Integrand size = 31, antiderivative size = 223 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx=\frac {35 x}{128 a^3 c^4}-\frac {i}{64 a^3 f (c-i c \tan (e+f x))^4}-\frac {i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac {i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac {5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac {5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}-\frac {5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac {15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )} \]
35/128*x/a^3/c^4-1/64*I/a^3/f/(c-I*c*tan(f*x+e))^4-1/24*I/a^3/c/f/(c-I*c*t an(f*x+e))^3+1/96*I/a^3/c/f/(c+I*c*tan(f*x+e))^3-5/64*I/a^3/f/(c^2-I*c^2*t an(f*x+e))^2+5/128*I/a^3/f/(c^2+I*c^2*tan(f*x+e))^2-5/32*I/a^3/f/(c^4-I*c^ 4*tan(f*x+e))+15/128*I/a^3/f/(c^4+I*c^4*tan(f*x+e))
Time = 1.40 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx=-\frac {\sec ^7(e+f x) (-525 \cos (e+f x)+126 \cos (3 (e+f x))+14 \cos (5 (e+f x))+\cos (7 (e+f x))-840 i \arctan (\tan (e+f x)) (\cos (e+f x)-i \sin (e+f x))-315 i \sin (e+f x)-378 i \sin (3 (e+f x))-70 i \sin (5 (e+f x))-7 i \sin (7 (e+f x)))}{3072 a^3 c^4 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^4} \]
-1/3072*(Sec[e + f*x]^7*(-525*Cos[e + f*x] + 126*Cos[3*(e + f*x)] + 14*Cos [5*(e + f*x)] + Cos[7*(e + f*x)] - (840*I)*ArcTan[Tan[e + f*x]]*(Cos[e + f *x] - I*Sin[e + f*x]) - (315*I)*Sin[e + f*x] - (378*I)*Sin[3*(e + f*x)] - (70*I)*Sin[5*(e + f*x)] - (7*I)*Sin[7*(e + f*x)]))/(a^3*c^4*f*(-I + Tan[e + f*x])^3*(I + Tan[e + f*x])^4)
Time = 0.45 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3042, 4005, 3042, 3968, 54, 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.
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4}dx\) |
| \(\Big \downarrow \) 4005 |
| \(\displaystyle \frac {\int \frac {\cos ^6(e+f x)}{c-i c \tan (e+f x)}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {1}{\sec (e+f x)^6 (c-i c \tan (e+f x))}dx}{a^3 c^3}\) |
| \(\Big \downarrow \) 3968 |
| \(\displaystyle \frac {i c^4 \int \frac {1}{(c-i c \tan (e+f x))^5 (i \tan (e+f x) c+c)^4}d(-i c \tan (e+f x))}{a^3 f}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {i c^4 \int \left (\frac {5}{32 c^7 (c-i c \tan (e+f x))^2}+\frac {15}{128 c^7 (i \tan (e+f x) c+c)^2}+\frac {5}{32 c^6 (c-i c \tan (e+f x))^3}+\frac {5}{64 c^6 (i \tan (e+f x) c+c)^3}+\frac {1}{8 c^5 (c-i c \tan (e+f x))^4}+\frac {1}{32 c^5 (i \tan (e+f x) c+c)^4}+\frac {1}{16 c^4 (c-i c \tan (e+f x))^5}+\frac {35}{128 c^7 \left (\tan ^2(e+f x) c^2+c^2\right )}\right )d(-i c \tan (e+f x))}{a^3 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i c^4 \left (-\frac {35 i \arctan (\tan (e+f x))}{128 c^8}-\frac {5}{32 c^7 (c-i c \tan (e+f x))}+\frac {15}{128 c^7 (c+i c \tan (e+f x))}-\frac {5}{64 c^6 (c-i c \tan (e+f x))^2}+\frac {5}{128 c^6 (c+i c \tan (e+f x))^2}-\frac {1}{24 c^5 (c-i c \tan (e+f x))^3}+\frac {1}{96 c^5 (c+i c \tan (e+f x))^3}-\frac {1}{64 c^4 (c-i c \tan (e+f x))^4}\right )}{a^3 f}\) |
(I*c^4*((((-35*I)/128)*ArcTan[Tan[e + f*x]])/c^8 - 1/(64*c^4*(c - I*c*Tan[ e + f*x])^4) - 1/(24*c^5*(c - I*c*Tan[e + f*x])^3) - 5/(64*c^6*(c - I*c*Ta n[e + f*x])^2) - 5/(32*c^7*(c - I*c*Tan[e + f*x])) + 1/(96*c^5*(c + I*c*Ta n[e + f*x])^3) + 5/(128*c^6*(c + I*c*Tan[e + f*x])^2) + 15/(128*c^7*(c + I *c*Tan[e + f*x]))))/(a^3*f)
3.10.55.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[a^m*c^m Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && EqQ[ b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] && !(IGtQ[n, 0] && (LtQ[ m, 0] || GtQ[m, n]))
Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(\frac {35 x}{128 a^{3} c^{4}}-\frac {i {\mathrm e}^{8 i \left (f x +e \right )}}{1024 a^{3} c^{4} f}-\frac {i \cos \left (6 f x +6 e \right )}{128 a^{3} c^{4} f}+\frac {\sin \left (6 f x +6 e \right )}{96 a^{3} c^{4} f}-\frac {7 i \cos \left (4 f x +4 e \right )}{256 a^{3} c^{4} f}+\frac {7 \sin \left (4 f x +4 e \right )}{128 a^{3} c^{4} f}-\frac {7 i \cos \left (2 f x +2 e \right )}{128 a^{3} c^{4} f}+\frac {7 \sin \left (2 f x +2 e \right )}{32 a^{3} c^{4} f}\) | \(155\) |
| derivativedivides | \(\frac {5 i}{64 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {i}{64 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {35 \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{4}}-\frac {1}{24 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {5}{32 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )}-\frac {5 i}{128 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {1}{96 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {15}{128 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )-i\right )}\) | \(177\) |
| default | \(\frac {5 i}{64 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )^{2}}-\frac {i}{64 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}}+\frac {35 \arctan \left (\tan \left (f x +e \right )\right )}{128 f \,a^{3} c^{4}}-\frac {1}{24 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )^{3}}+\frac {5}{32 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )+i\right )}-\frac {5 i}{128 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {1}{96 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {15}{128 f \,a^{3} c^{4} \left (\tan \left (f x +e \right )-i\right )}\) | \(177\) |
| norman | \(\frac {\frac {35 x}{128 a c}-\frac {i}{8 a c f}+\frac {93 \tan \left (f x +e \right )}{128 a c f}+\frac {511 \left (\tan ^{3}\left (f x +e \right )\right )}{384 a c f}+\frac {385 \left (\tan ^{5}\left (f x +e \right )\right )}{384 a c f}+\frac {35 \left (\tan ^{7}\left (f x +e \right )\right )}{128 a c f}+\frac {35 x \left (\tan ^{2}\left (f x +e \right )\right )}{32 a c}+\frac {105 x \left (\tan ^{4}\left (f x +e \right )\right )}{64 a c}+\frac {35 x \left (\tan ^{6}\left (f x +e \right )\right )}{32 a c}+\frac {35 x \left (\tan ^{8}\left (f x +e \right )\right )}{128 a c}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{4} a^{2} c^{3}}\) | \(184\) |
35/128*x/a^3/c^4-1/1024*I/a^3/c^4/f*exp(8*I*(f*x+e))-1/128*I/a^3/c^4/f*cos (6*f*x+6*e)+1/96/a^3/c^4/f*sin(6*f*x+6*e)-7/256*I/a^3/c^4/f*cos(4*f*x+4*e) +7/128/a^3/c^4/f*sin(4*f*x+4*e)-7/128*I/a^3/c^4/f*cos(2*f*x+2*e)+7/32/a^3/ c^4/f*sin(2*f*x+2*e)
Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx=\frac {{\left (840 \, f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, e^{\left (14 i \, f x + 14 i \, e\right )} - 28 i \, e^{\left (12 i \, f x + 12 i \, e\right )} - 126 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 420 i \, e^{\left (8 i \, f x + 8 i \, e\right )} + 252 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 42 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{3072 \, a^{3} c^{4} f} \]
1/3072*(840*f*x*e^(6*I*f*x + 6*I*e) - 3*I*e^(14*I*f*x + 14*I*e) - 28*I*e^( 12*I*f*x + 12*I*e) - 126*I*e^(10*I*f*x + 10*I*e) - 420*I*e^(8*I*f*x + 8*I* e) + 252*I*e^(4*I*f*x + 4*I*e) + 42*I*e^(2*I*f*x + 2*I*e) + 4*I)*e^(-6*I*f *x - 6*I*e)/(a^3*c^4*f)
