3.12.59 \(\int \frac {x (a+b \text {ArcTan}(c x))}{(d+e x^2)^2} \, dx\) [1159]

Optimal. Leaf size=91 \[ \frac {b c^2 \text {ArcTan}(c x)}{2 \left (c^2 d-e\right ) e}-\frac {a+b \text {ArcTan}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \left (c^2 d-e\right ) \sqrt {e}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5094, 400, 209, 211} \begin {gather*} -\frac {a+b \text {ArcTan}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c^2 \text {ArcTan}(c x)}{2 e \left (c^2 d-e\right )}-\frac {b c \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {e} \left (c^2 d-e\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 211

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

Rule 400

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

Rule 5094

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

Rubi steps

\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 e}\\ &=-\frac {a+b \tan ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {(b c) \int \frac {1}{d+e x^2} \, dx}{2 \left (c^2 d-e\right )}+\frac {\left (b c^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=\frac {b c^2 \tan ^{-1}(c x)}{2 \left (c^2 d-e\right ) e}-\frac {a+b \tan ^{-1}(c x)}{2 e \left (d+e x^2\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \left (c^2 d-e\right ) \sqrt {e}}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 98, normalized size = 1.08 \begin {gather*} \frac {a \sqrt {d} \left (c^2 d-e\right )-b \sqrt {d} e \left (1+c^2 x^2\right ) \text {ArcTan}(c x)+b c \sqrt {e} \left (d+e x^2\right ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e \left (-c^2 d+e\right ) \left (d+e x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.31, size = 115, normalized size = 1.26

