3.11.88 \(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx\) [1088]

Optimal. Leaf size=234 \[ \frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )} \]

[Out]

1/8*(c^4+4*I*c^3*d-6*c^2*d^2-4*I*c*d^3-7*d^4)*x/a^3/(c-I*d)/(c+I*d)^4+d^4*ln(c*cos(f*x+e)+d*sin(f*x+e))/a^3/(c
+I*d)^4/(I*c+d)/f-1/6/(I*c-d)/f/(a+I*a*tan(f*x+e))^3+1/8*(I*c-3*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^2+1/8*(c^2
+4*I*c*d-7*d^2)/(I*c-d)^3/f/(a^3+I*a^3*tan(f*x+e))

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Rubi [A]
time = 0.46, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3640, 3677, 3612, 3611} \begin {gather*} \frac {c^2+4 i c d-7 d^2}{8 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {x \left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right )}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^4 (d+i c)}+\frac {-3 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c^4 + (4*I)*c^3*d - 6*c^2*d^2 - (4*I)*c*d^3 - 7*d^4)*x)/(8*a^3*(c - I*d)*(c + I*d)^4) + (d^4*Log[c*Cos[e + f
*x] + d*Sin[e + f*x]])/(a^3*(c + I*d)^4*(I*c + d)*f) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3) + (I*c - 3*d
)/(8*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2) + (c^2 + (4*I)*c*d - 7*d^2)/(8*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e
+ f*x]))

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3640

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n + 1)/(2*f*m*(b*c - a*d))), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3677

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

Rubi steps

\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-3 a (i c-2 d)-3 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx}{6 a^2 (i c-d)}\\ &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-6 a^2 \left (c^2+3 i c d-4 d^2\right )-6 a^2 (c+3 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {\int \frac {6 a^3 \left (i c^3-4 c^2 d-7 i c d^2+8 d^3\right )+6 a^3 d \left (i c^2-4 c d-7 i d^2\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {d^4 \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^3 (c+i d)^4 (i c+d)}\\ &=\frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}

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Mathematica [A]
time = 2.02, size = 435, normalized size = 1.86 \begin {gather*} \frac {\sec ^3(e+f x) \left (-3 \left (9 c^4+28 i c^3 d-18 c^2 d^2+28 i c d^3-27 d^4\right ) \cos (e+f x)+2 \cos (3 (e+f x)) \left (-36 i c^2 d^2 f x+c^4 (-1+6 i f x)+d^4 (1-42 i f x)+2 c d^3 (-i+12 f x)-2 c^3 d (i+12 f x)+24 d^4 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )\right )-9 i c^4 \sin (e+f x)+36 c^3 d \sin (e+f x)+42 i c^2 d^2 \sin (e+f x)+36 c d^3 \sin (e+f x)+51 i d^4 \sin (e+f x)+2 i c^4 \sin (3 (e+f x))-4 c^3 d \sin (3 (e+f x))-4 c d^3 \sin (3 (e+f x))-2 i d^4 \sin (3 (e+f x))-12 c^4 f x \sin (3 (e+f x))-48 i c^3 d f x \sin (3 (e+f x))+72 c^2 d^2 f x \sin (3 (e+f x))+48 i c d^3 f x \sin (3 (e+f x))+84 d^4 f x \sin (3 (e+f x))+48 i d^4 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin (3 (e+f x))\right )}{96 a^3 (c-i d) (c+i d)^4 f (-i+\tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]^3*(-3*(9*c^4 + (28*I)*c^3*d - 18*c^2*d^2 + (28*I)*c*d^3 - 27*d^4)*Cos[e + f*x] + 2*Cos[3*(e + f*
x)]*((-36*I)*c^2*d^2*f*x + c^4*(-1 + (6*I)*f*x) + d^4*(1 - (42*I)*f*x) + 2*c*d^3*(-I + 12*f*x) - 2*c^3*d*(I +
12*f*x) + 24*d^4*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]) - (9*I)*c^4*Sin[e + f*x] + 36*c^3*d*Sin[e + f*x] +
(42*I)*c^2*d^2*Sin[e + f*x] + 36*c*d^3*Sin[e + f*x] + (51*I)*d^4*Sin[e + f*x] + (2*I)*c^4*Sin[3*(e + f*x)] - 4
*c^3*d*Sin[3*(e + f*x)] - 4*c*d^3*Sin[3*(e + f*x)] - (2*I)*d^4*Sin[3*(e + f*x)] - 12*c^4*f*x*Sin[3*(e + f*x)]
- (48*I)*c^3*d*f*x*Sin[3*(e + f*x)] + 72*c^2*d^2*f*x*Sin[3*(e + f*x)] + (48*I)*c*d^3*f*x*Sin[3*(e + f*x)] + 84
*d^4*f*x*Sin[3*(e + f*x)] + (48*I)*d^4*Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[3*(e + f*x)]))/(96*a^3*(c
- I*d)*(c + I*d)^4*f*(-I + Tan[e + f*x])^3)

