\(\int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx\) [688]

Optimal result

Integrand size = 41, antiderivative size = 91 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=\frac {a^2 (i A+B)}{3 c^6 f (i+\tan (e+f x))^6}-\frac {a^2 (A-3 i B)}{5 c^6 f (i+\tan (e+f x))^5}-\frac {a^2 B}{4 c^6 f (i+\tan (e+f x))^4} \]

[Out]

1/3*a^2*(I*A+B)/c^6/f/(I+tan(f*x+e))^6-1/5*a^2*(A-3*I*B)/c^6/f/(I+tan(f*x+e))^5-1/4*a^2*B/c^6/f/(I+tan(f*x+e))
^4

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 78} \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=-\frac {a^2 (A-3 i B)}{5 c^6 f (\tan (e+f x)+i)^5}+\frac {a^2 (B+i A)}{3 c^6 f (\tan (e+f x)+i)^6}-\frac {a^2 B}{4 c^6 f (\tan (e+f x)+i)^4} \]

[In]

Int[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^6,x]

[Out]

(a^2*(I*A + B))/(3*c^6*f*(I + Tan[e + f*x])^6) - (a^2*(A - (3*I)*B))/(5*c^6*f*(I + Tan[e + f*x])^5) - (a^2*B)/
(4*c^6*f*(I + Tan[e + f*x])^4)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 3669

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a*(c/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x
], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x) (A+B x)}{(c-i c x)^7} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (-\frac {2 i a (A-i B)}{c^7 (i+x)^7}+\frac {a (A-3 i B)}{c^7 (i+x)^6}+\frac {a B}{c^7 (i+x)^5}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a^2 (i A+B)}{3 c^6 f (i+\tan (e+f x))^6}-\frac {a^2 (A-3 i B)}{5 c^6 f (i+\tan (e+f x))^5}-\frac {a^2 B}{4 c^6 f (i+\tan (e+f x))^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.41 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=-\frac {a^2 \left (-8 i A+B+6 (2 A-i B) \tan (e+f x)+15 B \tan ^2(e+f x)\right )}{60 c^6 f (i+\tan (e+f x))^6} \]

[In]

Integrate[((a + I*a*Tan[e + f*x])^2*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^6,x]

[Out]

-1/60*(a^2*((-8*I)*A + B + 6*(2*A - I*B)*Tan[e + f*x] + 15*B*Tan[e + f*x]^2))/(c^6*f*(I + Tan[e + f*x])^6)

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {-3 i B +A}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {B}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {-2 i A -2 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}\right )}{f \,c^{6}}\)	\(66\)
default	\(\frac {a^{2} \left (-\frac {-3 i B +A}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {B}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}-\frac {-2 i A -2 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}\right )}{f \,c^{6}}\)	\(66\)
risch	\(-\frac {a^{2} {\mathrm e}^{12 i \left (f x +e \right )} B}{192 c^{6} f}-\frac {i a^{2} {\mathrm e}^{12 i \left (f x +e \right )} A}{192 c^{6} f}-\frac {{\mathrm e}^{10 i \left (f x +e \right )} B \,a^{2}}{80 c^{6} f}-\frac {i {\mathrm e}^{10 i \left (f x +e \right )} A \,a^{2}}{40 c^{6} f}-\frac {3 i A \,a^{2} {\mathrm e}^{8 i \left (f x +e \right )}}{64 c^{6} f}+\frac {{\mathrm e}^{6 i \left (f x +e \right )} B \,a^{2}}{48 c^{6} f}-\frac {i {\mathrm e}^{6 i \left (f x +e \right )} A \,a^{2}}{24 c^{6} f}+\frac {a^{2} {\mathrm e}^{4 i \left (f x +e \right )} B}{64 c^{6} f}-\frac {i a^{2} {\mathrm e}^{4 i \left (f x +e \right )} A}{64 c^{6} f}\)	\(196\)

[In]

int((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^6,x,method=_RETURNVERBOSE)

[Out]

