3.955 \(\int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx\)

Optimal. Leaf size=223 \[ -\frac{5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac{15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )}-\frac{5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac{5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}+\frac{35 x}{128 a^3 c^4}-\frac{i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac{i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac{i}{64 a^3 f (c-i c \tan (e+f x))^4} \]

[Out]

(35*x)/(128*a^3*c^4) - (I/64)/(a^3*f*(c - I*c*Tan[e + f*x])^4) - (I/24)/(a^3*c*f*(c - I*c*Tan[e + f*x])^3) + (
I/96)/(a^3*c*f*(c + I*c*Tan[e + f*x])^3) - ((5*I)/64)/(a^3*f*(c^2 - I*c^2*Tan[e + f*x])^2) + ((5*I)/128)/(a^3*
f*(c^2 + I*c^2*Tan[e + f*x])^2) - ((5*I)/32)/(a^3*f*(c^4 - I*c^4*Tan[e + f*x])) + ((15*I)/128)/(a^3*f*(c^4 + I
*c^4*Tan[e + f*x]))

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Rubi [A]  time = 0.208341, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {3522, 3487, 44, 206} \[ -\frac{5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac{15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )}-\frac{5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac{5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}+\frac{35 x}{128 a^3 c^4}-\frac{i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac{i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac{i}{64 a^3 f (c-i c \tan (e+f x))^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4),x]

[Out]

(35*x)/(128*a^3*c^4) - (I/64)/(a^3*f*(c - I*c*Tan[e + f*x])^4) - (I/24)/(a^3*c*f*(c - I*c*Tan[e + f*x])^3) + (
I/96)/(a^3*c*f*(c + I*c*Tan[e + f*x])^3) - ((5*I)/64)/(a^3*f*(c^2 - I*c^2*Tan[e + f*x])^2) + ((5*I)/128)/(a^3*
f*(c^2 + I*c^2*Tan[e + f*x])^2) - ((5*I)/32)/(a^3*f*(c^4 - I*c^4*Tan[e + f*x])) + ((15*I)/128)/(a^3*f*(c^4 + I
*c^4*Tan[e + f*x]))

Rule 3522

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]))

Rule 3487

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^4} \, dx &=\frac{\int \frac{\cos ^6(e+f x)}{c-i c \tan (e+f x)} \, dx}{a^3 c^3}\\ &=\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^4 (c+x)^5} \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=\frac{\left (i c^4\right ) \operatorname{Subst}\left (\int \left (\frac{1}{32 c^5 (c-x)^4}+\frac{5}{64 c^6 (c-x)^3}+\frac{15}{128 c^7 (c-x)^2}+\frac{1}{16 c^4 (c+x)^5}+\frac{1}{8 c^5 (c+x)^4}+\frac{5}{32 c^6 (c+x)^3}+\frac{5}{32 c^7 (c+x)^2}+\frac{35}{128 c^7 \left (c^2-x^2\right )}\right ) \, dx,x,-i c \tan (e+f x)\right )}{a^3 f}\\ &=-\frac{i}{64 a^3 f (c-i c \tan (e+f x))^4}-\frac{i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac{i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac{5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac{5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}-\frac{5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac{15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )}+\frac{(35 i) \operatorname{Subst}\left (\int \frac{1}{c^2-x^2} \, dx,x,-i c \tan (e+f x)\right )}{128 a^3 c^3 f}\\ &=\frac{35 x}{128 a^3 c^4}-\frac{i}{64 a^3 f (c-i c \tan (e+f x))^4}-\frac{i}{24 a^3 c f (c-i c \tan (e+f x))^3}+\frac{i}{96 a^3 c f (c+i c \tan (e+f x))^3}-\frac{5 i}{64 a^3 f \left (c^2-i c^2 \tan (e+f x)\right )^2}+\frac{5 i}{128 a^3 f \left (c^2+i c^2 \tan (e+f x)\right )^2}-\frac{5 i}{32 a^3 f \left (c^4-i c^4 \tan (e+f x)\right )}+\frac{15 i}{128 a^3 f \left (c^4+i c^4 \tan (e+f x)\right )}\\ \end{align*}