method	result	size
derivativedivides	\(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {b \,c^{4} \arctan \left (c x \right )}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e}-\frac {b \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \left (c^{2} d -e \right ) \sqrt {d e}}+\frac {b \,c^{4} \arctan \left (c x \right )}{2 e \left (c^{2} d -e \right )}}{c^{2}}\)	\(115\)
default	\(\frac {-\frac {a \,c^{4}}{2 e \left (e \,c^{2} x^{2}+c^{2} d \right )}-\frac {b \,c^{4} \arctan \left (c x \right )}{2 \left (e \,c^{2} x^{2}+c^{2} d \right ) e}-\frac {b \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \left (c^{2} d -e \right ) \sqrt {d e}}+\frac {b \,c^{4} \arctan \left (c x \right )}{2 e \left (c^{2} d -e \right )}}{c^{2}}\)	\(115\)
risch	\(\frac {i b \ln \left (i c x +1\right )}{4 e \left (e \,x^{2}+d \right )}-\frac {-i \ln \left (\left (d \,c^{3}-c e \right ) x +i d \,c^{2}-i e \right ) b \,c^{2} d e \,x^{2}+i \ln \left (\left (d \,c^{3}-c e \right ) x -i d \,c^{2}+i e \right ) b \,c^{2} d e \,x^{2}-\ln \left (\sqrt {-d e}\, x +d \right ) \sqrt {-d e}\, b c e \,x^{2}+\ln \left (\sqrt {-d e}\, x -d \right ) \sqrt {-d e}\, b c e \,x^{2}-i \ln \left (\left (d \,c^{3}-c e \right ) x +i d \,c^{2}-i e \right ) b \,c^{2} d^{2}+i \ln \left (\left (d \,c^{3}-c e \right ) x -i d \,c^{2}+i e \right ) b \,c^{2} d^{2}+i b \,c^{2} d^{2} \ln \left (-i c x +1\right )-\ln \left (\sqrt {-d e}\, x +d \right ) \sqrt {-d e}\, b c d +\ln \left (\sqrt {-d e}\, x -d \right ) \sqrt {-d e}\, b c d -i b d e \ln \left (-i c x +1\right )+2 d^{2} c^{2} a -2 d e a}{4 e \left (e \,x^{2}+d \right ) \left (c^{2} d -e \right ) d}\)	\(341\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.47, size = 90, normalized size = 0.99 \begin {gather*} \frac {1}{2} \, {\left (c {\left (\frac {c \arctan \left (c x\right )}{c^{2} d e - e^{2}} - \frac {\arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{{\left (c^{2} d - e\right )} \sqrt {d}}\right )} - \frac {\arctan \left (c x\right )}{x^{2} e^{2} + d e}\right )} b - \frac {a}{2 \, {\left (x^{2} e^{2} + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.29, size = 237, normalized size = 2.60 \begin {gather*} \left [-\frac {2 \, a c^{2} d^{2} - 2 \, a d e - 2 \, {\left (b c^{2} d x^{2} + b d\right )} \arctan \left (c x\right ) e - {\left (b c x^{2} e + b c d\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right )}{4 \, {\left (c^{2} d^{3} e - d x^{2} e^{3} + {\left (c^{2} d^{2} x^{2} - d^{2}\right )} e^{2}\right )}}, -\frac {a c^{2} d^{2} + {\left (b c x^{2} e + b c d\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} - a d e - {\left (b c^{2} d x^{2} + b d\right )} \arctan \left (c x\right ) e}{2 \, {\left (c^{2} d^{3} e - d x^{2} e^{3} + {\left (c^{2} d^{2} x^{2} - d^{2}\right )} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1054 vs. \(2 (76) = 152\).