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Maple [A]
time = 0.56, size = 248, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {\frac {\left (-i c^{3}+11 i c \,d^{2}+5 c^{2} d -15 d^{3}\right ) \ln \left (\tan \left (f x +e \right )-i\right )}{16 \left (i d +c \right )^{4}}-\frac {-5 i c^{2} d +7 i d^{3}-c^{3}+11 c \,d^{2}}{8 \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{3}-7 i c \,d^{2}-5 c^{2} d +3 d^{3}}{8 \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}}{6 \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (i d -c \right ) \left (i d +c \right )^{4}}-\frac {i \ln \left (\tan \left (f x +e \right )+i\right )}{16 i d -16 c}}{f \,a^{3}}\)	\(248\)
default	\(\frac {\frac {\left (-i c^{3}+11 i c \,d^{2}+5 c^{2} d -15 d^{3}\right ) \ln \left (\tan \left (f x +e \right )-i\right )}{16 \left (i d +c \right )^{4}}-\frac {-5 i c^{2} d +7 i d^{3}-c^{3}+11 c \,d^{2}}{8 \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{3}-7 i c \,d^{2}-5 c^{2} d +3 d^{3}}{8 \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}}{6 \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{\left (i d -c \right ) \left (i d +c \right )^{4}}-\frac {i \ln \left (\tan \left (f x +e \right )+i\right )}{16 i d -16 c}}{f \,a^{3}}\)	\(248\)
risch	\(-\frac {x}{8 a^{3} \left (i d -c \right )}-\frac {5 \,{\mathrm e}^{-2 i \left (f x +e \right )} c d}{8 a^{3} \left (i d +c \right )^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {11 i {\mathrm e}^{-2 i \left (f x +e \right )} d^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {5 \,{\mathrm e}^{-4 i \left (f x +e \right )} d}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{48 a^{3} \left (i d +c \right ) f}-\frac {2 d^{4} x}{a^{3} \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {2 d^{4} e}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {i d^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}\)	\(370\)
norman	\(\frac {\frac {-16 i c d -5 c^{2}+17 d^{2}}{12 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {\left (20 i c d +7 c^{2}-17 d^{2}\right ) \tan \left (f x +e \right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (5 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {i d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{2 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (c^{2}+5 d^{2}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{4 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{6}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}-\frac {d^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}\)	\(763\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/f/a^3*(1/16/(c+I*d)^4*(-I*c^3+11*I*c*d^2+5*c^2*d-15*d^3)*ln(tan(f*x+e)-I)-1/8*(-5*I*c^2*d+7*I*d^3-c^3+11*c*d
^2)/(c+I*d)^4/(tan(f*x+e)-I)-1/8*(I*c^3-7*I*c*d^2-5*c^2*d+3*d^3)/(c+I*d)^4/(tan(f*x+e)-I)^2-1/6*(3*I*c^2*d-I*d
^3+c^3-3*c*d^2)/(c+I*d)^4/(tan(f*x+e)-I)^3+I*d^4/(I*d-c)/(c+I*d)^4*ln(c+d*tan(f*x+e))-I/(16*I*d-16*c)*ln(tan(f
*x+e)+I))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 1.35, size = 275, normalized size = 1.18 \begin {gather*} -\frac {{\left (96 \, d^{4} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) - 2 \, c^{4} - 4 i \, c^{3} d - 4 i \, c d^{3} + 2 \, d^{4} - 12 \, {\left (-i \, c^{4} + 4 \, c^{3} d + 6 i \, c^{2} d^{2} - 4 \, c d^{3} + 15 i \, d^{4}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 6 \, {\left (3 \, c^{4} + 10 i \, c^{3} d - 8 \, c^{2} d^{2} + 10 i \, c d^{3} - 11 \, d^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 3 \, {\left (3 \, c^{4} + 8 i \, c^{3} d - 2 \, c^{2} d^{2} + 8 i \, c d^{3} - 5 \, d^{4}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, {\left (-i \, a^{3} c^{5} + 3 \, a^{3} c^{4} d + 2 i \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} + 3 i \, a^{3} c d^{4} - a^{3} d^{5}\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(96*d^4*e^(6*I*f*x + 6*I*e)*log(((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c - d)/(I*c + d)) - 2*c^4 - 4*I*c^3*d
- 4*I*c*d^3 + 2*d^4 - 12*(-I*c^4 + 4*c^3*d + 6*I*c^2*d^2 - 4*c*d^3 + 15*I*d^4)*f*x*e^(6*I*f*x + 6*I*e) - 6*(3
*c^4 + 10*I*c^3*d - 8*c^2*d^2 + 10*I*c*d^3 - 11*d^4)*e^(4*I*f*x + 4*I*e) - 3*(3*c^4 + 8*I*c^3*d - 2*c^2*d^2 +
8*I*c*d^3 - 5*d^4)*e^(2*I*f*x + 2*I*e))*e^(-6*I*f*x - 6*I*e)/((-I*a^3*c^5 + 3*a^3*c^4*d + 2*I*a^3*c^3*d^2 + 2*
a^3*c^2*d^3 + 3*I*a^3*c*d^4 - a^3*d^5)*f)

