\(\int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx\) [107]

Optimal result

Integrand size = 22, antiderivative size = 122 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {b^2}{2 c d^2 (i-c x)}-\frac {b^2 \arctan (c x)}{2 c d^2}+\frac {i b (a+b \arctan (c x))}{c d^2 (i-c x)}-\frac {i (a+b \arctan (c x))^2}{2 c d^2}+\frac {i (a+b \arctan (c x))^2}{c d^2 (1+i c x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4974, 4972, 641, 46, 209, 5004} \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {i b (a+b \arctan (c x))}{c d^2 (-c x+i)}+\frac {i (a+b \arctan (c x))^2}{c d^2 (1+i c x)}-\frac {i (a+b \arctan (c x))^2}{2 c d^2}-\frac {b^2 \arctan (c x)}{2 c d^2}+\frac {b^2}{2 c d^2 (-c x+i)} \]

[In]

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

[Out]

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

Rule 46

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] && Lt
Q[m + n + 2, 0])

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 641

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c/e)*x)^
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 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]

Rule 4974

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

Rule 5004

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

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.59 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=-\frac {-2 a^2+2 i a b+b^2+b (2 i a+b) (i+c x) \arctan (c x)+b^2 (-1+i c x) \arctan (c x)^2}{2 c d^2 (-i+c x)} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (110 ) = 220\).

Time = 1.94 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.94

method	result	size
derivativedivides	\(\frac {\frac {i a^{2}}{d^{2} \left (i c x +1\right )}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {i \arctan \left (c x \right )}{4}-\frac {i}{4 \left (c x -i\right )}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}+\frac {\ln \left (c x +i\right )^{2}}{16}\right )\right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (i c x +1\right )}-\frac {i a b \arctan \left (c x \right )}{d^{2}}-\frac {i a b}{d^{2} \left (c x -i\right )}}{c}\)	\(237\)
default	\(\frac {\frac {i a^{2}}{d^{2} \left (i c x +1\right )}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {i \arctan \left (c x \right )}{4}-\frac {i}{4 \left (c x -i\right )}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}+\frac {\ln \left (c x +i\right )^{2}}{16}\right )\right )}{d^{2}}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} \left (i c x +1\right )}-\frac {i a b \arctan \left (c x \right )}{d^{2}}-\frac {i a b}{d^{2} \left (c x -i\right )}}{c}\)	\(237\)
parts	\(\frac {i a^{2}}{d^{2} c \left (i c x +1\right )}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{i c x +1}-2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{4}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{4}+\frac {\arctan \left (c x \right )}{2 c x -2 i}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{8}+\frac {\ln \left (c x -i\right )^{2}}{16}-\frac {i \arctan \left (c x \right )}{4}-\frac {i}{4 \left (c x -i\right )}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{8}+\frac {\ln \left (c x +i\right )^{2}}{16}\right )\right )}{d^{2} c}+\frac {2 i a b \arctan \left (c x \right )}{d^{2} c \left (i c x +1\right )}-\frac {i a b \arctan \left (c x \right )}{d^{2} c}-\frac {i a b}{d^{2} c \left (c x -i\right )}\)	\(248\)
risch	\(\frac {i \left (c x +i\right ) b^{2} \ln \left (i c x +1\right )^{2}}{8 d^{2} \left (c x -i\right ) c}-\frac {i b \left (b c x \ln \left (-i c x +1\right )+i b \ln \left (-i c x +1\right )-2 i b +4 a \right ) \ln \left (i c x +1\right )}{4 d^{2} \left (c x -i\right ) c}+\frac {8 i \ln \left (-i c x +1\right ) a b +4 b^{2} \ln \left (-i c x +1\right )+i b^{2} c x \ln \left (-i c x +1\right )^{2}-b^{2} \ln \left (-i c x +1\right )^{2}+2 i \ln \left (\left (-i b c +2 a c \right ) x -b -2 i a \right ) b^{2} c x -2 i \ln \left (\left (i b c -2 a c \right ) x -b -2 i a \right ) b^{2} c x -4 \ln \left (\left (-i b c +2 a c \right ) x -b -2 i a \right ) a b c x +4 \ln \left (\left (i b c -2 a c \right ) x -b -2 i a \right ) a b c x +4 i \ln \left (\left (-i b c +2 a c \right ) x -b -2 i a \right ) a b -4 i \ln \left (\left (i b c -2 a c \right ) x -b -2 i a \right ) a b +2 \ln \left (\left (-i b c +2 a c \right ) x -b -2 i a \right ) b^{2}-2 \ln \left (\left (i b c -2 a c \right ) x -b -2 i a \right ) b^{2}-8 i a b +8 a^{2}-4 b^{2}}{8 d^{2} \left (c x -i\right ) c}\)	\(403\)

