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

Optimal. Leaf size=69 \[ \frac{i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}+\frac{i b}{2 c d^2 (-c x+i)}-\frac{i b \tan ^{-1}(c x)}{2 c d^2} \]

[Out]

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

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Rubi [A]  time = 0.0472076, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4862, 627, 44, 203} \[ \frac{i \left (a+b \tan ^{-1}(c x)\right )}{c d^2 (1+i c x)}+\frac{i b}{2 c d^2 (-c x+i)}-\frac{i b \tan ^{-1}(c x)}{2 c d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4862

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]

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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])

Rubi steps

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

Mathematica [A]  time = 0.032655, size = 42, normalized size = 0.61 \[ \frac{2 a+(b-i b c x) \tan ^{-1}(c x)-i b}{2 c d^2 (c x-i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 76, normalized size = 1.1 \begin{align*}{\frac{ia}{c{d}^{2} \left ( 1+icx \right ) }}+{\frac{i\arctan \left ( cx \right ) b}{c{d}^{2} \left ( 1+icx \right ) }}-{\frac{{\frac{i}{2}}b\arctan \left ( cx \right ) }{c{d}^{2}}}-{\frac{{\frac{i}{2}}b}{c{d}^{2} \left ( cx-i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.52786, size = 112, normalized size = 1.62 \begin{align*} \frac{{\left (b c x + i \, b\right )} \log \left (-\frac{c x + i}{c x - i}\right ) + 4 \, a - 2 i \, b}{4 \,{\left (c^{2} d^{2} x - i \, c d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 16.1471, size = 1698, normalized size = 24.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*b*log(-I*c*x + 1)/(2*c**2*d**2*x - 2*I*c*d**2) - I*b*log(I*c*x + 1)/(2*c**2*d**2*x - 2*I*c*d**2) + b*(log(-b
*(262144*a**16*d**2 - 2097152*I*a**15*b*d**2 - 7864320*a**14*b**2*d**2 + 18350080*I*a**13*b**3*d**2 + 29818880
*a**12*b**4*d**2 - 35782656*I*a**11*b**5*d**2 - 32800768*a**10*b**6*d**2 + 23429120*I*a**9*b**7*d**2 + 1317888
0*a**8*b**8*d**2 - 5857280*I*a**7*b**9*d**2 - 2050048*a**6*b**10*d**2 + 559104*I*a**5*b**11*d**2 + 116480*a**4
*b**12*d**2 - 17920*I*a**3*b**13*d**2 - 1920*a**2*b**14*d**2 + 128*I*a*b**15*d**2 + 4*b**16*d**2)/(4*c*d**2*(6
5536*I*a**16*b + 524288*a**15*b**2 - 1966080*I*a**14*b**3 - 4587520*a**13*b**4 + 7454720*I*a**12*b**5 + 894566
4*a**11*b**6 - 8200192*I*a**10*b**7 - 5857280*a**9*b**8 + 3294720*I*a**8*b**9 + 1464320*a**7*b**10 - 512512*I*
a**6*b**11 - 139776*a**5*b**12 + 29120*I*a**4*b**13 + 4480*a**3*b**14 - 480*I*a**2*b**15 - 32*a*b**16 + I*b**1
7)) + x)/4 - log(b*(262144*a**16*d**2 - 2097152*I*a**15*b*d**2 - 7864320*a**14*b**2*d**2 + 18350080*I*a**13*b*
*3*d**2 + 29818880*a**12*b**4*d**2 - 35782656*I*a**11*b**5*d**2 - 32800768*a**10*b**6*d**2 + 23429120*I*a**9*b
**7*d**2 + 13178880*a**8*b**8*d**2 - 5857280*I*a**7*b**9*d**2 - 2050048*a**6*b**10*d**2 + 559104*I*a**5*b**11*
d**2 + 116480*a**4*b**12*d**2 - 17920*I*a**3*b**13*d**2 - 1920*a**2*b**14*d**2 + 128*I*a*b**15*d**2 + 4*b**16*