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Sympy [A]
time = 14.34, size = 1192, normalized size = 5.09 \begin {gather*} \frac {x \left (c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}\right )}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \begin {cases} \frac {\left (512 i a^{6} c^{5} f^{2} e^{6 i e} - 2560 a^{6} c^{4} d f^{2} e^{6 i e} - 5120 i a^{6} c^{3} d^{2} f^{2} e^{6 i e} + 5120 a^{6} c^{2} d^{3} f^{2} e^{6 i e} + 2560 i a^{6} c d^{4} f^{2} e^{6 i e} - 512 a^{6} d^{5} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (2304 i a^{6} c^{5} f^{2} e^{8 i e} - 13056 a^{6} c^{4} d f^{2} e^{8 i e} - 29184 i a^{6} c^{3} d^{2} f^{2} e^{8 i e} + 32256 a^{6} c^{2} d^{3} f^{2} e^{8 i e} + 17664 i a^{6} c d^{4} f^{2} e^{8 i e} - 3840 a^{6} d^{5} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (4608 i a^{6} c^{5} f^{2} e^{10 i e} - 29184 a^{6} c^{4} d f^{2} e^{10 i e} - 76800 i a^{6} c^{3} d^{2} f^{2} e^{10 i e} + 101376 a^{6} c^{2} d^{3} f^{2} e^{10 i e} + 66048 i a^{6} c d^{4} f^{2} e^{10 i e} - 16896 a^{6} d^{5} f^{2} e^{10 i e}\right ) e^{- 2 i f x}}{24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e}} & \text {for}\: 24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac {c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \frac {c^{3} e^{6 i e} + 3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3} + 5 i c^{2} d e^{6 i e} + 13 i c^{2} d e^{4 i e} + 11 i c^{2} d e^{2 i e} + 3 i c^{2} d - 11 c d^{2} e^{6 i e} - 21 c d^{2} e^{4 i e} - 13 c d^{2} e^{2 i e} - 3 c d^{2} - 15 i d^{3} e^{6 i e} - 11 i d^{3} e^{4 i e} - 5 i d^{3} e^{2 i e} - i d^{3}}{8 a^{3} c^{4} e^{6 i e} + 32 i a^{3} c^{3} d e^{6 i e} - 48 a^{3} c^{2} d^{2} e^{6 i e} - 32 i a^{3} c d^{3} e^{6 i e} + 8 a^{3} d^{4} e^{6 i e}}\right ) & \text {otherwise} \end {cases} - \frac {i d^{4} \log {\left (\frac {c + i d}{c e^{2 i e} - i d e^{2 i e}} + e^{2 i f x} \right )}}{a^{3} f \left (c - i d\right ) \left (c + i d\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(c**3 + 5*I*c**2*d - 11*c*d**2 - 15*I*d**3)/(8*a**3*c**4 + 32*I*a**3*c**3*d - 48*a**3*c**2*d**2 - 32*I*a**3*
