3.7.90 \(\int (a+b \tan (c+d x))^{2/3} \, dx\) [690]

Optimal. Leaf size=415 \[ -\frac {1}{4} \left (a-\sqrt {-b^2}\right )^{2/3} x-\frac {1}{4} \left (a+\sqrt {-b^2}\right )^{2/3} x-\frac {\sqrt {3} b \left (a-\sqrt {-b^2}\right )^{2/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}+\frac {\sqrt {3} b \left (a+\sqrt {-b^2}\right )^{2/3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {b \left (a-\sqrt {-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \left (a+\sqrt {-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {3 b \left (a-\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \left (a+\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d} \]

[Out]

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

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Rubi [A]
time = 0.28, antiderivative size = 415, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3566, 726, 52, 57, 631, 210, 31} \begin {gather*} -\frac {\sqrt {3} b \left (a-\sqrt {-b^2}\right )^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}+\frac {\sqrt {3} b \left (a+\sqrt {-b^2}\right )^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+\sqrt {-b^2}}}+1}{\sqrt {3}}\right )}{2 \sqrt {-b^2} d}-\frac {3 b \left (a-\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a-\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}+\frac {3 b \left (a+\sqrt {-b^2}\right )^{2/3} \log \left (\sqrt [3]{a+\sqrt {-b^2}}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 \sqrt {-b^2} d}-\frac {b \left (a-\sqrt {-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}+\frac {b \left (a+\sqrt {-b^2}\right )^{2/3} \log (\cos (c+d x))}{4 \sqrt {-b^2} d}-\frac {1}{4} x \left (a-\sqrt {-b^2}\right )^{2/3}-\frac {1}{4} x \left (a+\sqrt {-b^2}\right )^{2/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(2/3),x]

[Out]

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

Rule 31

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

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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 726

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2
), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m]

Rule 3566

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.31, size = 224, normalized size = 0.54 \begin {gather*} \frac {\frac {(i a+b) \left (2 \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 )-\log (i+\tan (c+d x))+3 \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )\right )}{\sqrt [3]{a-i b}}+\frac {(-i a+b) \left (2 \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 )-\log (i-\tan (c+d x))+3 \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )\right )}{\sqrt [3]{a+i b}}}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[c + d*x])^(2/3),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.32, size = 60, normalized size = 0.14

method	result	size
derivativedivides	\(\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\)	\(60\)
default	\(\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\textit {\_R}^{4} \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}\right )}{2 d}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(d*x+c))^(2/3),x,method=_RETURNVERBOSE)

[Out]

1/2/d*b*sum(_R^4/(_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((a+b*tan(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^(2/3), 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((a+b*tan(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(d*x+c))**(2/3),x)

[Out]

Integral((a + b*tan(c + d*x))**(2/3), 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((a+b*tan(d*x+c))^(2/3),x, algorithm="giac")

[Out]

undef

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Mupad [B]
time = 8.22, size = 1229, normalized size = 2.96 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(c + d*x))^(2/3),x)

[Out]

log((((((1944*b^4*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^2 + 1944*a*b^4*(a^2 + b^2)*(-((a - b*1i)^2*1i)/d
^3)^(2/3))*(-((a - b*1i)^2*1i)/d^3)^(1/3))/2 + (972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3)*(-((a - b*1i)^2*1i)/d^
3)^(2/3))/4 + (486*a*b^5*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5)*(-(2*a*b + a^2*1i - b^2*1i)/(8*d^3))^(
1/3) + log(((((1944*a*b^4*(a^2 + b^2)*(-((a*1i - b)^2*1i)/d^3)^(2/3) + (1944*b^4*(a^2 + b^2)^2*(a + b*tan(c +
d*x))^(1/3))/d^2)*(-((a*1i - b)^2*1i)/d^3)^(1/3))/2 + (972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3)*(-((a*1i - b)^2
*1i)/d^3)^(2/3))/4 + (486*a*b^5*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5)*(-(2*a*b - a^2*1i + b^2*1i)/(8*
d^3))^(1/3) - log((((3^(1/2)*1i)/2 - 1/2)*((972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3 - (((3^(1/2)*1i)/2 + 1/2)*(
(1944*b^4*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^2 + 1944*a*b^4*((3^(1/2)*1i)/2 - 1/2)*(a^2 + b^2)*(-((a
- b*1i)^2*1i)/d^3)^(2/3))*(-((a - b*1i)^2*1i)/d^3)^(1/3))/2)*(-((a - b*1i)^2*1i)/d^3)^(2/3))/4 + (486*a*b^5*(a
^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5)*((3^(1/2)*1i)/2 + 1/2)*(-(2*a*b + a^2*1i - b^2*1i)/(8*d^3))^(1/3)
+ log((486*a*b^5*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5 - (((3^(1/2)*1i)/2 + 1/2)*((972*b^5*(3*a^4 - b
^4 + 2*a^2*b^2))/d^3 + (((3^(1/2)*1i)/2 - 1/2)*((1944*b^4*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^2 - 1944
*a*b^4*((3^(1/2)*1i)/2 + 1/2)*(a^2 + b^2)*(-((a - b*1i)^2*1i)/d^3)^(2/3))*(-((a - b*1i)^2*1i)/d^3)^(1/3))/2)*(
-((a - b*1i)^2*1i)/d^3)^(2/3))/4)*((3^(1/2)*1i)/2 - 1/2)*(-(2*a*b + a^2*1i - b^2*1i)/(8*d^3))^(1/3) - log((486
*a*b^5*(a^2 + b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5 - (((3^(1/2)*1i)/2 - 1/2)*((((1944*b^4*(a^2 + b^2)^2*(a +
b*tan(c + d*x))^(1/3))/d^2 + 1944*a*b^4*((3^(1/2)*1i)/2 - 1/2)*(a^2 + b^2)*(-((a*1i - b)^2*1i)/d^3)^(2/3))*((
3^(1/2)*1i)/2 + 1/2)*(-((a*1i - b)^2*1i)/d^3)^(1/3))/2 - (972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3)*(-((a*1i - b
)^2*1i)/d^3)^(2/3))/4)*((3^(1/2)*1i)/2 + 1/2)*(-(2*a*b - a^2*1i + b^2*1i)/(8*d^3))^(1/3) + log((486*a*b^5*(a^2
+ b^2)^2*(a + b*tan(c + d*x))^(1/3))/d^5 - (((3^(1/2)*1i)/2 + 1/2)*((((1944*b^4*(a^2 + b^2)^2*(a + b*tan(c +
d*x))^(1/3))/d^2 - 1944*a*b^4*((3^(1/2)*1i)/2 + 1/2)*(a^2 + b^2)*(-((a*1i - b)^2*1i)/d^3)^(2/3))*((3^(1/2)*1i)
/2 - 1/2)*(-((a*1i - b)^2*1i)/d^3)^(1/3))/2 + (972*b^5*(3*a^4 - b^4 + 2*a^2*b^2))/d^3)*(-((a*1i - b)^2*1i)/d^3
)^(2/3))/4)*((3^(1/2)*1i)/2 - 1/2)*(-(2*a*b - a^2*1i + b^2*1i)/(8*d^3))^(1/3)

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