\(\int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx\) [1045]

Optimal result

Integrand size = 31, antiderivative size = 64 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=\frac {2 i a^2 (c-i c \tan (e+f x))^n}{f n}-\frac {i a^2 (c-i c \tan (e+f x))^{1+n}}{c f (1+n)} \]

[Out]

2*I*a^2*(c-I*c*tan(f*x+e))^n/f/n-I*a^2*(c-I*c*tan(f*x+e))^(1+n)/c/f/(1+n)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 64, 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} \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=\frac {2 i a^2 (c-i c \tan (e+f x))^n}{f n}-\frac {i a^2 (c-i c \tan (e+f x))^{n+1}}{c f (n+1)} \]

[In]

Int[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^n,x]

[Out]

((2*I)*a^2*(c - I*c*Tan[e + f*x])^n)/(f*n) - (I*a^2*(c - I*c*Tan[e + f*x])^(1 + n))/(c*f*(1 + n))

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^2 c^2\right ) \int \sec ^4(e+f x) (c-i c \tan (e+f x))^{-2+n} \, dx \\ & = \frac {\left (i a^2\right ) \text {Subst}\left (\int (c-x) (c+x)^{-1+n} \, dx,x,-i c \tan (e+f x)\right )}{c f} \\ & = \frac {\left (i a^2\right ) \text {Subst}\left (\int \left (2 c (c+x)^{-1+n}-(c+x)^n\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f} \\ & = \frac {2 i a^2 (c-i c \tan (e+f x))^n}{f n}-\frac {i a^2 (c-i c \tan (e+f x))^{1+n}}{c f (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=-\frac {a^2 (c-i c \tan (e+f x))^n (-i (2+n)+n \tan (e+f x))}{f n (1+n)} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])^2*(c - I*c*Tan[e + f*x])^n,x]

[Out]

-((a^2*(c - I*c*Tan[e + f*x])^n*((-I)*(2 + n) + n*Tan[e + f*x]))/(f*n*(1 + n)))

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {i \left (a^{2} n +2 a^{2}\right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{f n \left (1+n \right )}-\frac {a^{2} \tan \left (f x +e \right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{f \left (1+n \right )}\)	\(78\)
default	\(\frac {i \left (a^{2} n +2 a^{2}\right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{f n \left (1+n \right )}-\frac {a^{2} \tan \left (f x +e \right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{f \left (1+n \right )}\)	\(78\)
norman	\(\frac {i \left (a^{2} n +2 a^{2}\right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{f n \left (1+n \right )}-\frac {a^{2} \tan \left (f x +e \right ) {\mathrm e}^{n \ln \left (c -i c \tan \left (f x +e \right )\right )}}{f \left (1+n \right )}\)	\(78\)
risch	\(\frac {2 i a^{2} c^{n} 2^{n} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{-n} \left (n \,{\mathrm e}^{-\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{3} \pi n}{2}} {\mathrm e}^{\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \operatorname {csgn}\left (i c \right ) \pi n}{2}} {\mathrm e}^{\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) n}{2}} {\mathrm e}^{-\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \operatorname {csgn}\left (i c \right ) \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) n}{2}} {\mathrm e}^{2 i f x} {\mathrm e}^{2 i e}+{\mathrm e}^{-\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{3} \pi n}{2}} {\mathrm e}^{\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \operatorname {csgn}\left (i c \right ) \pi n}{2}} {\mathrm e}^{\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) n}{2}} {\mathrm e}^{-\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \operatorname {csgn}\left (i c \right ) \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) n}{2}} {\mathrm e}^{2 i f x} {\mathrm e}^{2 i e}+{\mathrm e}^{\frac {i \operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \pi n \left (\operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )-\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )+\operatorname {csgn}\left (i c \right )\right )}{2}}\right )}{\left (1+n \right ) f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) n}\)	\(458\)

[In]

int((a+I*a*tan(f*x+e))^2*(c-I*c*tan(f*x+e))^n,x,method=_RETURNVERBOSE)

[Out]

I/f/n/(1+n)*(a^2*n+2*a^2)*exp(n*ln(c-I*c*tan(f*x+e)))-a^2/f/(1+n)*tan(f*x+e)*exp(n*ln(c-I*c*tan(f*x+e)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.22 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=-\frac {2 \, {\left (-i \, a^{2} + {\left (-i \, a^{2} n - i \, a^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}}{f n^{2} + f n + {\left (f n^{2} + f n\right )} e^{\left (2 i \, f x + 2 i \, e\right )}} \]

[In]

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

[Out]

-2*(-I*a^2 + (-I*a^2*n - I*a^2)*e^(2*I*f*x + 2*I*e))*(2*c/(e^(2*I*f*x + 2*I*e) + 1))^n/(f*n^2 + f*n + (f*n^2 +
f*n)*e^(2*I*f*x + 2*I*e))

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. 311 vs. \(2 (49) = 98\).

