\(\int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx\) [695]

Optimal result

Integrand size = 14, antiderivative size = 338 \[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=-\frac {x}{4 (a-i b)^{5/3}}-\frac {x}{4 (a+i b)^{5/3}}-\frac {i \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 (a-i b)^{5/3} d}+\frac {i \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 (a+i b)^{5/3} d}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{5/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{5/3} d}+\frac {3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{5/3} d}-\frac {3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{5/3} d}-\frac {3 b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}} \]

[Out]

-1/4*x/(a-I*b)^(5/3)-1/4*x/(a+I*b)^(5/3)+1/4*I*ln(cos(d*x+c))/(a-I*b)^(5/3)/d-1/4*I*ln(cos(d*x+c))/(a+I*b)^(5/
3)/d+3/4*I*ln((a-I*b)^(1/3)-(a+b*tan(d*x+c))^(1/3))/(a-I*b)^(5/3)/d-3/4*I*ln((a+I*b)^(1/3)-(a+b*tan(d*x+c))^(1
/3))/(a+I*b)^(5/3)/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)^(5/3
)/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)^(5/3)/d-3/2*b/(a^2+b^
2)/d/(a+b*tan(d*x+c))^(2/3)

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3564, 3620, 3618, 59, 631, 210, 31} \[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=-\frac {3 b}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{2/3}}-\frac {i \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a-i b}}}{\sqrt {3}}\right )}{2 d (a-i b)^{5/3}}+\frac {i \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b \tan (c+d x)}}{\sqrt [3]{a+i b}}}{\sqrt {3}}\right )}{2 d (a+i b)^{5/3}}+\frac {3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a-i b}\right )}{4 d (a-i b)^{5/3}}-\frac {3 i \log \left (-\sqrt [3]{a+b \tan (c+d x)}+\sqrt [3]{a+i b}\right )}{4 d (a+i b)^{5/3}}+\frac {i \log (\cos (c+d x))}{4 d (a-i b)^{5/3}}-\frac {i \log (\cos (c+d x))}{4 d (a+i b)^{5/3}}-\frac {x}{4 (a-i b)^{5/3}}-\frac {x}{4 (a+i b)^{5/3}} \]

[In]

Int[(a + b*Tan[c + d*x])^(-5/3),x]

[Out]

-1/4*x/(a - I*b)^(5/3) - x/(4*(a + I*b)^(5/3)) - ((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)^(5/3)*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)^(5/3)*d) + ((I/4)*Log[Cos[c + d*x]])/((a - I*b)^(5/3)*d) - ((I/4)*Log[Cos[c +
d*x]])/((a + I*b)^(5/3)*d) + (((3*I)/4)*Log[(a - I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/((a - I*b)^(5/3)*d)
- (((3*I)/4)*Log[(a + I*b)^(1/3) - (a + b*Tan[c + d*x])^(1/3)])/((a + I*b)^(5/3)*d) - (3*b)/(2*(a^2 + b^2)*d*
(a + b*Tan[c + d*x])^(2/3))

Rule 31

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

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), 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*} \text {integral}& = -\frac {3 b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}}+\frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^{2/3}} \, dx}{a^2+b^2} \\ & = -\frac {3 b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}}+\frac {\int \frac {1+i \tan (c+d x)}{(a+b \tan (c+d x))^{2/3}} \, dx}{2 (a-i b)}+\frac {\int \frac {1-i \tan (c+d x)}{(a+b \tan (c+d x))^{2/3}} \, dx}{2 (a+i b)} \\ & = -\frac {3 b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) (a+i b x)^{2/3}} \, dx,x,-i \tan (c+d x)\right )}{2 (i a-b) d}-\frac {\text {Subst}\left (\int \frac {1}{(-1+x) (a-i b x)^{2/3}} \, dx,x,i \tan (c+d x)\right )}{2 (i a+b) d} \\ & = -\frac {x}{4 (a-i b)^{5/3}}-\frac {x}{4 (a+i b)^{5/3}}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{5/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{5/3} d}-\frac {3 b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}}-\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)^{5/3} d}-\frac {(3 i) \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 (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)^{5/3} d}+\frac {(3 i) \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 (a+i b)^{4/3} d} \\ & = -\frac {x}{4 (a-i b)^{5/3}}-\frac {x}{4 (a+i b)^{5/3}}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{5/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{5/3} d}+\frac {3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{5/3} d}-\frac {3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{5/3} d}-\frac {3 b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}}+\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)^{5/3} 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)^{5/3} d} \\ & = -\frac {x}{4 (a-i b)^{5/3}}-\frac {x}{4 (a+i b)^{5/3}}-\frac {i \sqrt {3} \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)^{5/3} d}+\frac {i \sqrt {3} \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)^{5/3} d}+\frac {i \log (\cos (c+d x))}{4 (a-i b)^{5/3} d}-\frac {i \log (\cos (c+d x))}{4 (a+i b)^{5/3} d}+\frac {3 i \log \left (\sqrt [3]{a-i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a-i b)^{5/3} d}-\frac {3 i \log \left (\sqrt [3]{a+i b}-\sqrt [3]{a+b \tan (c+d x)}\right )}{4 (a+i b)^{5/3} d}-\frac {3 b}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.31 \[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=\frac {3 i \left ((a+i b) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},1,\frac {1}{3},\frac {a+b \tan (c+d x)}{a-i b}\right )-(a-i b) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},1,\frac {1}{3},\frac {a+b \tan (c+d x)}{a+i b}\right )\right )}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{2/3}} \]

[In]

Integrate[(a + b*Tan[c + d*x])^(-5/3),x]

[Out]

(((3*I)/4)*((a + I*b)*Hypergeometric2F1[-2/3, 1, 1/3, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*Hypergeometr
ic2F1[-2/3, 1, 1/3, (a + b*Tan[c + d*x])/(a + I*b)]))/((a^2 + b^2)*d*(a + b*Tan[c + d*x])^(2/3))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.55 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.30

method	result	size
derivativedivides	\(\frac {b \left (\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+2 a \right ) \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}}{2 a^{2}+2 b^{2}}-\frac {3}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}\right )}{d}\)	\(100\)
default	\(\frac {b \left (\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a \,\textit {\_Z}^{3}+a^{2}+b^{2}\right )}{\sum }\frac {\left (-\textit {\_R}^{3}+2 a \right ) \ln \left (\left (a +b \tan \left (d x +c \right )\right )^{\frac {1}{3}}-\textit {\_R} \right )}{\textit {\_R}^{5}-\textit {\_R}^{2} a}}{2 a^{2}+2 b^{2}}-\frac {3}{2 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {2}{3}}}\right )}{d}\)	\(100\)

[In]

int(1/(a+b*tan(d*x+c))^(5/3),x,method=_RETURNVERBOSE)

[Out]

1/d*b*(1/2/(a^2+b^2)*sum((-_R^3+2*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))-3/2/(a^2+b^2)/(a+b*tan(d*x+c))^(2/3))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4842 vs. \(2 (244) = 488\).

Time = 0.43 (sec) , antiderivative size = 4842, normalized size of antiderivative = 14.33 \[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*tan(d*x+c))^(5/3),x, algorithm="fricas")

[Out]

1/4*(2*((a^2*b + b^3)*d*tan(d*x + c) + (a^3 + a*b^2)*d)*((5*a^4*b - 10*a^2*b^3 + b^5 + (a^10 + 5*a^8*b^2 + 10*
a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8
)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^
14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)
*d^3))^(1/3)*log((a^5 - 10*a^3*b^2 + 5*a*b^4)*(b*tan(d*x + c) + a)^(1/3) - ((a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5
*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 +
10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4
*b^16 + 10*a^2*b^18 + b^20)*d^6)) - 2*(a^6*b - 10*a^4*b^3 + 5*a^2*b^5)*d)*((5*a^4*b - 10*a^2*b^3 + b^5 + (a^10
+ 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^
4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8
*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 +
5*a^2*b^8 + b^10)*d^3))^(1/3)) - (sqrt(-3)*(a^3 + a*b^2)*d + (a^3 + a*b^2)*d + (sqrt(-3)*(a^2*b + b^3)*d + (a
^2*b + b^3)*d)*tan(d*x + c))*((5*a^4*b - 10*a^2*b^3 + b^5 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^
2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45
*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b
^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3))^(1/3)*log((a^5 - 10*
a^3*b^2 + 5*a*b^4)*(b*tan(d*x + c) + a)^(1/3) - 1/2*(2*sqrt(-3)*(a^6*b - 10*a^4*b^3 + 5*a^2*b^5)*d + 2*(a^6*b
- 10*a^4*b^3 + 5*a^2*b^5)*d - (sqrt(-3)*(a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4 +
(a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4)*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 -
100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 2
10*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))*((5*a^4*b - 10*a^2*b^3 + b^5 + (a^10 + 5
*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^
6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^1
2 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a
^2*b^8 + b^10)*d^3))^(1/3)) + (sqrt(-3)*(a^3 + a*b^2)*d - (a^3 + a*b^2)*d + (sqrt(-3)*(a^2*b + b^3)*d - (a^2*b
+ b^3)*d)*tan(d*x + c))*((5*a^4*b - 10*a^2*b^3 + b^5 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^
8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^1
6*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18
+ b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3))^(1/3)*log((a^5 - 10*a^3*
b^2 + 5*a*b^4)*(b*tan(d*x + c) + a)^(1/3) + 1/2*(2*sqrt(-3)*(a^6*b - 10*a^4*b^3 + 5*a^2*b^5)*d - 2*(a^6*b - 10
*a^4*b^3 + 5*a^2*b^5)*d - (sqrt(-3)*(a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4 - (a^1
2 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4)*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*
a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a
^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))*((5*a^4*b - 10*a^2*b^3 + b^5 + (a^10 + 5*a^8
*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 +
25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 +
120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b
^8 + b^10)*d^3))^(1/3)) + 2*((a^2*b + b^3)*d*tan(d*x + c) + (a^3 + a*b^2)*d)*((5*a^4*b - 10*a^2*b^3 + b^5 - (a
^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100
*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*
a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^
6 + 5*a^2*b^8 + b^10)*d^3))^(1/3)*log((a^5 - 10*a^3*b^2 + 5*a*b^4)*(b*tan(d*x + c) + a)^(1/3) + ((a^12 + 4*a^1
0*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 +
25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 +
120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)) + 2*(a^6*b - 10*a^4*b^3 + 5*a^2*b^5)*d)*((5*a^4*b - 10*
a^2*b^3 + b^5 - (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 +
110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 25
2*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a
^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3))^(1/3)) - (sqrt(-3)*(a^3 + a*b^2)*d + (a^3 + a*b^2)*d + (sqrt(-3)
*(a^2*b + b^3)*d + (a^2*b + b^3)*d)*tan(d*x + c))*((5*a^4*b - 10*a^2*b^3 + b^5 - (a^10 + 5*a^8*b^2 + 10*a^6*b^
4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^
20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 4
5*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3))
^(1/3)*log((a^5 - 10*a^3*b^2 + 5*a*b^4)*(b*tan(d*x + c) + a)^(1/3) - 1/2*(2*sqrt(-3)*(a^6*b - 10*a^4*b^3 + 5*a
^2*b^5)*d + 2*(a^6*b - 10*a^4*b^3 + 5*a^2*b^5)*d + (sqrt(-3)*(a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^
2*b^10 - b^12)*d^4 + (a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4)*sqrt(-(a^10 - 20*a^8
*b^2 + 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^
8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))*((5*a^4*b - 10*a^2*
b^3 + b^5 - (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110
*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^
10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b
^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3))^(1/3)) + (sqrt(-3)*(a^3 + a*b^2)*d - (a^3 + a*b^2)*d + (sqrt(-3)*(a^
2*b + b^3)*d - (a^2*b + b^3)*d)*tan(d*x + c))*((5*a^4*b - 10*a^2*b^3 + b^5 - (a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 +
10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^
4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3))^(1/
3)*log((a^5 - 10*a^3*b^2 + 5*a*b^4)*(b*tan(d*x + c) + a)^(1/3) + 1/2*(2*sqrt(-3)*(a^6*b - 10*a^4*b^3 + 5*a^2*b
^5)*d - 2*(a^6*b - 10*a^4*b^3 + 5*a^2*b^5)*d + (sqrt(-3)*(a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^
10 - b^12)*d^4 - (a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d^4)*sqrt(-(a^10 - 20*a^8*b^2
+ 110*a^6*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 +
252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))*((5*a^4*b - 10*a^2*b^3
+ b^5 - (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3*sqrt(-(a^10 - 20*a^8*b^2 + 110*a^6
*b^4 - 100*a^4*b^6 + 25*a^2*b^8)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b
^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^6)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^3))^(1/3)) - 6*(b*tan(d*x + c) + a)^(1/3)*b)/((a^2*b + b^3)*d*tan(d*x + c) +
(a^3 + a*b^2)*d)

Sympy [F]

\[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=\int \frac {1}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{3}}}\, dx \]

[In]

integrate(1/(a+b*tan(d*x+c))**(5/3),x)

[Out]

Integral((a + b*tan(c + d*x))**(-5/3), x)

Maxima [F]

\[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]

[In]

integrate(1/(a+b*tan(d*x+c))^(5/3),x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c) + a)^(-5/3), x)

Giac [F]

\[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=\int { \frac {1}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}}} \,d x } \]

[In]

integrate(1/(a+b*tan(d*x+c))^(5/3),x, algorithm="giac")

[Out]

undef

Mupad [B] (verification not implemented)

Time = 7.32 (sec) , antiderivative size = 4348, normalized size of antiderivative = 12.86 \[ \int \frac {1}{(a+b \tan (c+d x))^{5/3}} \, dx=\text {Too large to display} \]

[In]

int(1/(a + b*tan(c + d*x))^(5/3),x)

[Out]

(log((a + b*tan(c + d*x))^(1/3)*(486*b^18*d^5 + 2430*a^2*b^16*d^5 + 4374*a^4*b^14*d^5 + 2430*a^6*b^12*d^5 - 24
30*a^8*b^10*d^5 - 4374*a^10*b^8*d^5 - 2430*a^12*b^6*d^5 - 486*a^14*b^4*d^5) + ((1/(a^5*d^3*1i + b^5*d^3 + a*b^
4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3)*(((((1/(a^5*d^3*1i + b^5*d^3 + a*b^4*d^3*5i
+ 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3)*(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 + 77760*a^19*b^6*d^9 + 7776*a^21*b^4*d^9))/2 + (a + b*tan(c + d*x))^(1/3)*(
108864*a^6*b^17*d^8 - 19440*a^2*b^21*d^8 - 15552*a^4*b^19*d^8 - 3888*b^23*d^8 + 381024*a^8*b^15*d^8 + 598752*a
^10*b^13*d^8 + 544320*a^12*b^11*d^8 + 295488*a^14*b^9*d^8 + 89424*a^16*b^7*d^8 + 11664*a^18*b^5*d^8))*(1/(a^5*
d^3*1i + b^5*d^3 + a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(2/3))/4 + 3888*a*b^19*d^6
+ 19440*a^3*b^17*d^6 + 34992*a^5*b^15*d^6 + 19440*a^7*b^13*d^6 - 19440*a^9*b^11*d^6 - 34992*a^11*b^9*d^6 - 194
40*a^13*b^7*d^6 - 3888*a^15*b^5*d^6))/2)*(1/(a^5*d^3*1i + b^5*d^3 + a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^
3 - a^3*b^2*d^3*10i))^(1/3))/2 + log((a + b*tan(c + d*x))^(1/3)*(486*b^18*d^5 + 2430*a^2*b^16*d^5 + 4374*a^4*b
^14*d^5 + 2430*a^6*b^12*d^5 - 2430*a^8*b^10*d^5 - 4374*a^10*b^8*d^5 - 2430*a^12*b^6*d^5 - 486*a^14*b^4*d^5) +
(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3)*(((1i/(8
*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3)*(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 + 77760*a^19*b^6*d^9 + 7776*a^21*b^4
*d^9) + (a + b*tan(c + d*x))^(1/3)*(108864*a^6*b^17*d^8 - 19440*a^2*b^21*d^8 - 15552*a^4*b^19*d^8 - 3888*b^23*
d^8 + 381024*a^8*b^15*d^8 + 598752*a^10*b^13*d^8 + 544320*a^12*b^11*d^8 + 295488*a^14*b^9*d^8 + 89424*a^16*b^7
*d^8 + 11664*a^18*b^5*d^8))*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a
^3*b^2*d^3)))^(2/3) + 3888*a*b^19*d^6 + 19440*a^3*b^17*d^6 + 34992*a^5*b^15*d^6 + 19440*a^7*b^13*d^6 - 19440*a
^9*b^11*d^6 - 34992*a^11*b^9*d^6 - 19440*a^13*b^7*d^6 - 3888*a^15*b^5*d^6))*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a
*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3) + (log((a + b*tan(c + d*x))^(1/3)*(486*b^1
8*d^5 + 2430*a^2*b^16*d^5 + 4374*a^4*b^14*d^5 + 2430*a^6*b^12*d^5 - 2430*a^8*b^10*d^5 - 4374*a^10*b^8*d^5 - 24
30*a^12*b^6*d^5 - 486*a^14*b^4*d^5) + ((3^(1/2)*1i - 1)*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3
*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3)*(3888*a*b^19*d^6 + 19440*a^3*b^17*d^6 + 34992*a^5*b^15*d^6 + 1
9440*a^7*b^13*d^6 - 19440*a^9*b^11*d^6 - 34992*a^11*b^9*d^6 - 19440*a^13*b^7*d^6 - 3888*a^15*b^5*d^6 + ((3^(1/
2)*1i - 1)^2*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(
2/3)*((a + b*tan(c + d*x))^(1/3)*(108864*a^6*b^17*d^8 - 19440*a^2*b^21*d^8 - 15552*a^4*b^19*d^8 - 3888*b^23*d^
8 + 381024*a^8*b^15*d^8 + 598752*a^10*b^13*d^8 + 544320*a^12*b^11*d^8 + 295488*a^14*b^9*d^8 + 89424*a^16*b^7*d
^8 + 11664*a^18*b^5*d^8) + ((3^(1/2)*1i - 1)*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b
^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3)*(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 + 34992
0*a^17*b^8*d^9 + 77760*a^19*b^6*d^9 + 7776*a^21*b^4*d^9))/2))/4))/2)*(3^(1/2)*1i - 1)*(1i/(8*(a^5*d^3 + b^5*d^
3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3))/2 - (log((a + b*tan(c + d*x))^(
1/3)*(486*b^18*d^5 + 2430*a^2*b^16*d^5 + 4374*a^4*b^14*d^5 + 2430*a^6*b^12*d^5 - 2430*a^8*b^10*d^5 - 4374*a^10
*b^8*d^5 - 2430*a^12*b^6*d^5 - 486*a^14*b^4*d^5) - ((3^(1/2)*1i + 1)*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^
3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3)*(3888*a*b^19*d^6 + 19440*a^3*b^17*d^6 + 34992*a^5
*b^15*d^6 + 19440*a^7*b^13*d^6 - 19440*a^9*b^11*d^6 - 34992*a^11*b^9*d^6 - 19440*a^13*b^7*d^6 - 3888*a^15*b^5*
d^6 + ((3^(1/2)*1i + 1)^2*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3
*b^2*d^3)))^(2/3)*((a + b*tan(c + d*x))^(1/3)*(108864*a^6*b^17*d^8 - 19440*a^2*b^21*d^8 - 15552*a^4*b^19*d^8 -
3888*b^23*d^8 + 381024*a^8*b^15*d^8 + 598752*a^10*b^13*d^8 + 544320*a^12*b^11*d^8 + 295488*a^14*b^9*d^8 + 894
24*a^16*b^7*d^8 + 11664*a^18*b^5*d^8) - ((3^(1/2)*1i + 1)*(1i/(8*(a^5*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d
^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3)*(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^1
0*d^9 + 349920*a^17*b^8*d^9 + 77760*a^19*b^6*d^9 + 7776*a^21*b^4*d^9))/2))/4))/2)*(3^(1/2)*1i + 1)*(1i/(8*(a^5
*d^3 + b^5*d^3*1i + 5*a*b^4*d^3 + a^4*b*d^3*5i - a^2*b^3*d^3*10i - 10*a^3*b^2*d^3)))^(1/3))/2 + (log((a + b*ta
n(c + d*x))^(1/3)*(486*b^18*d^5 + 2430*a^2*b^16*d^5 + 4374*a^4*b^14*d^5 + 2430*a^6*b^12*d^5 - 2430*a^8*b^10*d^
5 - 4374*a^10*b^8*d^5 - 2430*a^12*b^6*d^5 - 486*a^14*b^4*d^5) + ((3^(1/2)*1i - 1)*(1/(a^5*d^3*1i + b^5*d^3 + a
*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3)*(((3^(1/2)*1i - 1)^2*(1/(a^5*d^3*1i + b^5
*d^3 + a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(2/3)*((a + b*tan(c + d*x))^(1/3)*(1088
64*a^6*b^17*d^8 - 19440*a^2*b^21*d^8 - 15552*a^4*b^19*d^8 - 3888*b^23*d^8 + 381024*a^8*b^15*d^8 + 598752*a^10*
b^13*d^8 + 544320*a^12*b^11*d^8 + 295488*a^14*b^9*d^8 + 89424*a^16*b^7*d^8 + 11664*a^18*b^5*d^8) + ((3^(1/2)*1
i - 1)*(1/(a^5*d^3*1i + b^5*d^3 + a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3)*(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 + 77760*a^19*b^6*d^9 + 7776*
a^21*b^4*d^9))/4))/16 + 3888*a*b^19*d^6 + 19440*a^3*b^17*d^6 + 34992*a^5*b^15*d^6 + 19440*a^7*b^13*d^6 - 19440
*a^9*b^11*d^6 - 34992*a^11*b^9*d^6 - 19440*a^13*b^7*d^6 - 3888*a^15*b^5*d^6))/4)*(3^(1/2)*1i - 1)*(1/(a^5*d^3*
1i + b^5*d^3 + a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3))/4 - (log((a + b*tan(c +
d*x))^(1/3)*(486*b^18*d^5 + 2430*a^2*b^16*d^5 + 4374*a^4*b^14*d^5 + 2430*a^6*b^12*d^5 - 2430*a^8*b^10*d^5 - 43
74*a^10*b^8*d^5 - 2430*a^12*b^6*d^5 - 486*a^14*b^4*d^5) - ((3^(1/2)*1i + 1)*(1/(a^5*d^3*1i + b^5*d^3 + a*b^4*d
^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3)*(((3^(1/2)*1i + 1)^2*(1/(a^5*d^3*1i + b^5*d^3 +
a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(2/3)*((a + b*tan(c + d*x))^(1/3)*(108864*a^6
*b^17*d^8 - 19440*a^2*b^21*d^8 - 15552*a^4*b^19*d^8 - 3888*b^23*d^8 + 381024*a^8*b^15*d^8 + 598752*a^10*b^13*d
^8 + 544320*a^12*b^11*d^8 + 295488*a^14*b^9*d^8 + 89424*a^16*b^7*d^8 + 11664*a^18*b^5*d^8) - ((3^(1/2)*1i + 1)
*(1/(a^5*d^3*1i + b^5*d^3 + a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3)*(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 + 77760*a^19*b^6*d^9 + 7776*a^21*b
^4*d^9))/4))/16 + 3888*a*b^19*d^6 + 19440*a^3*b^17*d^6 + 34992*a^5*b^15*d^6 + 19440*a^7*b^13*d^6 - 19440*a^9*b
^11*d^6 - 34992*a^11*b^9*d^6 - 19440*a^13*b^7*d^6 - 3888*a^15*b^5*d^6))/4)*(3^(1/2)*1i + 1)*(1/(a^5*d^3*1i + b
^5*d^3 + a*b^4*d^3*5i + 5*a^4*b*d^3 - 10*a^2*b^3*d^3 - a^3*b^2*d^3*10i))^(1/3))/4 - (3*b)/(2*d*(a^2 + b^2)*(a
+ b*tan(c + d*x))^(2/3))