Mathematica [A]  time = 1.69047, size = 133, normalized size = 0.6 \[ \frac{(\cos (e+f x)+i \sin (e+f x)) (-840 i f x \sin (e+f x)+420 \sin (e+f x)+378 \sin (3 (e+f x))+70 \sin (5 (e+f x))+7 \sin (7 (e+f x))+420 (2 f x-i) \cos (e+f x)+126 i \cos (3 (e+f x))+14 i \cos (5 (e+f x))+i \cos (7 (e+f x)))}{3072 a^3 c^4 f} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^3*(c - I*c*Tan[e + f*x])^4),x]

[Out]

((Cos[e + f*x] + I*Sin[e + f*x])*(420*(-I + 2*f*x)*Cos[e + f*x] + (126*I)*Cos[3*(e + f*x)] + (14*I)*Cos[5*(e +
f*x)] + I*Cos[7*(e + f*x)] + 420*Sin[e + f*x] - (840*I)*f*x*Sin[e + f*x] + 378*Sin[3*(e + f*x)] + 70*Sin[5*(e
+ f*x)] + 7*Sin[7*(e + f*x)]))/(3072*a^3*c^4*f)

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Maple [A]  time = 0.046, size = 203, normalized size = 0.9 \begin{align*}{\frac{-{\frac{35\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) -i \right ) }{f{a}^{3}{c}^{4}}}-{\frac{{\frac{5\,i}{128}}}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{2}}}-{\frac{1}{96\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) ^{3}}}+{\frac{15}{128\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) -i \right ) }}+{\frac{{\frac{5\,i}{64}}}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{2}}}-{\frac{{\frac{i}{64}}}{f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{4}}}+{\frac{{\frac{35\,i}{256}}\ln \left ( \tan \left ( fx+e \right ) +i \right ) }{f{a}^{3}{c}^{4}}}-{\frac{1}{24\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) ^{3}}}+{\frac{5}{32\,f{a}^{3}{c}^{4} \left ( \tan \left ( fx+e \right ) +i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35/256*I/f/a^3/c^4*ln(tan(f*x+e)-I)-5/128*I/f/a^3/c^4/(tan(f*x+e)-I)^2-1/96/f/a^3/c^4/(tan(f*x+e)-I)^3+15/128
/f/a^3/c^4/(tan(f*x+e)-I)+5/64*I/f/a^3/c^4/(tan(f*x+e)+I)^2-1/64*I/f/a^3/c^4/(tan(f*x+e)+I)^4+35/256*I/f/a^3/c
^4*ln(tan(f*x+e)+I)-1/24/f/a^3/c^4/(tan(f*x+e)+I)^3+5/32/f/a^3/c^4/(tan(f*x+e)+I)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.43305, size = 333, normalized size = 1.49 \begin{align*} \frac{{\left (840 \, f x e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, e^{\left (14 i \, f x + 14 i \, e\right )} - 28 i \, e^{\left (12 i \, f x + 12 i \, e\right )} - 126 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 420 i \, e^{\left (8 i \, f x + 8 i \, e\right )} + 252 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 42 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{3072 \, a^{3} c^{4} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="fricas")

[Out]

1/3072*(840*f*x*e^(6*I*f*x + 6*I*e) - 3*I*e^(14*I*f*x + 14*I*e) - 28*I*e^(12*I*f*x + 12*I*e) - 126*I*e^(10*I*f
*x + 10*I*e) - 420*I*e^(8*I*f*x + 8*I*e) + 252*I*e^(4*I*f*x + 4*I*e) + 42*I*e^(2*I*f*x + 2*I*e) + 4*I)*e^(-6*I
*f*x - 6*I*e)/(a^3*c^4*f)