c*d**3 + 8*a**3*d**4) + Piecewise((((512*I*a**6*c**5*f**2*exp(6*I*e) - 2560*a**6*c**4*d*f**2*exp(6*I*e) - 5120
*I*a**6*c**3*d**2*f**2*exp(6*I*e) + 5120*a**6*c**2*d**3*f**2*exp(6*I*e) + 2560*I*a**6*c*d**4*f**2*exp(6*I*e) -
512*a**6*d**5*f**2*exp(6*I*e))*exp(-6*I*f*x) + (2304*I*a**6*c**5*f**2*exp(8*I*e) - 13056*a**6*c**4*d*f**2*exp
(8*I*e) - 29184*I*a**6*c**3*d**2*f**2*exp(8*I*e) + 32256*a**6*c**2*d**3*f**2*exp(8*I*e) + 17664*I*a**6*c*d**4*
f**2*exp(8*I*e) - 3840*a**6*d**5*f**2*exp(8*I*e))*exp(-4*I*f*x) + (4608*I*a**6*c**5*f**2*exp(10*I*e) - 29184*a
**6*c**4*d*f**2*exp(10*I*e) - 76800*I*a**6*c**3*d**2*f**2*exp(10*I*e) + 101376*a**6*c**2*d**3*f**2*exp(10*I*e)
+ 66048*I*a**6*c*d**4*f**2*exp(10*I*e) - 16896*a**6*d**5*f**2*exp(10*I*e))*exp(-2*I*f*x))/(24576*a**9*c**6*f*
*3*exp(12*I*e) + 147456*I*a**9*c**5*d*f**3*exp(12*I*e) - 368640*a**9*c**4*d**2*f**3*exp(12*I*e) - 491520*I*a**
9*c**3*d**3*f**3*exp(12*I*e) + 368640*a**9*c**2*d**4*f**3*exp(12*I*e) + 147456*I*a**9*c*d**5*f**3*exp(12*I*e)
- 24576*a**9*d**6*f**3*exp(12*I*e)), Ne(24576*a**9*c**6*f**3*exp(12*I*e) + 147456*I*a**9*c**5*d*f**3*exp(12*I*
e) - 368640*a**9*c**4*d**2*f**3*exp(12*I*e) - 491520*I*a**9*c**3*d**3*f**3*exp(12*I*e) + 368640*a**9*c**2*d**4
*f**3*exp(12*I*e) + 147456*I*a**9*c*d**5*f**3*exp(12*I*e) - 24576*a**9*d**6*f**3*exp(12*I*e), 0)), (x*(-(c**3
+ 5*I*c**2*d - 11*c*d**2 - 15*I*d**3)/(8*a**3*c**4 + 32*I*a**3*c**3*d - 48*a**3*c**2*d**2 - 32*I*a**3*c*d**3 +
8*a**3*d**4) + (c**3*exp(6*I*e) + 3*c**3*exp(4*I*e) + 3*c**3*exp(2*I*e) + c**3 + 5*I*c**2*d*exp(6*I*e) + 13*I
*c**2*d*exp(4*I*e) + 11*I*c**2*d*exp(2*I*e) + 3*I*c**2*d - 11*c*d**2*exp(6*I*e) - 21*c*d**2*exp(4*I*e) - 13*c*
d**2*exp(2*I*e) - 3*c*d**2 - 15*I*d**3*exp(6*I*e) - 11*I*d**3*exp(4*I*e) - 5*I*d**3*exp(2*I*e) - I*d**3)/(8*a*
*3*c**4*exp(6*I*e) + 32*I*a**3*c**3*d*exp(6*I*e) - 48*a**3*c**2*d**2*exp(6*I*e) - 32*I*a**3*c*d**3*exp(6*I*e)
+ 8*a**3*d**4*exp(6*I*e))), True)) - I*d**4*log((c + I*d)/(c*exp(2*I*e) - I*d*exp(2*I*e)) + exp(2*I*f*x))/(a**
3*f*(c - I*d)*(c + I*d)**4)

