3.7.94 \(\int \frac {1}{(a+b \tan (c+d x))^{4/3}} \, dx\) [694]

Optimal. Leaf size=336 \[ -\frac {x}{4 (a-i b)^{4/3}}-\frac {x}{4 (a+i b)^{4/3}}+\frac {i \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 (a-i b)^{4/3} d}-\frac {i \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 (a+i b)^{4/3} d}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{4/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{4/3} d}-\frac {3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}} \]

[Out]

-1/4*x/(a-I*b)^(4/3)-1/4*x/(a+I*b)^(4/3)+1/4*I*ln(cos(d*x+c))/(a-I*b)^(4/3)/d-1/4*I*ln(cos(d*x+c))/(a+I*b)^(4/
3)/d+3/4*I*ln((a-I*b)^(1/3)-(a+b*tan(d*x+c))^(1/3))/(a-I*b)^(4/3)/d-3/4*I*ln((a+I*b)^(1/3)-(a+b*tan(d*x+c))^(1
/3))/(a+I*b)^(4/3)/d+1/2*I*arctan(1/3*(1+2*(a+b*tan(d*x+c))^(1/3)/(a-I*b)^(1/3))*3^(1/2))*3^(1/2)/(a-I*b)^(4/3
)/d-1/2*I*arctan(1/3*(1+2*(a+b*tan(d*x+c))^(1/3)/(a+I*b)^(1/3))*3^(1/2))*3^(1/2)/(a+I*b)^(4/3)/d-3*b/(a^2+b^2)
/d/(a+b*tan(d*x+c))^(1/3)

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Rubi [A]
time = 0.27, antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3564, 3620, 3618, 57, 631, 210, 31} \begin {gather*} -\frac {3 b}{d \left (a^2+b^2\right ) \sqrt [3]{a+b \tan (c+d x)}}+\frac {i \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 d (a-i b)^{4/3}}-\frac {i \sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 d (a+i b)^{4/3}}+\frac {3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{4 d (a-i b)^{4/3}}-\frac {3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{4 d (a+i b)^{4/3}}+\frac {i \log (\cos (c+d x))}{4 d (a-i b)^{4/3}}-\frac {i \log (\cos (c+d x))}{4 d (a+i b)^{4/3}}-\frac {x}{4 (a-i b)^{4/3}}-\frac {x}{4 (a+i b)^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(-4/3),x]

[Out]

-1/4*x/(a - I*b)^(4/3) - x/(4*(a + I*b)^(4/3)) + ((I/2)*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a
- I*b)^(1/3))/Sqrt[3]])/((a - I*b)^(4/3)*d) - ((I/2)*Sqrt[3]*ArcTan[(1 + (2*(a + b*Tan[c + d*x])^(1/3))/(a + I
*b)^(1/3))/Sqrt[3]])/((a + I*b)^(4/3)*d) + ((I/4)*Log[Cos[c + d*x]])/((a - I*b)^(4/3)*d) - ((I/4)*Log[Cos[c +
d*x]])/((a + I*b)^(4/3)*d) + (((3*I)/4)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/((a - I*b)^(4/3)*d)
- (((3*I)/4)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/((a + I*b)^(4/3)*d) - (3*b)/((a^2 + b^2)*d*(a
+ b*Tan[c + d*x])^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/
3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ
[(b*c - a*d)/b]

Rule 210

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 3564

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*
(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3618

