\(\int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx\) [918]

Optimal result

Integrand size = 29, antiderivative size = 25 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {i a (c-i c \tan (e+f x))^4}{4 f} \]

[Out]

1/4*I*a*(c-I*c*tan(f*x+e))^4/f

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3603, 3568, 32} \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {i a (c-i c \tan (e+f x))^4}{4 f} \]

[In]

Int[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]

[Out]

((I/4)*a*(c - I*c*Tan[e + f*x])^4)/f

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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}& = (a c) \int \sec ^2(e+f x) (c-i c \tan (e+f x))^3 \, dx \\ & = \frac {(i a) \text {Subst}\left (\int (c+x)^3 \, dx,x,-i c \tan (e+f x)\right )}{f} \\ & = \frac {i a (c-i c \tan (e+f x))^4}{4 f} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(25)=50\).

Time = 0.38 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {a c^4 \tan (e+f x) \left (4-6 i \tan (e+f x)-4 \tan ^2(e+f x)+i \tan ^3(e+f x)\right )}{4 f} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {i a \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}}{4 f}\)	\(22\)
default	\(\frac {i a \,c^{4} \left (\tan \left (f x +e \right )+i\right )^{4}}{4 f}\)	\(22\)
risch	\(\frac {4 i a \,c^{4}}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}\)	\(24\)
parallelrisch	\(\frac {i a \,c^{4} \left (\tan ^{4}\left (f x +e \right )\right )-6 i a \,c^{4} \left (\tan ^{2}\left (f x +e \right )\right )-4 \left (\tan ^{3}\left (f x +e \right )\right ) a \,c^{4}+4 \tan \left (f x +e \right ) a \,c^{4}}{4 f}\)	\(63\)
norman	\(\frac {a \,c^{4} \tan \left (f x +e \right )}{f}-\frac {a \,c^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{f}-\frac {3 i a \,c^{4} \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {i a \,c^{4} \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}\)	\(69\)
parts	\(a \,c^{4} x -\frac {3 i a \,c^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}-\frac {2 i a \,c^{4} \left (\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}+\frac {i a \,c^{4} \left (\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}-\frac {2 a \,c^{4} \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}-\frac {3 a \,c^{4} \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\)	\(167\)

[In]

int((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

1/4*I/f*a*c^4*(tan(f*x+e)+I)^4

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (19) = 38\).

Time = 0.23 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

4*I*a*c^4/(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e)
+ 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. 82 vs. \(2 (19) = 38\).

Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.28 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {4 i a c^{4}}{f e^{8 i e} e^{8 i f x} + 4 f e^{6 i e} e^{6 i f x} + 6 f e^{4 i e} e^{4 i f x} + 4 f e^{2 i e} e^{2 i f x} + f} \]

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))**4,x)

[Out]

4*I*a*c**4/(f*exp(8*I*e)*exp(8*I*f*x) + 4*f*exp(6*I*e)*exp(6*I*f*x) + 6*f*exp(4*I*e)*exp(4*I*f*x) + 4*f*exp(2*
I*e)*exp(2*I*f*x) + f)

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. 60 vs. \(2 (19) = 38\).

Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {i \, a c^{4} \tan \left (f x + e\right )^{4} - 4 \, a c^{4} \tan \left (f x + e\right )^{3} - 6 i \, a c^{4} \tan \left (f x + e\right )^{2} + 4 \, a c^{4} \tan \left (f x + e\right )}{4 \, f} \]

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

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

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. 57 vs. \(2 (19) = 38\).

Time = 0.53 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.28 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {4 i \, a c^{4}}{f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f} \]

[In]

integrate((a+I*a*tan(f*x+e))*(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

4*I*a*c^4/(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e)
+ f)

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int (a+i a \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {a\,c^4\,\sin \left (e+f\,x\right )\,\left (-4\,{\cos \left (e+f\,x\right )}^3+{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )\,6{}\mathrm {i}+4\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^2-{\sin \left (e+f\,x\right )}^3\,1{}\mathrm {i}\right )}{4\,f\,{\cos \left (e+f\,x\right )}^4} \]

[In]

int((a + a*tan(e + f*x)*1i)*(c - c*tan(e + f*x)*1i)^4,x)

[Out]

-(a*c^4*sin(e + f*x)*(4*cos(e + f*x)*sin(e + f*x)^2 + cos(e + f*x)^2*sin(e + f*x)*6i - 4*cos(e + f*x)^3 - sin(
e + f*x)^3*1i))/(4*f*cos(e + f*x)^4)