\(\int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx\) [6]

Optimal result

Integrand size = 16, antiderivative size = 98 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx=\frac {b c^2 d \arctan (c x)}{e \left (c^2 d^2+e^2\right )}-\frac {a+b \arctan (c x)}{e (d+e x)}+\frac {b c \log (d+e x)}{c^2 d^2+e^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4972, 720, 31, 649, 209, 266} \[ \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx=-\frac {a+b \arctan (c x)}{e (d+e x)}+\frac {b c^2 d \arctan (c x)}{e \left (c^2 d^2+e^2\right )}-\frac {b c \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )}+\frac {b c \log (d+e x)}{c^2 d^2+e^2} \]

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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 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 649

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

Rule 720

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

Rule 4972

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

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx=\frac {-\frac {a+b \arctan (c x)}{d+e x}+\frac {b c \left (\left (\sqrt {-c^2} d-e\right ) \log \left (1-\sqrt {-c^2} x\right )-\left (\sqrt {-c^2} d+e\right ) \log \left (1+\sqrt {-c^2} x\right )+2 e \log (d+e x)\right )}{2 \left (c^2 d^2+e^2\right )}}{e} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.12

method	result	size
parts	\(-\frac {a}{\left (e x +d \right ) e}+\frac {b \left (-\frac {c^{2} \arctan \left (c x \right )}{\left (c e x +c d \right ) e}+\frac {c^{2} \left (\frac {e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}+\frac {-\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \arctan \left (c x \right )}{c^{2} d^{2}+e^{2}}\right )}{e}\right )}{c}\)	\(110\)
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\frac {e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}+\frac {-\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \arctan \left (c x \right )}{c^{2} d^{2}+e^{2}}}{e}\right )}{c}\)	\(114\)
default	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\arctan \left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\frac {e \ln \left (c e x +c d \right )}{c^{2} d^{2}+e^{2}}+\frac {-\frac {e \ln \left (c^{2} x^{2}+1\right )}{2}+d c \arctan \left (c x \right )}{c^{2} d^{2}+e^{2}}}{e}\right )}{c}\)	\(114\)
parallelrisch	\(-\frac {-2 b d x \arctan \left (c x \right ) c^{4} e +\ln \left (c^{2} x^{2}+1\right ) x b \,c^{3} e^{2}-2 \ln \left (e x +d \right ) x b \,c^{3} e^{2}+\ln \left (c^{2} x^{2}+1\right ) b \,c^{3} d e -2 \ln \left (e x +d \right ) b \,c^{3} d e +2 a \,c^{4} d^{2}+2 b \,e^{2} \arctan \left (c x \right ) c^{2}+2 a \,c^{2} e^{2}}{2 \left (e x +d \right ) c^{2} e \left (c^{2} d^{2}+e^{2}\right )}\)	\(140\)
risch	\(\frac {i b \ln \left (i c x +1\right )}{2 e \left (e x +d \right )}+\frac {-i b \,c^{2} d^{2} \ln \left (-i c x +1\right )-i b \,e^{2} \ln \left (-i c x +1\right )+2 \ln \left (-e x -d \right ) b c \,e^{2} x +2 \ln \left (-e x -d \right ) b c d e -2 a \,c^{2} d^{2}-2 a \,e^{2}-\ln \left (\left (-i c^{2} d +3 c e \right ) x +3 i e +c d \right ) b c \,e^{2} x -\ln \left (\left (-i c^{2} d +3 c e \right ) x +3 i e +c d \right ) b c d e +i \ln \left (\left (-i c^{2} d +3 c e \right ) x +3 i e +c d \right ) b \,c^{2} d e x +i \ln \left (\left (-i c^{2} d +3 c e \right ) x +3 i e +c d \right ) b \,c^{2} d^{2}-\ln \left (\left (-i c^{2} d -3 c e \right ) x +3 i e -c d \right ) b c \,e^{2} x -\ln \left (\left (-i c^{2} d -3 c e \right ) x +3 i e -c d \right ) b c d e -i \ln \left (\left (-i c^{2} d -3 c e \right ) x +3 i e -c d \right ) b \,c^{2} d e x -i \ln \left (\left (-i c^{2} d -3 c e \right ) x +3 i e -c d \right ) b \,c^{2} d^{2}}{2 \left (e x +d \right ) \left (i d c +e \right ) \left (-i d c +e \right ) e}\)	\(391\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.18 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx=-\frac {2 \, a c^{2} d^{2} + 2 \, a e^{2} - 2 \, {\left (b c^{2} d e x - b e^{2}\right )} \arctan \left (c x\right ) + {\left (b c e^{2} x + b c d e\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (b c e^{2} x + b c d e\right )} \log \left (e x + d\right )}{2 \, {\left (c^{2} d^{3} e + d e^{3} + {\left (c^{2} d^{2} e^{2} + e^{4}\right )} x\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.50 (sec) , antiderivative size = 658, normalized size of antiderivative = 6.71 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx=\begin {cases} \frac {a x}{d^{2}} & \text {for}\: c = 0 \wedge e = 0 \\\frac {a x + b x \operatorname {atan}{\left (c x \right )} - \frac {b \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c}}{d^{2}} & \text {for}\: e = 0 \\- \frac {a}{d e + e^{2} x} & \text {for}\: c = 0 \\- \frac {2 a d}{2 d^{2} e + 2 d e^{2} x} + \frac {i b d \operatorname {atanh}{\left (\frac {e x}{d} \right )}}{2 d^{2} e + 2 d e^{2} x} + \frac {i b d}{2 d^{2} e + 2 d e^{2} x} - \frac {i b e x \operatorname {atanh}{\left (\frac {e x}{d} \right )}}{2 d^{2} e + 2 d e^{2} x} & \text {for}\: c = - \frac {i e}{d} \\- \frac {2 a d}{2 d^{2} e + 2 d e^{2} x} - \frac {i b d \operatorname {atanh}{\left (\frac {e x}{d} \right )}}{2 d^{2} e + 2 d e^{2} x} - \frac {i b d}{2 d^{2} e + 2 d e^{2} x} + \frac {i b e x \operatorname {atanh}{\left (\frac {e x}{d} \right )}}{2 d^{2} e + 2 d e^{2} x} & \text {for}\: c = \frac {i e}{d} \\- \frac {2 a c^{2} d^{2}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} - \frac {2 a e^{2}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} + \frac {2 b c^{2} d e x \operatorname {atan}{\left (c x \right )}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} - \frac {b c d e \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} + \frac {2 b c d e \log {\left (\frac {d}{e} + x \right )}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} - \frac {b c e^{2} x \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} + \frac {2 b c e^{2} x \log {\left (\frac {d}{e} + x \right )}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} - \frac {2 b e^{2} \operatorname {atan}{\left (c x \right )}}{2 c^{2} d^{3} e + 2 c^{2} d^{2} e^{2} x + 2 d e^{3} + 2 e^{4} x} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*x/d**2, Eq(c, 0) & Eq(e, 0)), ((a*x + b*x*atan(c*x) - b*log(x**2 + c**(-2))/(2*c))/d**2, Eq(e, 0)
), (-a/(d*e + e**2*x), Eq(c, 0)), (-2*a*d/(2*d**2*e + 2*d*e**2*x) + I*b*d*atanh(e*x/d)/(2*d**2*e + 2*d*e**2*x)
+ I*b*d/(2*d**2*e + 2*d*e**2*x) - I*b*e*x*atanh(e*x/d)/(2*d**2*e + 2*d*e**2*x), Eq(c, -I*e/d)), (-2*a*d/(2*d*
*2*e + 2*d*e**2*x) - I*b*d*atanh(e*x/d)/(2*d**2*e + 2*d*e**2*x) - I*b*d/(2*d**2*e + 2*d*e**2*x) + I*b*e*x*atan
h(e*x/d)/(2*d**2*e + 2*d*e**2*x), Eq(c, I*e/d)), (-2*a*c**2*d**2/(2*c**2*d**3*e + 2*c**2*d**2*e**2*x + 2*d*e**
3 + 2*e**4*x) - 2*a*e**2/(2*c**2*d**3*e + 2*c**2*d**2*e**2*x + 2*d*e**3 + 2*e**4*x) + 2*b*c**2*d*e*x*atan(c*x)
/(2*c**2*d**3*e + 2*c**2*d**2*e**2*x + 2*d*e**3 + 2*e**4*x) - b*c*d*e*log(x**2 + c**(-2))/(2*c**2*d**3*e + 2*c
**2*d**2*e**2*x + 2*d*e**3 + 2*e**4*x) + 2*b*c*d*e*log(d/e + x)/(2*c**2*d**3*e + 2*c**2*d**2*e**2*x + 2*d*e**3
+ 2*e**4*x) - b*c*e**2*x*log(x**2 + c**(-2))/(2*c**2*d**3*e + 2*c**2*d**2*e**2*x + 2*d*e**3 + 2*e**4*x) + 2*b
*c*e**2*x*log(d/e + x)/(2*c**2*d**3*e + 2*c**2*d**2*e**2*x + 2*d*e**3 + 2*e**4*x) - 2*b*e**2*atan(c*x)/(2*c**2
*d**3*e + 2*c**2*d**2*e**2*x + 2*d*e**3 + 2*e**4*x), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx=\frac {1}{2} \, {\left ({\left (\frac {2 \, c d \arctan \left (c x\right )}{c^{2} d^{2} e + e^{3}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{2} d^{2} + e^{2}} + \frac {2 \, \log \left (e x + d\right )}{c^{2} d^{2} + e^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{e^{2} x + d e}\right )} b - \frac {a}{e^{2} x + d e} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 3.76 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \frac {a+b \arctan (c x)}{(d+e x)^2} \, dx=\frac {d^2\,\left (b\,c\,\ln \left (d+e\,x\right )-\frac {b\,c\,\ln \left (c^2\,x^2+1\right )}{2}+a\,c^2\,x+b\,c^2\,x\,\mathrm {atan}\left (c\,x\right )\right )-d\,e\,\left (b\,\mathrm {atan}\left (c\,x\right )-b\,c\,x\,\ln \left (d+e\,x\right )+\frac {b\,c\,x\,\ln \left (c^2\,x^2+1\right )}{2}\right )+a\,e^2\,x}{d\,\left (c^2\,d^2+e^2\right )\,\left (d+e\,x\right )} \]

[In]

int((a + b*atan(c*x))/(d + e*x)^2,x)

[Out]

(d^2*(b*c*log(d + e*x) - (b*c*log(c^2*x^2 + 1))/2 + a*c^2*x + b*c^2*x*atan(c*x)) - d*e*(b*atan(c*x) - b*c*x*lo
g(d + e*x) + (b*c*x*log(c^2*x^2 + 1))/2) + a*e^2*x)/(d*(e^2 + c^2*d^2)*(d + e*x))