Integrand size = 21, antiderivative size = 207 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=-\frac {b^2}{54 c (i-c x)^3}+\frac {5 i b^2}{144 c (i-c x)^2}+\frac {11 b^2}{144 c (i-c x)}-\frac {11 b^2 \arctan (c x)}{144 c}-\frac {i b (a+b \arctan (c x))}{9 c (i-c x)^3}-\frac {b (a+b \arctan (c x))}{12 c (i-c x)^2}+\frac {i b (a+b \arctan (c x))}{12 c (i-c x)}-\frac {i (a+b \arctan (c x))^2}{24 c}+\frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3} \]
-1/54*b^2/c/(I-c*x)^3+5/144*I*b^2/c/(I-c*x)^2+11/144*b^2/c/(I-c*x)-11/144* b^2*arctan(c*x)/c-1/9*I*b*(a+b*arctan(c*x))/c/(I-c*x)^3-1/12*b*(a+b*arctan (c*x))/c/(I-c*x)^2+1/12*I*b*(a+b*arctan(c*x))/c/(I-c*x)-1/24*I*(a+b*arctan (c*x))^2/c+1/3*I*(a+b*arctan(c*x))^2/c/(1+I*c*x)^3
Time = 0.35 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=-\frac {144 a^2+12 a b \left (-10 i+9 c x+3 i c^2 x^2\right )+b^2 \left (-56-81 i c x+33 c^2 x^2\right )+3 b (i+c x) \left (12 a \left (-7 i+4 c x+i c^2 x^2\right )+b \left (-29-32 i c x+11 c^2 x^2\right )\right ) \arctan (c x)+18 b^2 \left (7-3 i c x+3 c^2 x^2+i c^3 x^3\right ) \arctan (c x)^2}{432 c (-i+c x)^3} \]
-1/432*(144*a^2 + 12*a*b*(-10*I + 9*c*x + (3*I)*c^2*x^2) + b^2*(-56 - (81* I)*c*x + 33*c^2*x^2) + 3*b*(I + c*x)*(12*a*(-7*I + 4*c*x + I*c^2*x^2) + b* (-29 - (32*I)*c*x + 11*c^2*x^2))*ArcTan[c*x] + 18*b^2*(7 - (3*I)*c*x + 3*c ^2*x^2 + I*c^3*x^3)*ArcTan[c*x]^2)/(c*(-I + c*x)^3)
Time = 0.42 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5389, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx\) |
| \(\Big \downarrow \) 5389 |
| \(\displaystyle \frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}-\frac {2}{3} i b \int \left (\frac {a+b \arctan (c x)}{8 \left (c^2 x^2+1\right )}-\frac {a+b \arctan (c x)}{8 (i-c x)^2}-\frac {i (a+b \arctan (c x))}{4 (i-c x)^3}+\frac {a+b \arctan (c x)}{2 (i-c x)^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i (a+b \arctan (c x))^2}{3 c (1+i c x)^3}-\frac {2}{3} i b \left (\frac {(a+b \arctan (c x))^2}{16 b c}-\frac {a+b \arctan (c x)}{8 c (-c x+i)}-\frac {i (a+b \arctan (c x))}{8 c (-c x+i)^2}+\frac {a+b \arctan (c x)}{6 c (-c x+i)^3}-\frac {11 i b \arctan (c x)}{96 c}+\frac {11 i b}{96 c (-c x+i)}-\frac {5 b}{96 c (-c x+i)^2}-\frac {i b}{36 c (-c x+i)^3}\right )\) |
((I/3)*(a + b*ArcTan[c*x])^2)/(c*(1 + I*c*x)^3) - ((2*I)/3)*b*(((-1/36*I)* b)/(c*(I - c*x)^3) - (5*b)/(96*c*(I - c*x)^2) + (((11*I)/96)*b)/(c*(I - c* x)) - (((11*I)/96)*b*ArcTan[c*x])/c + (a + b*ArcTan[c*x])/(6*c*(I - c*x)^3 ) - ((I/8)*(a + b*ArcTan[c*x]))/(c*(I - c*x)^2) - (a + b*ArcTan[c*x])/(8*c *(I - c*x)) + (a + b*ArcTan[c*x])^2/(16*b*c))
3.2.18.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTan[c*x])^p/(e*(q + 1))), x] - S imp[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] && NeQ[q, -1]