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

Rule 3620

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

Rubi steps

\begin {align*} \int \frac {1}{(a+b \tan (c+d x))^{4/3}} \, dx &=-\frac {3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac {\int \frac {a-b \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx}{a^2+b^2}\\ &=-\frac {3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac {\int \frac {1+i \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx}{2 (a-i b)}+\frac {\int \frac {1-i \tan (c+d x)}{\sqrt [3]{a+b \tan (c+d x)}} \, dx}{2 (a+i b)}\\ &=-\frac {3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (i a-b) d}-\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt [3]{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (i a+b) d}\\ &=-\frac {x}{4 (a-i b)^{4/3}}-\frac {x}{4 (a+i b)^{4/3}}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{4/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{4/3} d}-\frac {3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}+\frac {3 \text {Subst}\left (\int \frac {1}{(a+i b)^{2/3}+\sqrt [3]{a+i b} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (i a-b) d}-\frac {(3 i) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a-i b}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{4/3} d}+\frac {(3 i) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+i b}-x} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{4/3} d}-\frac {3 \text {Subst}\left (\int \frac {1}{(a-i b)^{2/3}+\sqrt [3]{a-i b} x+x^2} \, dx,x,\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (i a+b) d}\\ &=-\frac {x}{4 (a-i b)^{4/3}}-\frac {x}{4 (a+i b)^{4/3}}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{4/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{4/3} d}-\frac {3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}-\frac {(3 i) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}\right )}{2 (a-i b)^{4/3} d}+\frac {(3 i) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}\right )}{2 (a+i b)^{4/3} d}\\ &=-\frac {x}{4 (a-i b)^{4/3}}-\frac {x}{4 (a+i b)^{4/3}}+\frac {i \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 (a-i b)^{4/3} d}-\frac {i \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 (a+i b)^{4/3} d}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{4/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{4/3} d}-\frac {3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{4/3} d}-\frac {3 b}{\left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.25, size = 106, normalized size = 0.32 \begin {gather*} \frac {3 i \left ((a+i b) \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {a+b \tan (c+d x)}{a-i b}\right )-(a-i b) \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {a+b \tan (c+d x)}{a+i b}\right )\right )}{2 \left (a^2+b^2\right ) d \sqrt [3]{a+b \tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x])^(-4/3),x]

[Out]

(((3*I)/2)*((a + I*b)*Hypergeometric2F1[-1/3, 1, 2/3, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*Hypergeometr
ic2F1[-1/3, 1, 2/3, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.22, size = 102, normalized size = 0.30

method	result	size
derivativedivides	\(\frac {3 b \left (-\frac {1}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R} a \right ) \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}}{6 a^{2}+6 b^{2}}\right )}{d}\)	\(102\)
default	\(\frac {3 b \left (-\frac {1}{\left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R} a \right ) \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}}{6 a^{2}+6 b^{2}}\right )}{d}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*tan(d*x+c))^(4/3),x,method=_RETURNVERBOSE)

[Out]

3/d*b*(-1/(a^2+b^2)/(a+b*tan(d*x+c))^(1/3)+1/6/(a^2+b^2)*sum((-_R^4+2*_R*a)/(_R^5-_R^2*a)*ln((a+b*tan(d*x+c))^
(1/3)-_R),_R=RootOf(_Z^6-2*_Z^3*a+a^2+b^2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^(-4/3), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))**(4/3),x)

[Out]

Integral((a + b*tan(c + d*x))**(-4/3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*tan(d*x+c))^(4/3),x, algorithm="giac")

[Out]

undef

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Mupad [B]
time = 6.59, size = 2500, normalized size = 7.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*tan(c + d*x))^(4/3),x)

[Out]