[In]

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

[Out]

1/c*(I*a^2/d^2/(1+I*c*x)+b^2/d^2*(I/(1+I*c*x)*arctan(c*x)^2-2*I*(1/4*I*arctan(c*x)*ln(c*x+I)-1/4*I*arctan(c*x)
*ln(c*x-I)+1/2*arctan(c*x)/(c*x-I)-1/8*ln(c*x-I)*ln(-1/2*I*(c*x+I))+1/16*ln(c*x-I)^2-1/4*I*arctan(c*x)-1/4*I/(
c*x-I)-1/8*(ln(c*x+I)-ln(-1/2*I*(c*x+I)))*ln(-1/2*I*(-c*x+I))+1/16*ln(c*x+I)^2))+2*I*a*b/d^2/(1+I*c*x)*arctan(
c*x)-I*a*b/d^2*arctan(c*x)-I*a*b/d^2/(c*x-I))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\frac {{\left (i \, b^{2} c x - b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + 8 \, a^{2} - 8 i \, a b - 4 \, b^{2} + 2 \, {\left ({\left (2 \, a b - i \, b^{2}\right )} c x + 2 i \, a b + b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{8 \, {\left (c^{2} d^{2} x - i \, c d^{2}\right )}} \]

[In]

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

[Out]

1/8*((I*b^2*c*x - b^2)*log(-(c*x + I)/(c*x - I))^2 + 8*a^2 - 8*I*a*b - 4*b^2 + 2*((2*a*b - I*b^2)*c*x + 2*I*a*
b + b^2)*log(-(c*x + I)/(c*x - I)))/(c^2*d^2*x - I*c*d^2)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (94) = 188\).

Time = 5.05 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.48 \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=- \frac {b \left (2 a - i b\right ) \log {\left (- \frac {i b \left (2 a - i b\right )}{c} + x \left (2 a b - i b^{2}\right ) \right )}}{4 c d^{2}} + \frac {b \left (2 a - i b\right ) \log {\left (\frac {i b \left (2 a - i b\right )}{c} + x \left (2 a b - i b^{2}\right ) \right )}}{4 c d^{2}} + \frac {\left (- 2 i a b - b^{2}\right ) \log {\left (i c x + 1 \right )}}{2 c^{2} d^{2} x - 2 i c d^{2}} + \frac {\left (i b^{2} c x - b^{2}\right ) \log {\left (- i c x + 1 \right )}^{2}}{8 c^{2} d^{2} x - 8 i c d^{2}} + \frac {\left (i b^{2} c x - b^{2}\right ) \log {\left (i c x + 1 \right )}^{2}}{8 c^{2} d^{2} x - 8 i c d^{2}} + \frac {\left (4 i a b - i b^{2} c x \log {\left (i c x + 1 \right )} + b^{2} \log {\left (i c x + 1 \right )} + 2 b^{2}\right ) \log {\left (- i c x + 1 \right )}}{4 c^{2} d^{2} x - 4 i c d^{2}} - \frac {- 2 a^{2} + 2 i a b + b^{2}}{2 c^{2} d^{2} x - 2 i c d^{2}} \]

[In]

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

[Out]

-b*(2*a - I*b)*log(-I*b*(2*a - I*b)/c + x*(2*a*b - I*b**2))/(4*c*d**2) + b*(2*a - I*b)*log(I*b*(2*a - I*b)/c +
x*(2*a*b - I*b**2))/(4*c*d**2) + (-2*I*a*b - b**2)*log(I*c*x + 1)/(2*c**2*d**2*x - 2*I*c*d**2) + (I*b**2*c*x
- b**2)*log(-I*c*x + 1)**2/(8*c**2*d**2*x - 8*I*c*d**2) + (I*b**2*c*x - b**2)*log(I*c*x + 1)**2/(8*c**2*d**2*x
- 8*I*c*d**2) + (4*I*a*b - I*b**2*c*x*log(I*c*x + 1) + b**2*log(I*c*x + 1) + 2*b**2)*log(-I*c*x + 1)/(4*c**2*
d**2*x - 4*I*c*d**2) - (-2*a**2 + 2*I*a*b + b**2)/(2*c**2*d**2*x - 2*I*c*d**2)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(d+i c d x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]

[In]

int((a + b*atan(c*x))^2/(d + c*d*x*1i)^2,x)

[Out]

int((a + b*atan(c*x))^2/(d + c*d*x*1i)^2, x)