time = 118.77, size = 1054, normalized size = 11.58 \begin {gather*} \begin {cases} \frac {\frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b x}{2 c} + \frac {b \operatorname {atan}{\left (c x \right )}}{2 c^{2}}}{d^{2}} & \text {for}\: e = 0 \\- \frac {2 a d}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac {b d x \sqrt {\frac {e}{d}}}{4 d^{2} e + 4 d e^{2} x^{2}} + \frac {b d \operatorname {atan}{\left (x \sqrt {\frac {e}{d}} \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac {b e x^{2} \operatorname {atan}{\left (x \sqrt {\frac {e}{d}} \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} & \text {for}\: c = - \sqrt {\frac {e}{d}} \\- \frac {2 a d}{4 d^{2} e + 4 d e^{2} x^{2}} + \frac {b d x \sqrt {\frac {e}{d}}}{4 d^{2} e + 4 d e^{2} x^{2}} - \frac {b d \operatorname {atan}{\left (x \sqrt {\frac {e}{d}} \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} + \frac {b e x^{2} \operatorname {atan}{\left (x \sqrt {\frac {e}{d}} \right )}}{4 d^{2} e + 4 d e^{2} x^{2}} & \text {for}\: c = \sqrt {\frac {e}{d}} \\\frac {- \frac {a}{2 x^{2}} - \frac {b c^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b c}{2 x} - \frac {b \operatorname {atan}{\left (c x \right )}}{2 x^{2}}}{e^{2}} & \text {for}\: d = 0 \\- \frac {2 a c^{2} d \sqrt {- \frac {d}{e}}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} + \frac {2 a e \sqrt {- \frac {d}{e}}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} + \frac {2 b c^{2} e x^{2} \sqrt {- \frac {d}{e}} \operatorname {atan}{\left (c x \right )}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} - \frac {b c d \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} + \frac {b c d \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} - \frac {b c e x^{2} \log {\left (x - \sqrt {- \frac {d}{e}} \right )}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} + \frac {b c e x^{2} \log {\left (x + \sqrt {- \frac {d}{e}} \right )}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} + \frac {2 b e \sqrt {- \frac {d}{e}} \operatorname {atan}{\left (c x \right )}}{4 c^{2} d^{2} e \sqrt {- \frac {d}{e}} + 4 c^{2} d e^{2} x^{2} \sqrt {- \frac {d}{e}} - 4 d e^{2} \sqrt {- \frac {d}{e}} - 4 e^{3} x^{2} \sqrt {- \frac {d}{e}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((a*x**2/2 + b*x**2*atan(c*x)/2 - b*x/(2*c) + b*atan(c*x)/(2*c**2))/d**2, Eq(e, 0)), (-2*a*d/(4*d**2
*e + 4*d*e**2*x**2) - b*d*x*sqrt(e/d)/(4*d**2*e + 4*d*e**2*x**2) + b*d*atan(x*sqrt(e/d))/(4*d**2*e + 4*d*e**2*
x**2) - b*e*x**2*atan(x*sqrt(e/d))/(4*d**2*e + 4*d*e**2*x**2), Eq(c, -sqrt(e/d))), (-2*a*d/(4*d**2*e + 4*d*e**
2*x**2) + b*d*x*sqrt(e/d)/(4*d**2*e + 4*d*e**2*x**2) - b*d*atan(x*sqrt(e/d))/(4*d**2*e + 4*d*e**2*x**2) + b*e*
x**2*atan(x*sqrt(e/d))/(4*d**2*e + 4*d*e**2*x**2), Eq(c, sqrt(e/d))), ((-a/(2*x**2) - b*c**2*atan(c*x)/2 - b*c
/(2*x) - b*atan(c*x)/(2*x**2))/e**2, Eq(d, 0)), (-2*a*c**2*d*sqrt(-d/e)/(4*c**2*d**2*e*sqrt(-d/e) + 4*c**2*d*e
**2*x**2*sqrt(-d/e) - 4*d*e**2*sqrt(-d/e) - 4*e**3*x**2*sqrt(-d/e)) + 2*a*e*sqrt(-d/e)/(4*c**2*d**2*e*sqrt(-d/
e) + 4*c**2*d*e**2*x**2*sqrt(-d/e) - 4*d*e**2*sqrt(-d/e) - 4*e**3*x**2*sqrt(-d/e)) + 2*b*c**2*e*x**2*sqrt(-d/e
)*atan(c*x)/(4*c**2*d**2*e*sqrt(-d/e) + 4*c**2*d*e**2*x**2*sqrt(-d/e) - 4*d*e**2*sqrt(-d/e) - 4*e**3*x**2*sqrt
(-d/e)) - b*c*d*log(x - sqrt(-d/e))/(4*c**2*d**2*e*sqrt(-d/e) + 4*c**2*d*e**2*x**2*sqrt(-d/e) - 4*d*e**2*sqrt(
-d/e) - 4*e**3*x**2*sqrt(-d/e)) + b*c*d*log(x + sqrt(-d/e))/(4*c**2*d**2*e*sqrt(-d/e) + 4*c**2*d*e**2*x**2*sqr