1/f*a^2/c^6*(-1/5*(A-3*I*B)/(I+tan(f*x+e))^5-1/4*B/(I+tan(f*x+e))^4-1/6*(-2*I*A-2*B)/(I+tan(f*x+e))^6)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.14 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=-\frac {5 \, {\left (i \, A + B\right )} a^{2} e^{\left (12 i \, f x + 12 i \, e\right )} + 12 \, {\left (2 i \, A + B\right )} a^{2} e^{\left (10 i \, f x + 10 i \, e\right )} + 45 i \, A a^{2} e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, {\left (2 i \, A - B\right )} a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, {\left (i \, A - B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{960 \, c^{6} f} \]

[In]

integrate((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^6,x, algorithm="fricas")

[Out]

-1/960*(5*(I*A + B)*a^2*e^(12*I*f*x + 12*I*e) + 12*(2*I*A + B)*a^2*e^(10*I*f*x + 10*I*e) + 45*I*A*a^2*e^(8*I*f
*x + 8*I*e) + 20*(2*I*A - B)*a^2*e^(6*I*f*x + 6*I*e) + 15*(I*A - B)*a^2*e^(4*I*f*x + 4*I*e))/(c^6*f)

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. 379 vs. \(2 (73) = 146\).

Time = 0.52 (sec) , antiderivative size = 379, normalized size of antiderivative = 4.16 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=\begin {cases} \frac {- 141557760 i A a^{2} c^{24} f^{4} e^{8 i e} e^{8 i f x} + \left (- 47185920 i A a^{2} c^{24} f^{4} e^{4 i e} + 47185920 B a^{2} c^{24} f^{4} e^{4 i e}\right ) e^{4 i f x} + \left (- 125829120 i A a^{2} c^{24} f^{4} e^{6 i e} + 62914560 B a^{2} c^{24} f^{4} e^{6 i e}\right ) e^{6 i f x} + \left (- 75497472 i A a^{2} c^{24} f^{4} e^{10 i e} - 37748736 B a^{2} c^{24} f^{4} e^{10 i e}\right ) e^{10 i f x} + \left (- 15728640 i A a^{2} c^{24} f^{4} e^{12 i e} - 15728640 B a^{2} c^{24} f^{4} e^{12 i e}\right ) e^{12 i f x}}{3019898880 c^{30} f^{5}} & \text {for}\: c^{30} f^{5} \neq 0 \\\frac {x \left (A a^{2} e^{12 i e} + 4 A a^{2} e^{10 i e} + 6 A a^{2} e^{8 i e} + 4 A a^{2} e^{6 i e} + A a^{2} e^{4 i e} - i B a^{2} e^{12 i e} - 2 i B a^{2} e^{10 i e} + 2 i B a^{2} e^{6 i e} + i B a^{2} e^{4 i e}\right )}{16 c^{6}} & \text {otherwise} \end {cases} \]

[In]

integrate((a+I*a*tan(f*x+e))**2*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))**6,x)

[Out]

Piecewise(((-141557760*I*A*a**2*c**24*f**4*exp(8*I*e)*exp(8*I*f*x) + (-47185920*I*A*a**2*c**24*f**4*exp(4*I*e)
+ 47185920*B*a**2*c**24*f**4*exp(4*I*e))*exp(4*I*f*x) + (-125829120*I*A*a**2*c**24*f**4*exp(6*I*e) + 62914560
*B*a**2*c**24*f**4*exp(6*I*e))*exp(6*I*f*x) + (-75497472*I*A*a**2*c**24*f**4*exp(10*I*e) - 37748736*B*a**2*c**
24*f**4*exp(10*I*e))*exp(10*I*f*x) + (-15728640*I*A*a**2*c**24*f**4*exp(12*I*e) - 15728640*B*a**2*c**24*f**4*e
xp(12*I*e))*exp(12*I*f*x))/(3019898880*c**30*f**5), Ne(c**30*f**5, 0)), (x*(A*a**2*exp(12*I*e) + 4*A*a**2*exp(
10*I*e) + 6*A*a**2*exp(8*I*e) + 4*A*a**2*exp(6*I*e) + A*a**2*exp(4*I*e) - I*B*a**2*exp(12*I*e) - 2*I*B*a**2*ex
p(10*I*e) + 2*I*B*a**2*exp(6*I*e) + I*B*a**2*exp(4*I*e))/(16*c**6), True))

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^6,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. 360 vs. \(2 (75) = 150\).