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (203) = 406\).
time = 0.69, size = 445, normalized size = 1.90 \begin {gather*} -\frac {\frac {96 i \, d^{5} \log \left (-i \, d \tan \left (f x + e\right ) - i \, c\right )}{a^{3} c^{5} d + 3 i \, a^{3} c^{4} d^{2} - 2 \, a^{3} c^{3} d^{3} + 2 i \, a^{3} c^{2} d^{4} - 3 \, a^{3} c d^{5} - i \, a^{3} d^{6}} - \frac {6 \, {\left (-i \, c^{3} + 5 \, c^{2} d + 11 i \, c d^{2} - 15 \, d^{3}\right )} \log \left (i \, \tan \left (f x + e\right ) + 1\right )}{a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}} - \frac {192 \, \log \left (\tan \left (f x + e\right ) + i\right )}{-32 i \, a^{3} c - 32 \, a^{3} d} - \frac {11 i \, c^{3} \tan \left (f x + e\right )^{3} - 55 \, c^{2} d \tan \left (f x + e\right )^{3} - 121 i \, c d^{2} \tan \left (f x + e\right )^{3} + 165 \, d^{3} \tan \left (f x + e\right )^{3} + 45 \, c^{3} \tan \left (f x + e\right )^{2} + 225 i \, c^{2} d \tan \left (f x + e\right )^{2} - 495 \, c d^{2} \tan \left (f x + e\right )^{2} - 579 i \, d^{3} \tan \left (f x + e\right )^{2} - 69 i \, c^{3} \tan \left (f x + e\right ) + 345 \, c^{2} d \tan \left (f x + e\right ) + 711 i \, c d^{2} \tan \left (f x + e\right ) - 699 \, d^{3} \tan \left (f x + e\right ) - 51 \, c^{3} - 223 i \, c^{2} d + 385 \, c d^{2} + 301 i \, d^{3}}{{\left (a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}\right )} {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(96*I*d^5*log(-I*d*tan(f*x + e) - I*c)/(a^3*c^5*d + 3*I*a^3*c^4*d^2 - 2*a^3*c^3*d^3 + 2*I*a^3*c^2*d^4 -
3*a^3*c*d^5 - I*a^3*d^6) - 6*(-I*c^3 + 5*c^2*d + 11*I*c*d^2 - 15*d^3)*log(I*tan(f*x + e) + 1)/(a^3*c^4 + 4*I*a
^3*c^3*d - 6*a^3*c^2*d^2 - 4*I*a^3*c*d^3 + a^3*d^4) - 192*log(tan(f*x + e) + I)/(-32*I*a^3*c - 32*a^3*d) - (11
*I*c^3*tan(f*x + e)^3 - 55*c^2*d*tan(f*x + e)^3 - 121*I*c*d^2*tan(f*x + e)^3 + 165*d^3*tan(f*x + e)^3 + 45*c^3
*tan(f*x + e)^2 + 225*I*c^2*d*tan(f*x + e)^2 - 495*c*d^2*tan(f*x + e)^2 - 579*I*d^3*tan(f*x + e)^2 - 69*I*c^3*
tan(f*x + e) + 345*c^2*d*tan(f*x + e) + 711*I*c*d^2*tan(f*x + e) - 699*d^3*tan(f*x + e) - 51*c^3 - 223*I*c^2*d
+ 385*c*d^2 + 301*I*d^3)/((a^3*c^4 + 4*I*a^3*c^3*d - 6*a^3*c^2*d^2 - 4*I*a^3*c*d^3 + a^3*d^4)*(tan(f*x + e) -
I)^3))/f