(log((((a + b*tan(c + d*x))^(1/3)*(38880*a^4*b^18*d^7 - 1944*a^2*b^20*d^7 - 1944*b^22*d^7 + 163296*a^6*b^16*d^
7 + 299376*a^8*b^14*d^7 + 299376*a^10*b^12*d^7 + 163296*a^12*b^10*d^7 + 38880*a^14*b^8*d^7 - 1944*a^16*b^6*d^7
- 1944*a^18*b^4*d^7) - ((-1/(a^4*d^3*1i + b^4*d^3*1i + 4*a*b^3*d^3 - 4*a^3*b*d^3 - a^2*b^2*d^3*6i))^(2/3)*(77
76*a*b^24*d^9 + 77760*a^3*b^22*d^9 + 349920*a^5*b^20*d^9 + 933120*a^7*b^18*d^9 + 1632960*a^9*b^16*d^9 + 195955
2*a^11*b^14*d^9 + 1632960*a^13*b^12*d^9 + 933120*a^15*b^10*d^9 + 349920*a^17*b^8*d^9 + 77760*a^19*b^6*d^9 + 77
76*a^21*b^4*d^9))/4)*(-1/(a^4*d^3*1i + b^4*d^3*1i + 4*a*b^3*d^3 - 4*a^3*b*d^3 - a^2*b^2*d^3*6i))^(1/3))/2 - 97
2*b^21*d^6 - 3888*a^2*b^19*d^6 + 27216*a^6*b^15*d^6 + 68040*a^8*b^13*d^6 + 81648*a^10*b^11*d^6 + 54432*a^12*b^
9*d^6 + 19440*a^14*b^7*d^6 + 2916*a^16*b^5*d^6)*(-1/(a^4*d^3*1i + b^4*d^3*1i + 4*a*b^3*d^3 - 4*a^3*b*d^3 - a^2
*b^2*d^3*6i))^(1/3))/2 + log(((a + b*tan(c + d*x))^(1/3)*(38880*a^4*b^18*d^7 - 1944*a^2*b^20*d^7 - 1944*b^22*d
^7 + 163296*a^6*b^16*d^7 + 299376*a^8*b^14*d^7 + 299376*a^10*b^12*d^7 + 163296*a^12*b^10*d^7 + 38880*a^14*b^8*
d^7 - 1944*a^16*b^6*d^7 - 1944*a^18*b^4*d^7) - (-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i - 6*a^
2*b^2*d^3)))^(2/3)*(7776*a*b^24*d^9 + 77760*a^3*b^22*d^9 + 349920*a^5*b^20*d^9 + 933120*a^7*b^18*d^9 + 1632960
*a^9*b^16*d^9 + 1959552*a^11*b^14*d^9 + 1632960*a^13*b^12*d^9 + 933120*a^15*b^10*d^9 + 349920*a^17*b^8*d^9 + 7
7760*a^19*b^6*d^9 + 7776*a^21*b^4*d^9))*(-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i - 6*a^2*b^2*d
^3)))^(1/3) - 972*b^21*d^6 - 3888*a^2*b^19*d^6 + 27216*a^6*b^15*d^6 + 68040*a^8*b^13*d^6 + 81648*a^10*b^11*d^6
+ 54432*a^12*b^9*d^6 + 19440*a^14*b^7*d^6 + 2916*a^16*b^5*d^6)*(-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^
3*b*d^3*4i - 6*a^2*b^2*d^3)))^(1/3) + (log(((-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i - 6*a^2*b
^2*d^3)))^(1/3)*(3^(1/2)*1i - 1)*((a + b*tan(c + d*x))^(1/3)*(38880*a^4*b^18*d^7 - 1944*a^2*b^20*d^7 - 1944*b^
22*d^7 + 163296*a^6*b^16*d^7 + 299376*a^8*b^14*d^7 + 299376*a^10*b^12*d^7 + 163296*a^12*b^10*d^7 + 38880*a^14*
b^8*d^7 - 1944*a^16*b^6*d^7 - 1944*a^18*b^4*d^7) - ((-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i -
6*a^2*b^2*d^3)))^(2/3)*(3^(1/2)*1i - 1)^2*(7776*a*b^24*d^9 + 77760*a^3*b^22*d^9 + 349920*a^5*b^20*d^9 + 93312
0*a^7*b^18*d^9 + 1632960*a^9*b^16*d^9 + 1959552*a^11*b^14*d^9 + 1632960*a^13*b^12*d^9 + 933120*a^15*b^10*d^9 +
349920*a^17*b^8*d^9 + 77760*a^19*b^6*d^9 + 7776*a^21*b^4*d^9))/4))/2 - 972*b^21*d^6 - 3888*a^2*b^19*d^6 + 272
