Integrand size = 31, antiderivative size = 124 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx=\frac {x}{4 a^3 c}-\frac {i}{16 a^3 f (c-i c \tan (e+f x))}+\frac {i c^2}{12 a^3 f (c+i c \tan (e+f x))^3}+\frac {i c}{8 a^3 f (c+i c \tan (e+f x))^2}+\frac {3 i}{16 a^3 f (c+i c \tan (e+f x))} \]
1/4*x/a^3/c-1/16*I/a^3/f/(c-I*c*tan(f*x+e))+1/12*I*c^2/a^3/f/(c+I*c*tan(f* x+e))^3+1/8*I*c/a^3/f/(c+I*c*tan(f*x+e))^2+3/16*I/a^3/f/(c+I*c*tan(f*x+e))
Time = 1.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx=-\frac {4 i+\tan (e+f x)+6 i \tan ^2(e+f x)-3 \tan ^3(e+f x)-3 \arctan (\tan (e+f x)) (-i+\tan (e+f x))^3 (i+\tan (e+f x))}{12 a^3 c f (-i+\tan (e+f x))^3 (i+\tan (e+f x))} \]
-1/12*(4*I + Tan[e + f*x] + (6*I)*Tan[e + f*x]^2 - 3*Tan[e + f*x]^3 - 3*Ar cTan[Tan[e + f*x]]*(-I + Tan[e + f*x])^3*(I + Tan[e + f*x]))/(a^3*c*f*(-I + Tan[e + f*x])^3*(I + Tan[e + f*x]))
Time = 0.40 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.95, 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.
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))}dx\) |
\(\Big \downarrow \) 4005 |
\(\displaystyle \frac {\int \cos ^6(e+f x) (c-i c \tan (e+f x))^2dx}{a^3 c^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {(c-i c \tan (e+f x))^2}{\sec (e+f x)^6}dx}{a^3 c^3}\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle \frac {i c^4 \int \frac {1}{(c-i c \tan (e+f x))^2 (i \tan (e+f x) c+c)^4}d(-i c \tan (e+f x))}{a^3 f}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {i c^4 \int \left (\frac {1}{16 c^4 (c-i c \tan (e+f x))^2}+\frac {3}{16 c^4 (i \tan (e+f x) c+c)^2}+\frac {1}{4 c^3 (i \tan (e+f x) c+c)^3}+\frac {1}{4 c^2 (i \tan (e+f x) c+c)^4}+\frac {1}{4 c^4 \left (\tan ^2(e+f x) c^2+c^2\right )}\right )d(-i c \tan (e+f x))}{a^3 f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i c^4 \left (-\frac {i \arctan (\tan (e+f x))}{4 c^5}-\frac {1}{16 c^4 (c-i c \tan (e+f x))}+\frac {3}{16 c^4 (c+i c \tan (e+f x))}+\frac {1}{8 c^3 (c+i c \tan (e+f x))^2}+\frac {1}{12 c^2 (c+i c \tan (e+f x))^3}\right )}{a^3 f}\) |
(I*c^4*(((-1/4*I)*ArcTan[Tan[e + f*x]])/c^5 - 1/(16*c^4*(c - I*c*Tan[e + f *x])) + 1/(12*c^2*(c + I*c*Tan[e + f*x])^3) + 1/(8*c^3*(c + I*c*Tan[e + f* x])^2) + 3/(16*c^4*(c + I*c*Tan[e + f*x]))))/(a^3*f)
3.10.30.3.1 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 = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76
method | result | size |
risch | \(\frac {x}{4 a^{3} c}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )}}{16 a^{3} c f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{96 a^{3} c f}+\frac {5 i \cos \left (2 f x +2 e \right )}{32 a^{3} c f}+\frac {7 \sin \left (2 f x +2 e \right )}{32 a^{3} c f}\) | \(94\) |
derivativedivides | \(\frac {\arctan \left (\tan \left (f x +e \right )\right )}{4 f \,a^{3} c}+\frac {1}{16 f \,a^{3} c \left (\tan \left (f x +e \right )+i\right )}-\frac {i}{8 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {1}{12 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {3}{16 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )}\) | \(109\) |
default | \(\frac {\arctan \left (\tan \left (f x +e \right )\right )}{4 f \,a^{3} c}+\frac {1}{16 f \,a^{3} c \left (\tan \left (f x +e \right )+i\right )}-\frac {i}{8 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {1}{12 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {3}{16 f \,a^{3} c \left (\tan \left (f x +e \right )-i\right )}\) | \(109\) |