t(-d/e) - 4*d*e**2*sqrt(-d/e) - 4*e**3*x**2*sqrt(-d/e)) - b*c*e*x**2*log(x - sqrt(-d/e))/(4*c**2*d**2*e*sqrt(-
d/e) + 4*c**2*d*e**2*x**2*sqrt(-d/e) - 4*d*e**2*sqrt(-d/e) - 4*e**3*x**2*sqrt(-d/e)) + b*c*e*x**2*log(x + sqrt
(-d/e))/(4*c**2*d**2*e*sqrt(-d/e) + 4*c**2*d*e**2*x**2*sqrt(-d/e) - 4*d*e**2*sqrt(-d/e) - 4*e**3*x**2*sqrt(-d/
e)) + 2*b*e*sqrt(-d/e)*atan(c*x)/(4*c**2*d**2*e*sqrt(-d/e) + 4*c**2*d*e**2*x**2*sqrt(-d/e) - 4*d*e**2*sqrt(-d/
e) - 4*e**3*x**2*sqrt(-d/e)), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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Mupad [B]
time = 0.85, size = 696, normalized size = 7.65 \begin {gather*} \frac {b\,c\,\ln \left (e\,x+\sqrt {-d\,e}\right )\,\sqrt {-d\,e}}{4\,d\,e^2-4\,c^2\,d^2\,e}-\frac {2\,b\,c^2\,\mathrm {atan}\left (-\frac {\frac {c^2\,\left (c^8\,e\,x+\frac {c^2\,\left (2\,c^5\,e^3-4\,c^7\,d\,e^2+2\,c^9\,d^2\,e+\frac {c^2\,x\,\left (8\,c^{10}\,d^3\,e^2-8\,c^8\,d^2\,e^3-8\,c^6\,d\,e^4+8\,c^4\,e^5\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )}{4\,e^2-4\,c^2\,d\,e}-\frac {c^2\,\left (-c^8\,e\,x+\frac {c^2\,\left (2\,c^5\,e^3-4\,c^7\,d\,e^2+2\,c^9\,d^2\,e-\frac {c^2\,x\,\left (8\,c^{10}\,d^3\,e^2-8\,c^8\,d^2\,e^3-8\,c^6\,d\,e^4+8\,c^4\,e^5\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )}{4\,e^2-4\,c^2\,d\,e}}{\frac {c^2\,\left (c^8\,e\,x+\frac {c^2\,\left (2\,c^5\,e^3-4\,c^7\,d\,e^2+2\,c^9\,d^2\,e+\frac {c^2\,x\,\left (8\,c^{10}\,d^3\,e^2-8\,c^8\,d^2\,e^3-8\,c^6\,d\,e^4+8\,c^4\,e^5\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}+\frac {c^2\,\left (-c^8\,e\,x+\frac {c^2\,\left (2\,c^5\,e^3-4\,c^7\,d\,e^2+2\,c^9\,d^2\,e-\frac {c^2\,x\,\left (8\,c^{10}\,d^3\,e^2-8\,c^8\,d^2\,e^3-8\,c^6\,d\,e^4+8\,c^4\,e^5\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}\right )\,1{}\mathrm {i}}{4\,e^2-4\,c^2\,d\,e}}\right )}{4\,e^2-4\,c^2\,d\,e}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{2\,e\,\left (e\,x^2+d\right )}-\frac {b\,c\,\ln \left (e\,x-\sqrt {-d\,e}\right )\,\sqrt {-d\,e}}{4\,\left (d\,e^2-c^2\,d^2\,e\right )}-\frac {a}{2\,e^2\,x^2+2\,d\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*c*log(e*x + (-d*e)^(1/2))*(-d*e)^(1/2))/(4*d*e^2 - 4*c^2*d^2*e) - (2*b*c^2*atan(-((c^2*((c^2*(2*c^5*e^3 - 4
*c^7*d*e^2 + 2*c^9*d^2*e + (c^2*x*(8*c^4*e^5 - 8*c^6*d*e^4 - 8*c^8*d^2*e^3 + 8*c^10*d^3*e^2)*1i)/(4*e^2 - 4*c^
2*d*e))*1i)/(4*e^2 - 4*c^2*d*e) + c^8*e*x))/(4*e^2 - 4*c^2*d*e) - (c^2*((c^2*(2*c^5*e^3 - 4*c^7*d*e^2 + 2*c^9*
d^2*e - (c^2*x*(8*c^4*e^5 - 8*c^6*d*e^4 - 8*c^8*d^2*e^3 + 8*c^10*d^3*e^2)*1i)/(4*e^2 - 4*c^2*d*e))*1i)/(4*e^2
- 4*c^2*d*e) - c^8*e*x))/(4*e^2 - 4*c^2*d*e))/((c^2*((c^2*(2*c^5*e^3 - 4*c^7*d*e^2 + 2*c^9*d^2*e + (c^2*x*(8*c
^4*e^5 - 8*c^6*d*e^4 - 8*c^8*d^2*e^3 + 8*c^10*d^3*e^2)*1i)/(4*e^2 - 4*c^2*d*e))*1i)/(4*e^2 - 4*c^2*d*e) + c^8*
e*x)*1i)/(4*e^2 - 4*c^2*d*e) + (c^2*((c^2*(2*c^5*e^3 - 4*c^7*d*e^2 + 2*c^9*d^2*e - (c^2*x*(8*c^4*e^5 - 8*c^6*d
*e^4 - 8*c^8*d^2*e^3 + 8*c^10*d^3*e^2)*1i)/(4*e^2 - 4*c^2*d*e))*1i)/(4*e^2 - 4*c^2*d*e) - c^8*e*x)*1i)/(4*e^2
- 4*c^2*d*e))))/(4*e^2 - 4*c^2*d*e) - (b*atan(c*x))/(2*e*(d + e*x^2)) - (b*c*log(e*x - (-d*e)^(1/2))*(-d*e)^(1
/2))/(4*(d*e^2 - c^2*d^2*e)) - a/(2*d*e + 2*e^2*x^2)

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