\(\int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx\) [938]

Optimal result

Integrand size = 31, antiderivative size = 154 \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=-\frac {40 a^6 x}{c^3}+\frac {40 i a^6 \log (\cos (e+f x))}{c^3 f}+\frac {9 a^6 \tan (e+f x)}{c^3 f}+\frac {i a^6 \tan ^2(e+f x)}{2 c^3 f}-\frac {32 i a^6}{3 f (c-i c \tan (e+f x))^3}+\frac {40 i a^6}{c f (c-i c \tan (e+f x))^2}-\frac {80 i a^6}{f \left (c^3-i c^3 \tan (e+f x)\right )} \]

[Out]

-40*a^6*x/c^3+40*I*a^6*ln(cos(f*x+e))/c^3/f+9*a^6*tan(f*x+e)/c^3/f+1/2*I*a^6*tan(f*x+e)^2/c^3/f-32/3*I*a^6/f/(
c-I*c*tan(f*x+e))^3+40*I*a^6/c/f/(c-I*c*tan(f*x+e))^2-80*I*a^6/f/(c^3-I*c^3*tan(f*x+e))

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 3568, 45} \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=\frac {i a^6 \tan ^2(e+f x)}{2 c^3 f}+\frac {9 a^6 \tan (e+f x)}{c^3 f}-\frac {80 i a^6}{f \left (c^3-i c^3 \tan (e+f x)\right )}+\frac {40 i a^6 \log (\cos (e+f x))}{c^3 f}-\frac {40 a^6 x}{c^3}+\frac {40 i a^6}{c f (c-i c \tan (e+f x))^2}-\frac {32 i a^6}{3 f (c-i c \tan (e+f x))^3} \]

[In]

Int[(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^3,x]

[Out]

(-40*a^6*x)/c^3 + ((40*I)*a^6*Log[Cos[e + f*x]])/(c^3*f) + (9*a^6*Tan[e + f*x])/(c^3*f) + ((I/2)*a^6*Tan[e + f
*x]^2)/(c^3*f) - (((32*I)/3)*a^6)/(f*(c - I*c*Tan[e + f*x])^3) + ((40*I)*a^6)/(c*f*(c - I*c*Tan[e + f*x])^2) -
((80*I)*a^6)/(f*(c^3 - I*c^3*Tan[e + f*x]))

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 3568

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rule 3603

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[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]))

Rubi steps \begin{align*} \text {integral}& = \left (a^6 c^6\right ) \int \frac {\sec ^{12}(e+f x)}{(c-i c \tan (e+f x))^9} \, dx \\ & = \frac {\left (i a^6\right ) \text {Subst}\left (\int \frac {(c-x)^5}{(c+x)^4} \, dx,x,-i c \tan (e+f x)\right )}{c^5 f} \\ & = \frac {\left (i a^6\right ) \text {Subst}\left (\int \left (9 c-x+\frac {32 c^5}{(c+x)^4}-\frac {80 c^4}{(c+x)^3}+\frac {80 c^3}{(c+x)^2}-\frac {40 c^2}{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^5 f} \\ & = -\frac {40 a^6 x}{c^3}+\frac {40 i a^6 \log (\cos (e+f x))}{c^3 f}+\frac {9 a^6 \tan (e+f x)}{c^3 f}+\frac {i a^6 \tan ^2(e+f x)}{2 c^3 f}-\frac {32 i a^6}{3 f (c-i c \tan (e+f x))^3}+\frac {40 i a^6}{c f (c-i c \tan (e+f x))^2}-\frac {80 i a^6}{f \left (c^3-i c^3 \tan (e+f x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.98 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.56 \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=\frac {i a^6 \left (-240 \log (i+\tan (e+f x))-54 i \tan (e+f x)+3 \tan ^2(e+f x)+\frac {16 \left (19 i+45 \tan (e+f x)-30 i \tan ^2(e+f x)\right )}{(i+\tan (e+f x))^3}\right )}{6 c^3 f} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])^6/(c - I*c*Tan[e + f*x])^3,x]

[Out]