16*a^6*b^15*d^6 + 68040*a^8*b^13*d^6 + 81648*a^10*b^11*d^6 + 54432*a^12*b^9*d^6 + 19440*a^14*b^7*d^6 + 2916*a^
16*b^5*d^6)*(-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i - 6*a^2*b^2*d^3)))^(1/3)*(3^(1/2)*1i - 1)
)/2 - (log(27216*a^6*b^15*d^6 - ((-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i - 6*a^2*b^2*d^3)))^(
1/3)*(3^(1/2)*1i + 1)*((a + b*tan(c + d*x))^(1/3)*(38880*a^4*b^18*d^7 - 1944*a^2*b^20*d^7 - 1944*b^22*d^7 + 16
3296*a^6*b^16*d^7 + 299376*a^8*b^14*d^7 + 299376*a^10*b^12*d^7 + 163296*a^12*b^10*d^7 + 38880*a^14*b^8*d^7 - 1
944*a^16*b^6*d^7 - 1944*a^18*b^4*d^7) - ((-1i/(8*(a^4*d^3 + b^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i - 6*a^2*b^2*
d^3)))^(2/3)*(3^(1/2)*1i + 1)^2*(7776*a*b^24*d^9 + 77760*a^3*b^22*d^9 + 349920*a^5*b^20*d^9 + 933120*a^7*b^18*
d^9 + 1632960*a^9*b^16*d^9 + 1959552*a^11*b^14*d^9 + 1632960*a^13*b^12*d^9 + 933120*a^15*b^10*d^9 + 349920*a^1
7*b^8*d^9 + 77760*a^19*b^6*d^9 + 7776*a^21*b^4*d^9))/4))/2 - 3888*a^2*b^19*d^6 - 972*b^21*d^6 + 68040*a^8*b^13
*d^6 + 81648*a^10*b^11*d^6 + 54432*a^12*b^9*d^6 + 19440*a^14*b^7*d^6 + 2916*a^16*b^5*d^6)*(-1i/(8*(a^4*d^3 + b
^4*d^3 + a*b^3*d^3*4i - a^3*b*d^3*4i - 6*a^2*b^2*d^3)))^(1/3)*(3^(1/2)*1i + 1))/2 + (log(((-1/(8*(a^4*d^3*1i +
b^4*d^3*1i + 4*a*b^3*d^3 - 4*a^3*b*d^3 - a^2*b^2*d^3*6i)))^(1/3)*(3^(1/2)*1i - 1)*((a + b*tan(c + d*x))^(1/3)
*(38880*a^4*b^18*d^7 - 1944*a^2*b^20*d^7 - 1944*b^22*d^7 + 163296*a^6*b^16*d^7 + 299376*a^8*b^14*d^7 + 299376*
a^10*b^12*d^7 + 163296*a^12*b^10*d^7 + 38880*a^14*b^8*d^7 - 1944*a^16*b^6*d^7 - 1944*a^18*b^4*d^7) - ((-1/(8*(
a^4*d^3*1i + b^4*d^3*1i + 4*a*b^3*d^3 - 4*a^3*b*d^3 - a^2*b^2*d^3*6i)))^(2/3)*(3^(1/2)*1i - 1)^2*(7776*a*b^24*
d^9 + 77760*a^3*b^22*d^9 + 349920*a^5*b^20*d^9 + 933120*a^7*b^18*d^9 + 1632960*a^9*b^16*d^9 + 1959552*a^11*b^1
4*d^9 + 1632960*a^13*b^12*d^9 + 933120*a^15*b^10*d^9 + 349920*a^17*b^8*d^9 + 77760*a^19*b^6*d^9 + 7776*a^21*b^
4*d^9))/4))/2 - 972*b^21*d^6 - 3888*a^2*b^19*d^6 + 27216*a^6*b^15*d^6 + 68040*a^8*b^13*d^6 + 81648*a^10*b^11*d
^6 + 54432*a^12*b^9*d^6 + 19440*a^14*b^7*d^6 + 2916*a^16*b^5*d^6)*(-1/(8*(a^4*d^3*1i + b^4*d^3*1i + 4*a*b^3*d^
3 - 4*a^3*b*d^3 - a^2*b^2*d^3*6i)))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(27216*a^6*b^15*d^6 - ((-1/(8*(a^4*d^3*1i
+ b^4*d^3*1i + 4*a*b^3*d^3 - 4*a^3*b*d^3 - a^2*b^2*d^3*6i)))^(1/3)*(3^(1/2)*1i + 1)*((a + b*tan(c + d*x))^(1/3
)*(38880*a^4*b^18*d^7 - 1944*a^2*b^20*d^7 - 1944*b^22*d^7 + 163296*a^6*b^16*d^7 + 299376*a^8*b^14*d^7 + 299376
*a^10*b^12*d^7 + 163296*a^12*b^10*d^7 + 38880*a...

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