Time = 2.41 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\frac {i a^{2}}{3 \left (i c x +1\right )^{3}}+b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{3 \left (i c x +1\right )^{3}}-\frac {2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{16}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{16}-\frac {i \arctan \left (c x \right )}{8 \left (c x -i\right )^{2}}-\frac {\arctan \left (c x \right )}{6 \left (c x -i\right )^{3}}+\frac {\arctan \left (c x \right )}{8 c x -8 i}-\frac {11 i \arctan \left (c x \right )}{96}+\frac {i}{36 \left (c x -i\right )^{3}}-\frac {11 i}{96 \left (c x -i\right )}-\frac {5}{96 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x -i\right )^{2}}{64}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x +i\right )^{2}}{64}\right )}{3}\right )+\frac {2 i a b \arctan \left (c x \right )}{3 \left (i c x +1\right )^{3}}-\frac {i a b \arctan \left (c x \right )}{12}-\frac {a b}{12 \left (c x -i\right )^{2}}+\frac {i a b}{9 \left (c x -i\right )^{3}}-\frac {i a b}{12 \left (c x -i\right )}}{c}\) | \(297\) |
| default | \(\frac {\frac {i a^{2}}{3 \left (i c x +1\right )^{3}}+b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{3 \left (i c x +1\right )^{3}}-\frac {2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{16}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{16}-\frac {i \arctan \left (c x \right )}{8 \left (c x -i\right )^{2}}-\frac {\arctan \left (c x \right )}{6 \left (c x -i\right )^{3}}+\frac {\arctan \left (c x \right )}{8 c x -8 i}-\frac {11 i \arctan \left (c x \right )}{96}+\frac {i}{36 \left (c x -i\right )^{3}}-\frac {11 i}{96 \left (c x -i\right )}-\frac {5}{96 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x -i\right )^{2}}{64}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x +i\right )^{2}}{64}\right )}{3}\right )+\frac {2 i a b \arctan \left (c x \right )}{3 \left (i c x +1\right )^{3}}-\frac {i a b \arctan \left (c x \right )}{12}-\frac {a b}{12 \left (c x -i\right )^{2}}+\frac {i a b}{9 \left (c x -i\right )^{3}}-\frac {i a b}{12 \left (c x -i\right )}}{c}\) | \(297\) |
| parts | \(\frac {i a^{2}}{3 \left (i c x +1\right )^{3} c}+\frac {b^{2} \left (\frac {i \arctan \left (c x \right )^{2}}{3 \left (i c x +1\right )^{3}}-\frac {2 i \left (\frac {i \arctan \left (c x \right ) \ln \left (c x +i\right )}{16}-\frac {i \arctan \left (c x \right ) \ln \left (c x -i\right )}{16}-\frac {i \arctan \left (c x \right )}{8 \left (c x -i\right )^{2}}-\frac {\arctan \left (c x \right )}{6 \left (c x -i\right )^{3}}+\frac {\arctan \left (c x \right )}{8 c x -8 i}-\frac {11 i \arctan \left (c x \right )}{96}+\frac {i}{36 \left (c x -i\right )^{3}}-\frac {11 i}{96 \left (c x -i\right )}-\frac {5}{96 \left (c x -i\right )^{2}}-\frac {\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x -i\right )^{2}}{64}-\frac {\left (\ln \left (c x +i\right )-\ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right ) \ln \left (-\frac {i \left (-c x +i\right )}{2}\right )}{32}+\frac {\ln \left (c x +i\right )^{2}}{64}\right )}{3}\right )}{c}+\frac {2 i a b \arctan \left (c x \right )}{3 c \left (i c x +1\right )^{3}}-\frac {i a b \arctan \left (c x \right )}{12 c}-\frac {a b}{12 c \left (c x -i\right )^{2}}+\frac {i a b}{9 c \left (c x -i\right )^{3}}-\frac {i a b}{12 c \left (c x -i\right )}\) | \(314\) |
| risch | \(\frac {i b^{2} \left (c^{3} x^{3}-3 i c^{2} x^{2}-3 c x -7 i\right ) \ln \left (i c x +1\right )^{2}}{96 \left (c x -i\right )^{3} c}+\frac {i b \left (-3 b \,c^{3} x^{3} \ln \left (-i c x +1\right )+9 i b \,x^{2} \ln \left (-i c x +1\right ) c^{2}+6 i b \,c^{2} x^{2}+9 b c x \ln \left (-i c x +1\right )+21 i b \ln \left (-i c x +1\right )+18 x b c -20 i b +48 a \right ) \ln \left (i c x +1\right )}{144 \left (c x -i\right )^{3} c}-\frac {288 a^{2}+66 b^{2} c^{2} x^{2}+288 i \ln \left (-i c x +1\right ) a b -112 b^{2}+216 a b c x +120 b^{2} \ln \left (-i c x +1\right )+27 i b^{2} c x \ln \left (-i c x +1\right )^{2}-240 i a b -63 b^{2} \ln \left (-i c x +1\right )^{2}-162 i b^{2} c x -36 b^{2} \ln \left (-i c x +1\right ) c^{2} x^{2}+108 i \ln \left (-i c x +1\right ) b^{2} c x -27 b^{2} c^{2} x^{2} \ln \left (-i c x +1\right )^{2}+99 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{2} x^{2}-99 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{2} x^{2}-36 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b +36 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b -108 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b \,c^{2} x^{2}+108 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b \,c^{2} x^{2}-33 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2}+33 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2}-9 i c^{3} b^{2} x^{3} \ln \left (-i c x +1\right )^{2}+33 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{3} x^{3}-33 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2} c^{3} x^{3}-99 i \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) b^{2} c x +99 i \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) b^{2} c x -36 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b \,c^{3} x^{3}+36 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b \,c^{3} x^{3}+108 \ln \left (\left (11 i b c -12 a c \right ) x -12 i a -11 b \right ) a b c x -108 \ln \left (\left (-11 i b c +12 a c \right ) x -12 i a -11 b \right ) a b c x +72 i a b \,c^{2} x^{2}}{864 \left (c x -i\right )^{3} c}\) | \(832\) |