Time = 0.47 (sec) , antiderivative size = 311, normalized size of antiderivative = 4.86 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=\begin {cases} x \left (i a \tan {\left (e \right )} + a\right )^{2} \left (- i c \tan {\left (e \right )} + c\right )^{n} & \text {for}\: f = 0 \\- \frac {2 a^{2} f x \tan {\left (e + f x \right )}}{2 c f \tan {\left (e + f x \right )} + 2 i c f} - \frac {2 i a^{2} f x}{2 c f \tan {\left (e + f x \right )} + 2 i c f} - \frac {i a^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \tan {\left (e + f x \right )}}{2 c f \tan {\left (e + f x \right )} + 2 i c f} + \frac {a^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 c f \tan {\left (e + f x \right )} + 2 i c f} + \frac {4 a^{2}}{2 c f \tan {\left (e + f x \right )} + 2 i c f} & \text {for}\: n = -1 \\2 a^{2} x + \frac {i a^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{f} - \frac {a^{2} \tan {\left (e + f x \right )}}{f} & \text {for}\: n = 0 \\- \frac {a^{2} n \left (- i c \tan {\left (e + f x \right )} + c\right )^{n} \tan {\left (e + f x \right )}}{f n^{2} + f n} + \frac {i a^{2} n \left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{f n^{2} + f n} + \frac {2 i a^{2} \left (- i c \tan {\left (e + f x \right )} + c\right )^{n}}{f n^{2} + f n} & \text {otherwise} \end {cases} \]

[In]

integrate((a+I*a*tan(f*x+e))**2*(c-I*c*tan(f*x+e))**n,x)

[Out]

Piecewise((x*(I*a*tan(e) + a)**2*(-I*c*tan(e) + c)**n, Eq(f, 0)), (-2*a**2*f*x*tan(e + f*x)/(2*c*f*tan(e + f*x
) + 2*I*c*f) - 2*I*a**2*f*x/(2*c*f*tan(e + f*x) + 2*I*c*f) - I*a**2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)/(2*c
*f*tan(e + f*x) + 2*I*c*f) + a**2*log(tan(e + f*x)**2 + 1)/(2*c*f*tan(e + f*x) + 2*I*c*f) + 4*a**2/(2*c*f*tan(
e + f*x) + 2*I*c*f), Eq(n, -1)), (2*a**2*x + I*a**2*log(tan(e + f*x)**2 + 1)/f - a**2*tan(e + f*x)/f, Eq(n, 0)
), (-a**2*n*(-I*c*tan(e + f*x) + c)**n*tan(e + f*x)/(f*n**2 + f*n) + I*a**2*n*(-I*c*tan(e + f*x) + c)**n/(f*n*
*2 + f*n) + 2*I*a**2*(-I*c*tan(e + f*x) + c)**n/(f*n**2 + f*n), True))

Maxima [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. \(2 (56) = 112\).

Time = 0.81 (sec) , antiderivative size = 272, normalized size of antiderivative = 4.25 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=\frac {2^{n + 1} a^{2} c^{n} \cos \left (n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - i \cdot 2^{n + 1} a^{2} c^{n} \sin \left (n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + 2 \, {\left (a^{2} c^{n} n + a^{2} c^{n}\right )} 2^{n} \cos \left (-2 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 2 \, e\right ) + 2 \, {\left (-i \, a^{2} c^{n} n - i \, a^{2} c^{n}\right )} 2^{n} \sin \left (-2 \, f x + n \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 2 \, e\right )}{{\left (-i \, n^{2} + {\left (-i \, n^{2} - i \, n\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (n^{2} + n\right )} \sin \left (2 \, f x + 2 \, e\right ) - i \, n\right )} {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {1}{2} \, n} f} \]

[In]

integrate((a+I*a*tan(f*x+e))^2*(c-I*c*tan(f*x+e))^n,x, algorithm="maxima")

[Out]

(2^(n + 1)*a^2*c^n*cos(n*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) - I*2^(n + 1)*a^2*c^n*sin(n*arctan2(
sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1)) + 2*(a^2*c^n*n + a^2*c^n)*2^n*cos(-2*f*x + n*arctan2(sin(2*f*x + 2*e)
, cos(2*f*x + 2*e) + 1) - 2*e) + 2*(-I*a^2*c^n*n - I*a^2*c^n)*2^n*sin(-2*f*x + n*arctan2(sin(2*f*x + 2*e), cos
(2*f*x + 2*e) + 1) - 2*e))/((-I*n^2 + (-I*n^2 - I*n)*cos(2*f*x + 2*e) + (n^2 + n)*sin(2*f*x + 2*e) - I*n)*(cos
(2*f*x + 2*e)^2 + sin(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1)^(1/2*n)*f)

Giac [F]

\[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{2} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

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

[Out]

integrate((I*a*tan(f*x + e) + a)^2*(-I*c*tan(f*x + e) + c)^n, x)

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.75 \[ \int (a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^n \, dx=\frac {a^2\,{\left (\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}\right )}^n\,\left (n\,1{}\mathrm {i}+\cos \left (2\,e+2\,f\,x\right )\,2{}\mathrm {i}+n\,\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-n\,\sin \left (2\,e+2\,f\,x\right )+2{}\mathrm {i}\right )}{f\,n\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )\,\left (n+1\right )} \]

[In]

int((a + a*tan(e + f*x)*1i)^2*(c - c*tan(e + f*x)*1i)^n,x)

[Out]

(a^2*((c*(cos(2*e + 2*f*x) - sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^n*(n*1i + cos(2*e + 2*f*x)*2i +
n*cos(2*e + 2*f*x)*1i - n*sin(2*e + 2*f*x) + 2i))/(f*n*(cos(2*e + 2*f*x) + 1)*(n + 1))