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Mupad [B]
time = 10.00, size = 1952, normalized size = 8.34 \begin {gather*} \frac {\sum _{k=1}^3\ln \left (-\left (c^5\,d+c^4\,d^2\,8{}\mathrm {i}-30\,c^3\,d^3-c^2\,d^4\,64{}\mathrm {i}+81\,c\,d^5+d^6\,56{}\mathrm {i}\right )\,\left (-a^3\,c^6\,d^2+a^3\,c^5\,d^3\,6{}\mathrm {i}+15\,a^3\,c^4\,d^4-a^3\,c^3\,d^5\,20{}\mathrm {i}-15\,a^3\,c^2\,d^6+a^3\,c\,d^7\,6{}\mathrm {i}+a^3\,d^8\right )-\mathrm {root}\left (a^9\,c^5\,d^5\,e^3\,7168{}\mathrm {i}+3584\,a^9\,c^6\,d^4\,e^3-3584\,a^9\,c^4\,d^6\,e^3+3328\,a^9\,c^8\,d^2\,e^3-3328\,a^9\,c^2\,d^8\,e^3+a^9\,c^7\,d^3\,e^3\,2048{}\mathrm {i}+a^9\,c^3\,d^7\,e^3\,2048{}\mathrm {i}-a^9\,c^9\,d\,e^3\,1536{}\mathrm {i}-a^9\,c\,d^9\,e^3\,1536{}\mathrm {i}+256\,a^9\,d^{10}\,e^3-256\,a^9\,c^{10}\,e^3-a^3\,c\,d^7\,e\,56{}\mathrm {i}-a^3\,c^7\,d\,e\,8{}\mathrm {i}-68\,a^3\,c^2\,d^6\,e+a^3\,c^5\,d^3\,e\,56{}\mathrm {i}-54\,a^3\,c^4\,d^4\,e+28\,a^3\,c^6\,d^2\,e+a^3\,c^3\,d^5\,e\,8{}\mathrm {i}-241\,a^3\,d^8\,e-a^3\,c^8\,e-c^3\,d^4\,1{}\mathrm {i}+5\,c^2\,d^5+c\,d^6\,11{}\mathrm {i}-15\,d^7,e,k\right )\,\left (\left (-a^3\,c^6\,d^2+a^3\,c^5\,d^3\,6{}\mathrm {i}+15\,a^3\,c^4\,d^4-a^3\,c^3\,d^5\,20{}\mathrm {i}-15\,a^3\,c^2\,d^6+a^3\,c\,d^7\,6{}\mathrm {i}+a^3\,d^8\right )\,\left (8\,a^3\,c^7+a^3\,c^6\,d\,56{}\mathrm {i}-184\,a^3\,c^5\,d^2-a^3\,c^4\,d^3\,392{}\mathrm {i}+568\,a^3\,c^3\,d^4+a^3\,c^2\,d^5\,520{}\mathrm {i}-264\,a^3\,c\,d^6-a^3\,d^7\,56{}\mathrm {i}\right )+\mathrm {root}\left (a^9\,c^5\,d^5\,e^3\,7168{}\mathrm {i}+3584\,a^9\,c^6\,d^4\,e^3-3584\,a^9\,c^4\,d^6\,e^3+3328\,a^9\,c^8\,d^2\,e^3-3328\,a^9\,c^2\,d^8\,e^3+a^9\,c^7\,d^3\,e^3\,2048{}\mathrm {i}+a^9\,c^3\,d^7\,e^3\,2048{}\mathrm {i}-a^9\,c^9\,d\,e^3\,1536{}\mathrm {i}-a^9\,c\,d^9\,e^3\,1536{}\mathrm {i}+256\,a^9\,d^{10}\,e^3-256\,a^9\,c^{10}\,e^3-a^3\,c\,d^7\,e\,56{}\mathrm {i}-a^3\,c^7\,d\,e\,8{}\mathrm {i}-68\,a^3\,c^2\,d^6\,e+a^3\,c^5\,d^3\,e\,56{}\mathrm {i}-54\,a^3\,c^4\,d^4\,e+28\,a^3\,c^6\,d^2\,e+a^3\,c^3\,d^5\,e\,8{}\mathrm {i}-241\,a^3\,d^8\,e-a^3\,c^8\,e-c^3\,d^4\,1{}\mathrm {i}+5\,c^2\,d^5+c\,d^6\,11{}\mathrm {i}-15\,d^7,e,k\right )\,\left (\left (512\,a^6\,c^7\,d+a^6\,c^6\,d^2\,3072{}\mathrm {i}-7680\,a^6\,c^5\,d^3-a^6\,c^4\,d^4\,10240{}\mathrm {i}+7680\,a^6\,c^3\,d^5+a^6\,c^2\,d^6\,3072{}\mathrm {i}-512\,a^6\,c\,d^7\right )\,\left (-a^3\,c^6\,d^2+a^3\,c^5\,d^3\,6{}\mathrm {i}+15\,a^3\,c^4\,d^4-a^3\,c^3\,d^5\,20{}\mathrm {i}-15\,a^3\,c^2\,d^6+a^3\,c\,d^7\,6{}\mathrm {i}+a^3\,d^8\right )-\mathrm {tan}\left (e+f\,x\right )\,\left (-a^3\,c^6\,d^2+a^3\,c^5\,d^3\,6{}\mathrm {i}+15\,a^3\,c^4\,d^4-a^3\,c^3\,d^5\,20{}\mathrm {i}-15\,a^3\,c^2\,d^6+a^3\,c\,d^7\,6{}\mathrm {i}+a^3\,d^8\right )\,\left (128\,a^6\,c^8+a^6\,c^7\,d\,768{}\mathrm {i}-2304\,a^6\,c^6\,d^2-a^6\,c^5\,d^3\,4864{}\mathrm {i}+7680\,a^6\,c^4\,d^4+a^6\,c^3\,d^5\,8448{}\mathrm {i}-5888\,a^6\,c^2\,d^6-a^6\,c\,d^7\,2304{}\mathrm {i}+384\,a^6\,d^8\right )\right )+\mathrm {tan}\left (e+f\,x\right )\,\left (16\,a^3\,c^6\,d+a^3\,c^5\,d^2\,112{}\mathrm {i}-352\,a^3\,c^4\,d^3-a^3\,c^3\,d^4\,736{}\mathrm {i}+976\,a^3\,c^2\,d^5+a^3\,c\,d^6\,688{}\mathrm {i}-192\,a^3\,d^7\right )\,\left (-a^3\,c^6\,d^2+a^3\,c^5\,d^3\,6{}\mathrm {i}+15\,a^3\,c^4\,d^4-a^3\,c^3\,d^5\,20{}\mathrm {i}-15\,a^3\,c^2\,d^6+a^3\,c\,d^7\,6{}\mathrm {i}+a^3\,d^8\right )\right )-\mathrm {tan}\left (e+f\,x\right )\,\left (c^4\,d^2+c^3\,d^3\,8{}\mathrm {i}-30\,c^2\,d^4-c\,d^5\,56{}\mathrm {i}+49\,d^6\right )\,\left (-a^3\,c^6\,d^2+a^3\,c^5\,d^3\,6{}\mathrm {i}+15\,a^3\,c^4\,d^4-a^3\,c^3\,d^5\,20{}\mathrm {i}-15\,a^3\,c^2\,d^6+a^3\,c\,d^7\,6{}\mathrm {i}+a^3\,d^8\right )\right )\,\mathrm {root}\left (a^9\,c^5\,d^5\,e^3\,7168{}\mathrm {i}+3584\,a^9\,c^6\,d^4\,e^3-3584\,a^9\,c^4\,d^6\,e^3+3328\,a^9\,c^8\,d^2\,e^3-3328\,a^9\,c^2\,d^8\,e^3+a^9\,c^7\,d^3\,e^3\,2048{}\mathrm {i}+a^9\,c^3\,d^7\,e^3\,2048{}\mathrm {i}-a^9\,c^9\,d\,e^3\,1536{}\mathrm {i}-a^9\,c\,d^9\,e^3\,1536{}\mathrm {i}+256\,a^9\,d^{10}\,e^3-256\,a^9\,c^{10}\,e^3-a^3\,c\,d^7\,e\,56{}\mathrm {i}-a^3\,c^7\,d\,e\,8{}\mathrm {i}-68\,a^3\,c^2\,d^6\,e+a^3\,c^5\,d^3\,e\,56{}\mathrm {i}-54\,a^3\,c^4\,d^4\,e+28\,a^3\,c^6\,d^2\,e+a^3\,c^3\,d^5\,e\,8{}\mathrm {i}-241\,a^3\,d^8\,e-a^3\,c^8\,e-c^3\,d^4\,1{}\mathrm {i}+5\,c^2\,d^5+c\,d^6\,11{}\mathrm {i}-15\,d^7,e,k\right )}{f}-\frac {\frac {10\,c^2+c\,d\,32{}\mathrm {i}-34\,d^2}{24\,a^3\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (3\,c^2+c\,d\,12{}\mathrm {i}-17\,d^2\right )\,1{}\mathrm {i}}{8\,a^3\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}-\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (c^2+c\,d\,4{}\mathrm {i}-7\,d^2\right )}{8\,a^3\,\left (-c^3-c^2\,d\,3{}\mathrm {i}+3\,c\,d^2+d^3\,1{}\mathrm {i}\right )}}{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 )-\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(- (81*c*d^5 + c^5*d + d^6*56i - c^2*d^4*64i - 30*c^3*d^3 + c^4*d^2*8i)*(a^3*d^8 + a^3*c*d^7*6i - 15
*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2) - root(a^9*c^5*d^5*e^3*7168i +
3584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d^6*e^3 + 3328*a^9*c^8*d^2*e^3 - 3328*a^9*c^2*d^8*e^3 + a^9*c^7*d^3*e^3*2
048i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d*e^3*1536i - a^9*c*d^9*e^3*1536i + 256*a^9*d^10*e^3 - 256*a^9*c^10*e^3
- a^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68*a^3*c^2*d^6*e + a^3*c^5*d^3*e*56i - 54*a^3*c^4*d^4*e + 28*a^3*c^6*d^2