((I/6)*a^6*(-240*Log[I + Tan[e + f*x]] - (54*I)*Tan[e + f*x] + 3*Tan[e + f*x]^2 + (16*(19*I + 45*Tan[e + f*x]
- (30*I)*Tan[e + f*x]^2))/(I + Tan[e + f*x])^3))/(c^3*f)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {4 i a^{6} {\mathrm e}^{6 i \left (f x +e \right )}}{3 c^{3} f}+\frac {6 i a^{6} {\mathrm e}^{4 i \left (f x +e \right )}}{c^{3} f}-\frac {24 i a^{6} {\mathrm e}^{2 i \left (f x +e \right )}}{c^{3} f}+\frac {80 a^{6} e}{f \,c^{3}}+\frac {2 i a^{6} \left (10 \,{\mathrm e}^{2 i \left (f x +e \right )}+9\right )}{f \,c^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2}}+\frac {40 i a^{6} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f \,c^{3}}\)	\(139\)
derivativedivides	\(\frac {9 a^{6} \tan \left (f x +e \right )}{c^{3} f}+\frac {i a^{6} \left (\tan ^{2}\left (f x +e \right )\right )}{2 c^{3} f}-\frac {40 a^{6} \arctan \left (\tan \left (f x +e \right )\right )}{f \,c^{3}}-\frac {20 i a^{6} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f \,c^{3}}+\frac {80 a^{6}}{f \,c^{3} \left (\tan \left (f x +e \right )+i\right )}-\frac {32 a^{6}}{3 f \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{3}}-\frac {40 i a^{6}}{f \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{2}}\)	\(147\)
default	\(\frac {9 a^{6} \tan \left (f x +e \right )}{c^{3} f}+\frac {i a^{6} \left (\tan ^{2}\left (f x +e \right )\right )}{2 c^{3} f}-\frac {40 a^{6} \arctan \left (\tan \left (f x +e \right )\right )}{f \,c^{3}}-\frac {20 i a^{6} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f \,c^{3}}+\frac {80 a^{6}}{f \,c^{3} \left (\tan \left (f x +e \right )+i\right )}-\frac {32 a^{6}}{3 f \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{3}}-\frac {40 i a^{6}}{f \,c^{3} \left (\tan \left (f x +e \right )+i\right )^{2}}\)	\(147\)
norman	\(\frac {-\frac {40 a^{6} x}{c}-\frac {313 i a^{6}}{6 c f}-\frac {120 a^{6} x \left (\tan ^{2}\left (f x +e \right )\right )}{c}-\frac {120 a^{6} x \left (\tan ^{4}\left (f x +e \right )\right )}{c}-\frac {40 a^{6} x \left (\tan ^{6}\left (f x +e \right )\right )}{c}+\frac {41 a^{6} \tan \left (f x +e \right )}{c f}+\frac {289 a^{6} \left (\tan ^{3}\left (f x +e \right )\right )}{3 c f}+\frac {107 a^{6} \left (\tan ^{5}\left (f x +e \right )\right )}{c f}+\frac {9 a^{6} \left (\tan ^{7}\left (f x +e \right )\right )}{c f}-\frac {132 i a^{6} \left (\tan ^{2}\left (f x +e \right )\right )}{c f}-\frac {123 i a^{6} \left (\tan ^{4}\left (f x +e \right )\right )}{c f}+\frac {i a^{6} \left (\tan ^{8}\left (f x +e \right )\right )}{2 c f}}{c^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {20 i a^{6} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f \,c^{3}}\)	\(248\)

[In]

int((a+I*a*tan(f*x+e))^6/(c-I*c*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

-4/3*I/c^3/f*a^6*exp(6*I*(f*x+e))+6*I/c^3/f*a^6*exp(4*I*(f*x+e))-24*I/c^3/f*a^6*exp(2*I*(f*x+e))+80/f*a^6/c^3*
e+2*I*a^6*(10*exp(2*I*(f*x+e))+9)/f/c^3/(exp(2*I*(f*x+e))+1)^2+40*I/f*a^6/c^3*ln(exp(2*I*(f*x+e))+1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.06 \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=-\frac {2 \, {\left (2 i \, a^{6} e^{\left (10 i \, f x + 10 i \, e\right )} - 5 i \, a^{6} e^{\left (8 i \, f x + 8 i \, e\right )} + 20 i \, a^{6} e^{\left (6 i \, f x + 6 i \, e\right )} + 63 i \, a^{6} e^{\left (4 i \, f x + 4 i \, e\right )} + 6 i \, a^{6} e^{\left (2 i \, f x + 2 i \, e\right )} - 27 i \, a^{6} + 60 \, {\left (-i \, a^{6} e^{\left (4 i \, f x + 4 i \, e\right )} - 2 i \, a^{6} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{6}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )\right )}}{3 \, {\left (c^{3} f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, c^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{3} f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))^6/(c-I*c*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

-2/3*(2*I*a^6*e^(10*I*f*x + 10*I*e) - 5*I*a^6*e^(8*I*f*x + 8*I*e) + 20*I*a^6*e^(6*I*f*x + 6*I*e) + 63*I*a^6*e^
(4*I*f*x + 4*I*e) + 6*I*a^6*e^(2*I*f*x + 2*I*e) - 27*I*a^6 + 60*(-I*a^6*e^(4*I*f*x + 4*I*e) - 2*I*a^6*e^(2*I*f
*x + 2*I*e) - I*a^6)*log(e^(2*I*f*x + 2*I*e) + 1))/(c^3*f*e^(4*I*f*x + 4*I*e) + 2*c^3*f*e^(2*I*f*x + 2*I*e) +
c^3*f)