Time = 1.25 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.96 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=-\frac {2 \, {\left (15 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} + 60 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 15 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 235 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 20 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 480 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 90 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 822 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 84 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 904 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 158 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 822 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 84 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 480 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 90 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 235 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 20 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 15 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{15 \, c^{6} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{12}} \]

[In]

integrate((a+I*a*tan(f*x+e))^2*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^6,x, algorithm="giac")

[Out]

-2/15*(15*A*a^2*tan(1/2*f*x + 1/2*e)^11 + 60*I*A*a^2*tan(1/2*f*x + 1/2*e)^10 - 15*B*a^2*tan(1/2*f*x + 1/2*e)^1
0 - 235*A*a^2*tan(1/2*f*x + 1/2*e)^9 - 20*I*B*a^2*tan(1/2*f*x + 1/2*e)^9 - 480*I*A*a^2*tan(1/2*f*x + 1/2*e)^8
+ 90*B*a^2*tan(1/2*f*x + 1/2*e)^8 + 822*A*a^2*tan(1/2*f*x + 1/2*e)^7 + 84*I*B*a^2*tan(1/2*f*x + 1/2*e)^7 + 904
*I*A*a^2*tan(1/2*f*x + 1/2*e)^6 - 158*B*a^2*tan(1/2*f*x + 1/2*e)^6 - 822*A*a^2*tan(1/2*f*x + 1/2*e)^5 - 84*I*B
*a^2*tan(1/2*f*x + 1/2*e)^5 - 480*I*A*a^2*tan(1/2*f*x + 1/2*e)^4 + 90*B*a^2*tan(1/2*f*x + 1/2*e)^4 + 235*A*a^2
*tan(1/2*f*x + 1/2*e)^3 + 20*I*B*a^2*tan(1/2*f*x + 1/2*e)^3 + 60*I*A*a^2*tan(1/2*f*x + 1/2*e)^2 - 15*B*a^2*tan
(1/2*f*x + 1/2*e)^2 - 15*A*a^2*tan(1/2*f*x + 1/2*e))/(c^6*f*(tan(1/2*f*x + 1/2*e) + I)^12)

Mupad [B] (verification not implemented)

Time = 8.48 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.30 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^6} \, dx=-\frac {\frac {B\,a^2\,{\mathrm {tan}\left (e+f\,x\right )}^2}{4}+\frac {a^2\,\mathrm {tan}\left (e+f\,x\right )\,\left (12\,A-B\,6{}\mathrm {i}\right )}{60}-\frac {a^2\,\left (-B+A\,8{}\mathrm {i}\right )}{60}}{c^6\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^6+{\mathrm {tan}\left (e+f\,x\right )}^5\,6{}\mathrm {i}-15\,{\mathrm {tan}\left (e+f\,x\right )}^4-{\mathrm {tan}\left (e+f\,x\right )}^3\,20{}\mathrm {i}+15\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,6{}\mathrm {i}-1\right )} \]

[In]

int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^2)/(c - c*tan(e + f*x)*1i)^6,x)

[Out]

-((a^2*tan(e + f*x)*(12*A - B*6i))/60 - (a^2*(A*8i - B))/60 + (B*a^2*tan(e + f*x)^2)/4)/(c^6*f*(tan(e + f*x)*6
i + 15*tan(e + f*x)^2 - tan(e + f*x)^3*20i - 15*tan(e + f*x)^4 + tan(e + f*x)^5*6i + tan(e + f*x)^6 - 1))