1/c*(1/3*I*a^2/(1+I*c*x)^3+b^2*(1/3*I/(1+I*c*x)^3*arctan(c*x)^2-2/3*I*(1/1 6*I*arctan(c*x)*ln(c*x+I)-1/16*I*arctan(c*x)*ln(c*x-I)-1/8*I*arctan(c*x)/( c*x-I)^2-1/6*arctan(c*x)/(c*x-I)^3+1/8*arctan(c*x)/(c*x-I)-11/96*I*arctan( c*x)+1/36*I/(c*x-I)^3-11/96*I/(c*x-I)-5/96/(c*x-I)^2-1/32*ln(c*x-I)*ln(-1/ 2*I*(c*x+I))+1/64*ln(c*x-I)^2-1/32*(ln(c*x+I)-ln(-1/2*I*(c*x+I)))*ln(-1/2* I*(-c*x+I))+1/64*ln(c*x+I)^2))+2/3*I*a*b/(1+I*c*x)^3*arctan(c*x)-1/12*I*a* b*arctan(c*x)-1/12*a*b/(c*x-I)^2+1/9*I*a*b/(c*x-I)^3-1/12*I*a*b/(c*x-I))
Time = 0.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=-\frac {6 \, {\left (12 i \, a b + 11 \, b^{2}\right )} c^{2} x^{2} + 54 \, {\left (4 \, a b - 3 i \, b^{2}\right )} c x + 9 \, {\left (-i \, b^{2} c^{3} x^{3} - 3 \, b^{2} c^{2} x^{2} + 3 i \, b^{2} c x - 7 \, b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + 288 \, a^{2} - 240 i \, a b - 112 \, b^{2} - 3 \, {\left ({\left (12 \, a b - 11 i \, b^{2}\right )} c^{3} x^{3} - 3 \, {\left (12 i \, a b + 7 \, b^{2}\right )} c^{2} x^{2} - 3 \, {\left (12 \, a b + i \, b^{2}\right )} c x - 84 i \, a b - 29 \, b^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{864 \, {\left (c^{4} x^{3} - 3 i \, c^{3} x^{2} - 3 \, c^{2} x + i \, c\right )}} \]
-1/864*(6*(12*I*a*b + 11*b^2)*c^2*x^2 + 54*(4*a*b - 3*I*b^2)*c*x + 9*(-I*b ^2*c^3*x^3 - 3*b^2*c^2*x^2 + 3*I*b^2*c*x - 7*b^2)*log(-(c*x + I)/(c*x - I) )^2 + 288*a^2 - 240*I*a*b - 112*b^2 - 3*((12*a*b - 11*I*b^2)*c^3*x^3 - 3*( 12*I*a*b + 7*b^2)*c^2*x^2 - 3*(12*a*b + I*b^2)*c*x - 84*I*a*b - 29*b^2)*lo g(-(c*x + I)/(c*x - I)))/(c^4*x^3 - 3*I*c^3*x^2 - 3*c^2*x + I*c)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (158) = 316\).
Time = 21.42 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.67 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=- \frac {b \left (12 a - 11 i b\right ) \log {\left (- \frac {i b \left (12 a - 11 i b\right )}{c} + x \left (12 a b - 11 i b^{2}\right ) \right )}}{288 c} + \frac {b \left (12 a - 11 i b\right ) \log {\left (\frac {i b \left (12 a - 11 i b\right )}{c} + x \left (12 a b - 11 i b^{2}\right ) \right )}}{288 c} + \frac {- 144 a^{2} + 120 i a b + 56 b^{2} + x^{2} \left (- 36 i a b c^{2} - 33 b^{2} c^{2}\right ) + x \left (- 108 a b c + 81 i b^{2} c\right )}{432 c^{4} x^{3} - 1296 i c^{3} x^{2} - 1296 c^{2} x + 432 i c} + \frac {\left (- 48 i a b - 3 i b^{2} c^{3} x^{3} \log {\left (i c x + 1 \right )} - 9 b^{2} c^{2} x^{2} \log {\left (i c x + 1 \right )} + 6 b^{2} c^{2} x^{2} + 9 i b^{2} c x \log {\left (i c x + 1 \right )} - 18 i b^{2} c x - 21 b^{2} \log {\left (i c x + 1 \right )} - 20 b^{2}\right ) \log {\left (- i c x + 1 \right )}}{144 c^{4} x^{3} - 432 i c^{3} x^{2} - 432 c^{2} x + 144 i c} + \frac {\left (i b^{2} c^{3} x^{3} + 3 b^{2} c^{2} x^{2} - 3 i b^{2} c x + 7 b^{2}\right ) \log {\left (- i c x + 1 \right )}^{2}}{96 c^{4} x^{3} - 288 i c^{3} x^{2} - 288 c^{2} x + 96 i c} + \frac {\left (i b^{2} c^{3} x^{3} + 3 b^{2} c^{2} x^{2} - 3 i b^{2} c x + 7 b^{2}\right ) \log {\left (i c x + 1 \right )}^{2}}{96 c^{4} x^{3} - 288 i c^{3} x^{2} - 288 c^{2} x + 96 i c} + \frac {\left (24 i a b - 3 b^{2} c^{2} x^{2} + 9 i b^{2} c x + 10 b^{2}\right ) \log {\left (i c x + 1 \right )}}{72 c^{4} x^{3} - 216 i c^{3} x^{2} - 216 c^{2} x + 72 i c} \]
-b*(12*a - 11*I*b)*log(-I*b*(12*a - 11*I*b)/c + x*(12*a*b - 11*I*b**2))/(2 88*c) + b*(12*a - 11*I*b)*log(I*b*(12*a - 11*I*b)/c + x*(12*a*b - 11*I*b** 2))/(288*c) + (-144*a**2 + 120*I*a*b + 56*b**2 + x**2*(-36*I*a*b*c**2 - 33 *b**2*c**2) + x*(-108*a*b*c + 81*I*b**2*c))/(432*c**4*x**3 - 1296*I*c**3*x **2 - 1296*c**2*x + 432*I*c) + (-48*I*a*b - 3*I*b**2*c**3*x**3*log(I*c*x + 1) - 9*b**2*c**2*x**2*log(I*c*x + 1) + 6*b**2*c**2*x**2 + 9*I*b**2*c*x*lo g(I*c*x + 1) - 18*I*b**2*c*x - 21*b**2*log(I*c*x + 1) - 20*b**2)*log(-I*c* x + 1)/(144*c**4*x**3 - 432*I*c**3*x**2 - 432*c**2*x + 144*I*c) + (I*b**2* c**3*x**3 + 3*b**2*c**2*x**2 - 3*I*b**2*c*x + 7*b**2)*log(-I*c*x + 1)**2/( 96*c**4*x**3 - 288*I*c**3*x**2 - 288*c**2*x + 96*I*c) + (I*b**2*c**3*x**3 + 3*b**2*c**2*x**2 - 3*I*b**2*c*x + 7*b**2)*log(I*c*x + 1)**2/(96*c**4*x** 3 - 288*I*c**3*x**2 - 288*c**2*x + 96*I*c) + (24*I*a*b - 3*b**2*c**2*x**2 + 9*I*b**2*c*x + 10*b**2)*log(I*c*x + 1)/(72*c**4*x**3 - 216*I*c**3*x**2 - 216*c**2*x + 72*I*c)
Time = 0.25 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=\frac {3 \, {\left (-12 i \, a b - 11 \, b^{2}\right )} c^{2} x^{2} - 27 \, {\left (4 \, a b - 3 i \, b^{2}\right )} c x + 18 \, {\left (-i \, b^{2} c^{3} x^{3} - 3 \, b^{2} c^{2} x^{2} + 3 i \, b^{2} c x - 7 \, b^{2}\right )} \arctan \left (c x\right )^{2} - 144 \, a^{2} + 120 i \, a b + 56 \, b^{2} + 3 \, {\left ({\left (-12 i \, a b - 11 \, b^{2}\right )} c^{3} x^{3} - 3 \, {\left (12 \, a b - 7 i \, b^{2}\right )} c^{2} x^{2} + 3 \, {\left (12 i \, a b - b^{2}\right )} c x - 84 \, a b + 29 i \, b^{2}\right )} \arctan \left (c x\right )}{432 \, {\left (c^{4} x^{3} - 3 i \, c^{3} x^{2} - 3 \, c^{2} x + i \, c\right )}} \]
1/432*(3*(-12*I*a*b - 11*b^2)*c^2*x^2 - 27*(4*a*b - 3*I*b^2)*c*x + 18*(-I* b^2*c^3*x^3 - 3*b^2*c^2*x^2 + 3*I*b^2*c*x - 7*b^2)*arctan(c*x)^2 - 144*a^2 + 120*I*a*b + 56*b^2 + 3*((-12*I*a*b - 11*b^2)*c^3*x^3 - 3*(12*a*b - 7*I* b^2)*c^2*x^2 + 3*(12*I*a*b - b^2)*c*x - 84*a*b + 29*I*b^2)*arctan(c*x))/(c ^4*x^3 - 3*I*c^3*x^2 - 3*c^2*x + I*c)
\[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{{\left (i \, c x + 1\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \arctan (c x))^2}{(1+i c x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (1+c\,x\,1{}\mathrm {i}\right )}^4} \,d x \]