3.1159 \(\int \frac {x (a+b \tan ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=91 \[ -\frac {a+b \tan ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c^2 \tan ^{-1}(c x)}{2 e \left (c^2 d-e\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {e} \left (c^2 d-e\right )} \]

[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 = 0.07, 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 = {4974, 391, 203, 205} \[ -\frac {a+b \tan ^{-1}(c x)}{2 e \left (d+e x^2\right )}+\frac {b c^2 \tan ^{-1}(c x)}{2 e \left (c^2 d-e\right )}-\frac {b c \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} \sqrt {e} \left (c^2 d-e\right )} \]

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 203

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

Rule 205

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

Rule 391

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 4974

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 = 0.16, size = 98, normalized size = 1.08 \[ \frac {a \sqrt {d} \left (c^2 d-e\right )-b \sqrt {d} e \left (c^2 x^2+1\right ) \tan ^{-1}(c x)+b c \sqrt {e} \left (d+e x^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e \left (e-c^2 d\right ) \left (d+e x^2\right )} \]

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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fricas [A]  time = 0.53, size = 234, normalized size = 2.57 \[ \left [-\frac {2 \, a c^{2} d^{2} - 2 \, a d e - {\left (b c e x^{2} + b c d\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (b c^{2} d e x^{2} + b d e\right )} \arctan \left (c x\right )}{4 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}, -\frac {a c^{2} d^{2} - a d e + {\left (b c e x^{2} + b c d\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (b c^{2} d e x^{2} + b d e\right )} \arctan \left (c x\right )}{2 \, {\left (c^{2} d^{3} e - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - d e^{3}\right )} x^{2}\right )}}\right ] \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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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maple [A]  time = 0.04, size = 109, normalized size = 1.20 \[ -\frac {c^{2} a}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c^{2} b \arctan \left (c x \right )}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c b \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \left (c^{2} d -e \right ) \sqrt {d e}}+\frac {b \,c^{2} \arctan \left (c x \right )}{2 \left (c^{2} d -e \right ) e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.43, size = 90, normalized size = 0.99 \[ \frac {1}{2} \, {\left (c {\left (\frac {c \arctan \left (c x\right )}{c^{2} d e - e^{2}} - \frac {\arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c^{2} d - e\right )} \sqrt {d e}}\right )} - \frac {\arctan \left (c x\right )}{e^{2} x^{2} + d e}\right )} b - \frac {a}{2 \, {\left (e^{2} x^{2} + d e\right )}} \]

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

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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