*x))/(f^2 + c^2*f^2 + d^2*e^2 - 2*c*d*e*f)^2 + (b*c*d^2*f*log(e + f*x))/(f^2
+ c^2*f^2 + d^2*e^2 - 2*c*d*e*f)^2 + (a*c*d*e)/((e + f*x)^2*(f^2 + c^2*f^2 + d^2*e^2 - 2*c*d*e*f)) + (b*d*f*x
)/(2*(e + f*x)^2*(f^2 + c^2*f^2 + d^2*e^2 - 2*c*d*e*f)) - (a*d^2*e^2)/(2*f*(e + f*x)^2*(f^2 + c^2*f^2 + d^2*e^
2 - 2*c*d*e*f))