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.60 \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=\frac {40 i a^{6} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{3} f} + \frac {20 i a^{6} e^{2 i e} e^{2 i f x} + 18 i a^{6}}{c^{3} f e^{4 i e} e^{4 i f x} + 2 c^{3} f e^{2 i e} e^{2 i f x} + c^{3} f} + \begin {cases} \frac {- 4 i a^{6} c^{6} f^{2} e^{6 i e} e^{6 i f x} + 18 i a^{6} c^{6} f^{2} e^{4 i e} e^{4 i f x} - 72 i a^{6} c^{6} f^{2} e^{2 i e} e^{2 i f x}}{3 c^{9} f^{3}} & \text {for}\: c^{9} f^{3} \neq 0 \\\frac {x \left (8 a^{6} e^{6 i e} - 24 a^{6} e^{4 i e} + 48 a^{6} e^{2 i e}\right )}{c^{3}} & \text {otherwise} \end {cases} \]

[In]

integrate((a+I*a*tan(f*x+e))**6/(c-I*c*tan(f*x+e))**3,x)

[Out]

40*I*a**6*log(exp(2*I*f*x) + exp(-2*I*e))/(c**3*f) + (20*I*a**6*exp(2*I*e)*exp(2*I*f*x) + 18*I*a**6)/(c**3*f*e
xp(4*I*e)*exp(4*I*f*x) + 2*c**3*f*exp(2*I*e)*exp(2*I*f*x) + c**3*f) + Piecewise(((-4*I*a**6*c**6*f**2*exp(6*I*
e)*exp(6*I*f*x) + 18*I*a**6*c**6*f**2*exp(4*I*e)*exp(4*I*f*x) - 72*I*a**6*c**6*f**2*exp(2*I*e)*exp(2*I*f*x))/(
3*c**9*f**3), Ne(c**9*f**3, 0)), (x*(8*a**6*exp(6*I*e) - 24*a**6*exp(4*I*e) + 48*a**6*exp(2*I*e))/c**3, True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a+I*a*tan(f*x+e))^6/(c-I*c*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (134) = 268\).

Time = 1.06 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.77 \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=-\frac {2 \, {\left (-\frac {60 i \, a^{6} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c^{3}} + \frac {120 i \, a^{6} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c^{3}} - \frac {60 i \, a^{6} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c^{3}} - \frac {3 \, {\left (-30 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 9 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 61 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 9 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 30 i \, a^{6}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} c^{3}} + \frac {2 \, {\left (-147 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 930 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 2421 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3340 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2421 i \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 930 \, a^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 147 i \, a^{6}\right )}}{c^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{6}}\right )}}{3 \, f} \]

[In]

integrate((a+I*a*tan(f*x+e))^6/(c-I*c*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-2/3*(-60*I*a^6*log(tan(1/2*f*x + 1/2*e) + 1)/c^3 + 120*I*a^6*log(tan(1/2*f*x + 1/2*e) + I)/c^3 - 60*I*a^6*log
(tan(1/2*f*x + 1/2*e) - 1)/c^3 - 3*(-30*I*a^6*tan(1/2*f*x + 1/2*e)^4 - 9*a^6*tan(1/2*f*x + 1/2*e)^3 + 61*I*a^6
*tan(1/2*f*x + 1/2*e)^2 + 9*a^6*tan(1/2*f*x + 1/2*e) - 30*I*a^6)/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*c^3) + 2*(-14
7*I*a^6*tan(1/2*f*x + 1/2*e)^6 + 930*a^6*tan(1/2*f*x + 1/2*e)^5 + 2421*I*a^6*tan(1/2*f*x + 1/2*e)^4 - 3340*a^6
*tan(1/2*f*x + 1/2*e)^3 - 2421*I*a^6*tan(1/2*f*x + 1/2*e)^2 + 930*a^6*tan(1/2*f*x + 1/2*e) + 147*I*a^6)/(c^3*(
tan(1/2*f*x + 1/2*e) + I)^6))/f

Mupad [B] (verification not implemented)

Time = 6.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {(a+i a \tan (e+f x))^6}{(c-i c \tan (e+f x))^3} \, dx=\frac {9\,a^6\,\mathrm {tan}\left (e+f\,x\right )}{c^3\,f}-\frac {\frac {80\,a^6\,{\mathrm {tan}\left (e+f\,x\right )}^2}{c^3}-\frac {152\,a^6}{3\,c^3}+\frac {a^6\,\mathrm {tan}\left (e+f\,x\right )\,120{}\mathrm {i}}{c^3}}{f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}+\frac {a^6\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}}{2\,c^3\,f}-\frac {a^6\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,40{}\mathrm {i}}{c^3\,f} \]

[In]

int((a + a*tan(e + f*x)*1i)^6/(c - c*tan(e + f*x)*1i)^3,x)

[Out]

(9*a^6*tan(e + f*x))/(c^3*f) - ((a^6*tan(e + f*x)*120i)/c^3 - (152*a^6)/(3*c^3) + (80*a^6*tan(e + f*x)^2)/c^3)
/(f*(3*tan(e + f*x) - tan(e + f*x)^2*3i - tan(e + f*x)^3 + 1i)) + (a^6*tan(e + f*x)^2*1i)/(2*c^3*f) - (a^6*log
(tan(e + f*x) + 1i)*40i)/(c^3*f)