d**2)/(4*c*d**2*(65536*I*a**16*b + 524288*a**15*b**2 - 1966080*I*a**14*b**3 - 4587520*a**13*b**4 + 7454720*I*a
**12*b**5 + 8945664*a**11*b**6 - 8200192*I*a**10*b**7 - 5857280*a**9*b**8 + 3294720*I*a**8*b**9 + 1464320*a**7
*b**10 - 512512*I*a**6*b**11 - 139776*a**5*b**12 + 29120*I*a**4*b**13 + 4480*a**3*b**14 - 480*I*a**2*b**15 - 3
2*a*b**16 + I*b**17)) + x)/4)/(c*d**2) + (2097152*I*a**21*c + 22020096*a**20*b*c - 110100480*I*a**19*b**2*c -
348651520*a**18*b**3*c + 784465920*I*a**17*b**4*c + 1333592064*a**16*b**5*c - 1778122752*I*a**15*b**6*c - 1905
131520*a**14*b**7*c + 1666990080*I*a**13*b**8*c + 1203937280*a**12*b**9*c - 722362368*I*a**11*b**10*c - 361181
184*a**10*b**11*c + 150492160*I*a**9*b**12*c + 52093440*a**8*b**13*c - 14883840*I*a**7*b**14*c - 3472896*a**6*
b**15*c + 651168*I*a**5*b**16*c + 95760*a**4*b**17*c - 10640*I*a**3*b**18*c - 840*a**2*b**19*c + 42*I*a*b**20*
c + b**21*c)/(2097152*a**20*c**2*d**2 - 20971520*I*a**19*b*c**2*d**2 - 99614720*a**18*b**2*c**2*d**2 + 2988441
60*I*a**17*b**3*c**2*d**2 + 635043840*a**16*b**4*c**2*d**2 - 1016070144*I*a**15*b**5*c**2*d**2 - 1270087680*a*
*14*b**6*c**2*d**2 + 1270087680*I*a**13*b**7*c**2*d**2 + 1031946240*a**12*b**8*c**2*d**2 - 687964160*I*a**11*b
**9*c**2*d**2 - 378380288*a**10*b**10*c**2*d**2 + 171991040*I*a**9*b**11*c**2*d**2 + 64496640*a**8*b**12*c**2*
d**2 - 19845120*I*a**7*b**13*c**2*d**2 - 4961280*a**6*b**14*c**2*d**2 + 992256*I*a**5*b**15*c**2*d**2 + 155040
*a**4*b**16*c**2*d**2 - 18240*I*a**3*b**17*c**2*d**2 - 1520*a**2*b**18*c**2*d**2 + 80*I*a*b**19*c**2*d**2 + 2*
b**20*c**2*d**2 + x*(2097152*I*a**20*c**3*d**2 + 20971520*a**19*b*c**3*d**2 - 99614720*I*a**18*b**2*c**3*d**2
- 298844160*a**17*b**3*c**3*d**2 + 635043840*I*a**16*b**4*c**3*d**2 + 1016070144*a**15*b**5*c**3*d**2 - 127008
7680*I*a**14*b**6*c**3*d**2 - 1270087680*a**13*b**7*c**3*d**2 + 1031946240*I*a**12*b**8*c**3*d**2 + 687964160*
a**11*b**9*c**3*d**2 - 378380288*I*a**10*b**10*c**3*d**2 - 171991040*a**9*b**11*c**3*d**2 + 64496640*I*a**8*b*
*12*c**3*d**2 + 19845120*a**7*b**13*c**3*d**2 - 4961280*I*a**6*b**14*c**3*d**2 - 992256*a**5*b**15*c**3*d**2 +
155040*I*a**4*b**16*c**3*d**2 + 18240*a**3*b**17*c**3*d**2 - 1520*I*a**2*b**18*c**3*d**2 - 80*a*b**19*c**3*d*
*2 + 2*I*b**20*c**3*d**2))

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Giac [A]  time = 1.12151, size = 150, normalized size = 2.17 \begin{align*} -\frac{1}{4} \,{\left (c d^{2} i{\left (\frac{i \log \left (\frac{2 \, d i}{c d i x + d} - i\right )}{c^{2} d^{4}} + \frac{2 \, i}{{\left (c d i x + d\right )} c^{2} d^{3}}\right )} - \frac{4 \, i \arctan \left (\frac{{\left ({\left (c d i x + d\right )} i^{2} + d\right )} i}{d}\right )}{{\left (c d i x + d\right )} c d}\right )} b + \frac{a i}{{\left (c d i x + d\right )} c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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