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Sympy [A]  time = 2.04197, size = 335, normalized size = 1.5 \begin{align*} \begin{cases} \frac{\left (- 10133099161583616 i a^{18} c^{24} f^{6} e^{20 i e} e^{8 i f x} - 94575592174780416 i a^{18} c^{24} f^{6} e^{18 i e} e^{6 i f x} - 425590164786511872 i a^{18} c^{24} f^{6} e^{16 i e} e^{4 i f x} - 1418633882621706240 i a^{18} c^{24} f^{6} e^{14 i e} e^{2 i f x} + 851180329573023744 i a^{18} c^{24} f^{6} e^{10 i e} e^{- 2 i f x} + 141863388262170624 i a^{18} c^{24} f^{6} e^{8 i e} e^{- 4 i f x} + 13510798882111488 i a^{18} c^{24} f^{6} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{10376293541461622784 a^{21} c^{28} f^{7}} & \text{for}\: 10376293541461622784 a^{21} c^{28} f^{7} e^{12 i e} \neq 0 \\x \left (\frac{\left (e^{14 i e} + 7 e^{12 i e} + 21 e^{10 i e} + 35 e^{8 i e} + 35 e^{6 i e} + 21 e^{4 i e} + 7 e^{2 i e} + 1\right ) e^{- 6 i e}}{128 a^{3} c^{4}} - \frac{35}{128 a^{3} c^{4}}\right ) & \text{otherwise} \end{cases} + \frac{35 x}{128 a^{3} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))**3/(c-I*c*tan(f*x+e))**4,x)

[Out]

Piecewise(((-10133099161583616*I*a**18*c**24*f**6*exp(20*I*e)*exp(8*I*f*x) - 94575592174780416*I*a**18*c**24*f
**6*exp(18*I*e)*exp(6*I*f*x) - 425590164786511872*I*a**18*c**24*f**6*exp(16*I*e)*exp(4*I*f*x) - 14186338826217
06240*I*a**18*c**24*f**6*exp(14*I*e)*exp(2*I*f*x) + 851180329573023744*I*a**18*c**24*f**6*exp(10*I*e)*exp(-2*I
*f*x) + 141863388262170624*I*a**18*c**24*f**6*exp(8*I*e)*exp(-4*I*f*x) + 13510798882111488*I*a**18*c**24*f**6*
exp(6*I*e)*exp(-6*I*f*x))*exp(-12*I*e)/(10376293541461622784*a**21*c**28*f**7), Ne(10376293541461622784*a**21*
c**28*f**7*exp(12*I*e), 0)), (x*((exp(14*I*e) + 7*exp(12*I*e) + 21*exp(10*I*e) + 35*exp(8*I*e) + 35*exp(6*I*e)
+ 21*exp(4*I*e) + 7*exp(2*I*e) + 1)*exp(-6*I*e)/(128*a**3*c**4) - 35/(128*a**3*c**4)), True)) + 35*x/(128*a**
3*c**4)

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Giac [A]  time = 1.35806, size = 216, normalized size = 0.97 \begin{align*} -\frac{\frac{420 i \, \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{4}} - \frac{420 i \, \log \left (i \, \tan \left (f x + e\right ) - 1\right )}{a^{3} c^{4}} - \frac{2 \,{\left (385 \, \tan \left (f x + e\right )^{3} - 1335 i \, \tan \left (f x + e\right )^{2} - 1575 \, \tan \left (f x + e\right ) + 641 i\right )}}{a^{3} c^{4}{\left (i \, \tan \left (f x + e\right ) + 1\right )}^{3}} + \frac{875 i \, \tan \left (f x + e\right )^{4} - 3980 \, \tan \left (f x + e\right )^{3} - 6930 i \, \tan \left (f x + e\right )^{2} + 5548 \, \tan \left (f x + e\right ) + 1771 i}{a^{3} c^{4}{\left (\tan \left (f x + e\right ) + i\right )}^{4}}}{3072 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c-I*c*tan(f*x+e))^4,x, algorithm="giac")

[Out]

-1/3072*(420*I*log(tan(f*x + e) - I)/(a^3*c^4) - 420*I*log(I*tan(f*x + e) - 1)/(a^3*c^4) - 2*(385*tan(f*x + e)
^3 - 1335*I*tan(f*x + e)^2 - 1575*tan(f*x + e) + 641*I)/(a^3*c^4*(I*tan(f*x + e) + 1)^3) + (875*I*tan(f*x + e)
^4 - 3980*tan(f*x + e)^3 - 6930*I*tan(f*x + e)^2 + 5548*tan(f*x + e) + 1771*I)/(a^3*c^4*(tan(f*x + e) + I)^4))
/f