Time = 0.41 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx=\begin {cases} \frac {\left (- 10133099161583616 i a^{18} c^{24} f^{6} e^{20 i e} e^{8 i f x} - 94575592174780416 i a^{18} c^{24} f^{6} e^{18 i e} e^{6 i f x} - 425590164786511872 i a^{18} c^{24} f^{6} e^{16 i e} e^{4 i f x} - 1418633882621706240 i a^{18} c^{24} f^{6} e^{14 i e} e^{2 i f x} + 851180329573023744 i a^{18} c^{24} f^{6} e^{10 i e} e^{- 2 i f x} + 141863388262170624 i a^{18} c^{24} f^{6} e^{8 i e} e^{- 4 i f x} + 13510798882111488 i a^{18} c^{24} f^{6} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{10376293541461622784 a^{21} c^{28} f^{7}} & \text {for}\: a^{21} c^{28} f^{7} e^{12 i e} \neq 0 \\x \left (\frac {\left (e^{14 i e} + 7 e^{12 i e} + 21 e^{10 i e} + 35 e^{8 i e} + 35 e^{6 i e} + 21 e^{4 i e} + 7 e^{2 i e} + 1\right ) e^{- 6 i e}}{128 a^{3} c^{4}} - \frac {35}{128 a^{3} c^{4}}\right ) & \text {otherwise} \end {cases} + \frac {35 x}{128 a^{3} c^{4}} \]
Piecewise(((-10133099161583616*I*a**18*c**24*f**6*exp(20*I*e)*exp(8*I*f*x) - 94575592174780416*I*a**18*c**24*f**6*exp(18*I*e)*exp(6*I*f*x) - 4255901 64786511872*I*a**18*c**24*f**6*exp(16*I*e)*exp(4*I*f*x) - 1418633882621706 240*I*a**18*c**24*f**6*exp(14*I*e)*exp(2*I*f*x) + 851180329573023744*I*a** 18*c**24*f**6*exp(10*I*e)*exp(-2*I*f*x) + 141863388262170624*I*a**18*c**24 *f**6*exp(8*I*e)*exp(-4*I*f*x) + 13510798882111488*I*a**18*c**24*f**6*exp( 6*I*e)*exp(-6*I*f*x))*exp(-12*I*e)/(10376293541461622784*a**21*c**28*f**7) , Ne(a**21*c**28*f**7*exp(12*I*e), 0)), (x*((exp(14*I*e) + 7*exp(12*I*e) + 21*exp(10*I*e) + 35*exp(8*I*e) + 35*exp(6*I*e) + 21*exp(4*I*e) + 7*exp(2* I*e) + 1)*exp(-6*I*e)/(128*a**3*c**4) - 35/(128*a**3*c**4)), True)) + 35*x /(128*a**3*c**4)
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx=\text {Exception raised: RuntimeError} \]
Time = 0.76 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx=-\frac {-\frac {420 i \, \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c^{4}} + \frac {420 i \, \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{4}} - \frac {2 \, {\left (385 \, \tan \left (f x + e\right )^{3} - 1335 i \, \tan \left (f x + e\right )^{2} - 1575 \, \tan \left (f x + e\right ) + 641 i\right )}}{a^{3} c^{4} {\left (i \, \tan \left (f x + e\right ) + 1\right )}^{3}} + \frac {875 i \, \tan \left (f x + e\right )^{4} - 3980 \, \tan \left (f x + e\right )^{3} - 6930 i \, \tan \left (f x + e\right )^{2} + 5548 \, \tan \left (f x + e\right ) + 1771 i}{a^{3} c^{4} {\left (\tan \left (f x + e\right ) + i\right )}^{4}}}{3072 \, f} \]
-1/3072*(-420*I*log(tan(f*x + e) + I)/(a^3*c^4) + 420*I*log(tan(f*x + e) - I)/(a^3*c^4) - 2*(385*tan(f*x + e)^3 - 1335*I*tan(f*x + e)^2 - 1575*tan(f *x + e) + 641*I)/(a^3*c^4*(I*tan(f*x + e) + 1)^3) + (875*I*tan(f*x + e)^4 - 3980*tan(f*x + e)^3 - 6930*I*tan(f*x + e)^2 + 5548*tan(f*x + e) + 1771*I )/(a^3*c^4*(tan(f*x + e) + I)^4))/f
Time = 8.42 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.49 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx=\frac {35\,x}{128\,a^3\,c^4}-\frac {\frac {{\mathrm {tan}\left (e+f\,x\right )}^6\,35{}\mathrm {i}}{128}-\frac {35\,{\mathrm {tan}\left (e+f\,x\right )}^5}{128}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,35{}\mathrm {i}}{48}-\frac {35\,{\mathrm {tan}\left (e+f\,x\right )}^3}{48}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,77{}\mathrm {i}}{128}-\frac {77\,\mathrm {tan}\left (e+f\,x\right )}{128}+\frac {1}{8}{}\mathrm {i}}{a^3\,c^4\,f\,{\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^4} \]