*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^3*c^8*e - c^3*d^4*1i + 5*c^2*d^5 + c*d^6*11i - 15*d^7, e, k)*((a^3*d
^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)*(8*a^3*c
^7 - a^3*d^7*56i - 264*a^3*c*d^6 + a^3*c^6*d*56i + a^3*c^2*d^5*520i + 568*a^3*c^3*d^4 - a^3*c^4*d^3*392i - 184
*a^3*c^5*d^2) + root(a^9*c^5*d^5*e^3*7168i + 3584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d^6*e^3 + 3328*a^9*c^8*d^2*e^
3 - 3328*a^9*c^2*d^8*e^3 + a^9*c^7*d^3*e^3*2048i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d*e^3*1536i - a^9*c*d^9*e^3
*1536i + 256*a^9*d^10*e^3 - 256*a^9*c^10*e^3 - a^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68*a^3*c^2*d^6*e + a^3*c^5*d
^3*e*56i - 54*a^3*c^4*d^4*e + 28*a^3*c^6*d^2*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^3*c^8*e - c^3*d^4*1i + 5
*c^2*d^5 + c*d^6*11i - 15*d^7, e, k)*((512*a^6*c^7*d - 512*a^6*c*d^7 + a^6*c^2*d^6*3072i + 7680*a^6*c^3*d^5 -
a^6*c^4*d^4*10240i - 7680*a^6*c^5*d^3 + a^6*c^6*d^2*3072i)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*
d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2) - tan(e + f*x)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d
^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)*(128*a^6*c^8 + 384*a^6*d^8 - a^6*c*d^7*2
304i + a^6*c^7*d*768i - 5888*a^6*c^2*d^6 + a^6*c^3*d^5*8448i + 7680*a^6*c^4*d^4 - a^6*c^5*d^3*4864i - 2304*a^6
*c^6*d^2)) + tan(e + f*x)*(a^3*c*d^6*688i - 192*a^3*d^7 + 16*a^3*c^6*d + 976*a^3*c^2*d^5 - a^3*c^3*d^4*736i -
352*a^3*c^4*d^3 + a^3*c^5*d^2*112i)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^
4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)) - tan(e + f*x)*(49*d^6 - c*d^5*56i - 30*c^2*d^4 + c^3*d^3*8i + c^4*d^2)*(a^
3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2))*root
(a^9*c^5*d^5*e^3*7168i + 3584*a^9*c^6*d^4*e^3 - 3584*a^9*c^4*d^6*e^3 + 3328*a^9*c^8*d^2*e^3 - 3328*a^9*c^2*d^8
*e^3 + a^9*c^7*d^3*e^3*2048i + a^9*c^3*d^7*e^3*2048i - a^9*c^9*d*e^3*1536i - a^9*c*d^9*e^3*1536i + 256*a^9*d^1
0*e^3 - 256*a^9*c^10*e^3 - a^3*c*d^7*e*56i - a^3*c^7*d*e*8i - 68*a^3*c^2*d^6*e + a^3*c^5*d^3*e*56i - 54*a^3*c^
4*d^4*e + 28*a^3*c^6*d^2*e + a^3*c^3*d^5*e*8i - 241*a^3*d^8*e - a^3*c^8*e - c^3*d^4*1i + 5*c^2*d^5 + c*d^6*11i
- 15*d^7, e, k), k, 1, 3)/f - ((c*d*32i + 10*c^2 - 34*d^2)/(24*a^3*(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)) + (ta
n(e + f*x)*(c*d*12i + 3*c^2 - 17*d^2)*1i)/(8*a^3*(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)) - (tan(e + f*x)^2*(c*d*4
i + c^2 - 7*d^2))/(8*a^3*(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)))/(f*(3*tan(e + f*x) + tan(e + f*x)^2*3i - tan(e
+ f*x)^3 - 1i))

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