norman | \(\frac {\frac {x}{4 a c}+\frac {3 \tan \left (f x +e \right )}{4 a c f}+\frac {2 \left (\tan ^{3}\left (f x +e \right )\right )}{3 a c f}+\frac {\tan ^{5}\left (f x +e \right )}{4 a c f}+\frac {3 x \left (\tan ^{2}\left (f x +e \right )\right )}{4 a c}+\frac {3 x \left (\tan ^{4}\left (f x +e \right )\right )}{4 a c}+\frac {x \left (\tan ^{6}\left (f x +e \right )\right )}{4 a c}+\frac {i}{3 a c f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{3} a^{2}}\) | \(145\) |
1/4*x/a^3/c+1/16*I/a^3/c/f*exp(-4*I*(f*x+e))+1/96*I/a^3/c/f*exp(-6*I*(f*x+ e))+5/32*I/a^3/c/f*cos(2*f*x+2*e)+7/32/a^3/c/f*sin(2*f*x+2*e)
Time = 0.23 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx=\frac {{\left (24 \, f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, e^{\left (8 i \, f x + 8 i \, e\right )} + 18 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 6 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, a^{3} c f} \]
1/96*(24*f*x*e^(6*I*f*x + 6*I*e) - 3*I*e^(8*I*f*x + 8*I*e) + 18*I*e^(4*I*f *x + 4*I*e) + 6*I*e^(2*I*f*x + 2*I*e) + I)*e^(-6*I*f*x - 6*I*e)/(a^3*c*f)
Time = 0.23 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.73 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx=\begin {cases} \frac {\left (- 24576 i a^{9} c^{3} f^{3} e^{14 i e} e^{2 i f x} + 147456 i a^{9} c^{3} f^{3} e^{10 i e} e^{- 2 i f x} + 49152 i a^{9} c^{3} f^{3} e^{8 i e} e^{- 4 i f x} + 8192 i a^{9} c^{3} f^{3} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{786432 a^{12} c^{4} f^{4}} & \text {for}\: a^{12} c^{4} f^{4} e^{12 i e} \neq 0 \\x \left (\frac {\left (e^{8 i e} + 4 e^{6 i e} + 6 e^{4 i e} + 4 e^{2 i e} + 1\right ) e^{- 6 i e}}{16 a^{3} c} - \frac {1}{4 a^{3} c}\right ) & \text {otherwise} \end {cases} + \frac {x}{4 a^{3} c} \]
Piecewise(((-24576*I*a**9*c**3*f**3*exp(14*I*e)*exp(2*I*f*x) + 147456*I*a* *9*c**3*f**3*exp(10*I*e)*exp(-2*I*f*x) + 49152*I*a**9*c**3*f**3*exp(8*I*e) *exp(-4*I*f*x) + 8192*I*a**9*c**3*f**3*exp(6*I*e)*exp(-6*I*f*x))*exp(-12*I *e)/(786432*a**12*c**4*f**4), Ne(a**12*c**4*f**4*exp(12*I*e), 0)), (x*((ex p(8*I*e) + 4*exp(6*I*e) + 6*exp(4*I*e) + 4*exp(2*I*e) + 1)*exp(-6*I*e)/(16 *a**3*c) - 1/(4*a**3*c)), True)) + x/(4*a**3*c)
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]
Time = 0.49 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx=-\frac {-\frac {6 i \, \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3} c} + \frac {6 i \, \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c} + \frac {3 \, {\left (2 i \, \tan \left (f x + e\right ) - 3\right )}}{a^{3} c {\left (\tan \left (f x + e\right ) + i\right )}} + \frac {-11 i \, \tan \left (f x + e\right )^{3} - 42 \, \tan \left (f x + e\right )^{2} + 57 i \, \tan \left (f x + e\right ) + 30}{a^{3} c {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{48 \, f} \]
-1/48*(-6*I*log(tan(f*x + e) + I)/(a^3*c) + 6*I*log(tan(f*x + e) - I)/(a^3 *c) + 3*(2*I*tan(f*x + e) - 3)/(a^3*c*(tan(f*x + e) + I)) + (-11*I*tan(f*x + e)^3 - 42*tan(f*x + e)^2 + 57*I*tan(f*x + e) + 30)/(a^3*c*(tan(f*x + e) - I)^3))/f
Time = 6.44 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))} \, dx=\frac {x}{4\,a^3\,c}-\frac {\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{4}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{2}-\frac {\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}{12}+\frac {1}{3}}{a^3\,c\,f\,{\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \]