\(\int (a+a \tan ^2(c+d x))^{5/2} \, dx\) [272]
Optimal result
Integrand size = 16, antiderivative size = 98 \[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{8 d}+\frac {3 a^2 \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{8 d}+\frac {a \left (a \sec ^2(c+d x)\right )^{3/2} \tan (c+d x)}{4 d}
\]
[Out]
3/8*a^(5/2)*arctanh(a^(1/2)*tan(d*x+c)/(a*sec(d*x+c)^2)^(1/2))/d+1/4*a*(a*sec(d*x+c)^2)^(3/2)*tan(d*x+c)/d+3/8
*a^2*(a*sec(d*x+c)^2)^(1/2)*tan(d*x+c)/d
Rubi [A] (verified)
Time = 0.06 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of
steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3738, 4207, 201, 223, 212}
\[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{8 d}+\frac {3 a^2 \tan (c+d x) \sqrt {a \sec ^2(c+d x)}}{8 d}+\frac {a \tan (c+d x) \left (a \sec ^2(c+d x)\right )^{3/2}}{4 d}
\]
[In]
Int[(a + a*Tan[c + d*x]^2)^(5/2),x]
[Out]
(3*a^(5/2)*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/(8*d) + (3*a^2*Sqrt[a*Sec[c + d*x]^2]*Tan[c
+ d*x])/(8*d) + (a*(a*Sec[c + d*x]^2)^(3/2)*Tan[c + d*x])/(4*d)
Rule 201
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 223
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 3738
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]
Rule 4207
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p
]
Rubi steps \begin{align*}
\text {integral}& = \int \left (a \sec ^2(c+d x)\right )^{5/2} \, dx \\ & = \frac {a \text {Subst}\left (\int \left (a+a x^2\right )^{3/2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {a \left (a \sec ^2(c+d x)\right )^{3/2} \tan (c+d x)}{4 d}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\tan (c+d x)\right )}{4 d} \\ & = \frac {3 a^2 \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{8 d}+\frac {a \left (a \sec ^2(c+d x)\right )^{3/2} \tan (c+d x)}{4 d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\tan (c+d x)\right )}{8 d} \\ & = \frac {3 a^2 \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{8 d}+\frac {a \left (a \sec ^2(c+d x)\right )^{3/2} \tan (c+d x)}{4 d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{8 d} \\ & = \frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{8 d}+\frac {3 a^2 \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{8 d}+\frac {a \left (a \sec ^2(c+d x)\right )^{3/2} \tan (c+d x)}{4 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.60
\[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {a^2 \sqrt {a \sec ^2(c+d x)} \left (3 \text {arctanh}(\sin (c+d x)) \cos (c+d x)+\left (3+2 \sec ^2(c+d x)\right ) \tan (c+d x)\right )}{8 d}
\]
[In]
Integrate[(a + a*Tan[c + d*x]^2)^(5/2),x]
[Out]
(a^2*Sqrt[a*Sec[c + d*x]^2]*(3*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + (3 + 2*Sec[c + d*x]^2)*Tan[c + d*x]))/(8*d
)
Maple [A] (verified)
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.92
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {a \tan \left (d x +c \right ) \left (a +a \tan \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}{4 d}+\frac {3 a^{2} \tan \left (d x +c \right ) \sqrt {a +a \tan \left (d x +c \right )^{2}}}{8 d}+\frac {3 a^{\frac {5}{2}} \ln \left (\sqrt {a}\, \tan \left (d x +c \right )+\sqrt {a +a \tan \left (d x +c \right )^{2}}\right )}{8 d}\) |
\(90\) |
| default |
\(\frac {a \tan \left (d x +c \right ) \left (a +a \tan \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}{4 d}+\frac {3 a^{2} \tan \left (d x +c \right ) \sqrt {a +a \tan \left (d x +c \right )^{2}}}{8 d}+\frac {3 a^{\frac {5}{2}} \ln \left (\sqrt {a}\, \tan \left (d x +c \right )+\sqrt {a +a \tan \left (d x +c \right )^{2}}\right )}{8 d}\) |
\(90\) |
| risch |
\(-\frac {i a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 i \left (d x +c \right )}+11 \,{\mathrm e}^{4 i \left (d x +c \right )}-11 \,{\mathrm e}^{2 i \left (d x +c \right )}-3\right )}{4 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} d}+\frac {3 \ln \left ({\mathrm e}^{i d x}+i {\mathrm e}^{-i c}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, a^{2} \cos \left (d x +c \right )}{4 d}-\frac {3 \ln \left ({\mathrm e}^{i d x}-i {\mathrm e}^{-i c}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, a^{2} \cos \left (d x +c \right )}{4 d}\) |
\(197\) |
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[In]
int((a+a*tan(d*x+c)^2)^(5/2),x,method=_RETURNVERBOSE)
[Out]
1/4/d*a*tan(d*x+c)*(a+a*tan(d*x+c)^2)^(3/2)+3/8/d*a^2*tan(d*x+c)*(a+a*tan(d*x+c)^2)^(1/2)+3/8/d*a^(5/2)*ln(a^(
1/2)*tan(d*x+c)+(a+a*tan(d*x+c)^2)^(1/2))
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93
\[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {3 \, a^{\frac {5}{2}} \log \left (2 \, a \tan \left (d x + c\right )^{2} + 2 \, \sqrt {a \tan \left (d x + c\right )^{2} + a} \sqrt {a} \tan \left (d x + c\right ) + a\right ) + 2 \, {\left (2 \, a^{2} \tan \left (d x + c\right )^{3} + 5 \, a^{2} \tan \left (d x + c\right )\right )} \sqrt {a \tan \left (d x + c\right )^{2} + a}}{16 \, d}
\]
[In]
integrate((a+a*tan(d*x+c)^2)^(5/2),x, algorithm="fricas")
[Out]
1/16*(3*a^(5/2)*log(2*a*tan(d*x + c)^2 + 2*sqrt(a*tan(d*x + c)^2 + a)*sqrt(a)*tan(d*x + c) + a) + 2*(2*a^2*tan
(d*x + c)^3 + 5*a^2*tan(d*x + c))*sqrt(a*tan(d*x + c)^2 + a))/d
Sympy [F]
\[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\int \left (a \tan ^{2}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}\, dx
\]
[In]
integrate((a+a*tan(d*x+c)**2)**(5/2),x)
[Out]
Integral((a*tan(c + d*x)**2 + a)**(5/2), x)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1769 vs. \(2 (82) = 164\).
Time = 0.58 (sec) , antiderivative size = 1769, normalized size of antiderivative = 18.05
\[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*tan(d*x+c)^2)^(5/2),x, algorithm="maxima")
[Out]
1/16*(176*a^2*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 48*a^2*cos(d*x + c)*sin(2*d*x + 2*c) - 48*a^2*cos(2*d*x + 2*
c)*sin(d*x + c) - 12*a^2*sin(d*x + c) + 4*(3*a^2*sin(7*d*x + 7*c) + 11*a^2*sin(5*d*x + 5*c) - 11*a^2*sin(3*d*x
+ 3*c) - 3*a^2*sin(d*x + c))*cos(8*d*x + 8*c) - 24*(2*a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 2*a^2*s
in(2*d*x + 2*c))*cos(7*d*x + 7*c) + 16*(11*a^2*sin(5*d*x + 5*c) - 11*a^2*sin(3*d*x + 3*c) - 3*a^2*sin(d*x + c)
)*cos(6*d*x + 6*c) - 88*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*cos(5*d*x + 5*c) - 24*(11*a^2*sin(3*
d*x + 3*c) + 3*a^2*sin(d*x + c))*cos(4*d*x + 4*c) + 3*(a^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36
*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*
a^2*sin(4*d*x + 4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*
x + 2*c) + a^2 + 2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x
+ 8*c) + 8*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*
c) + a^2)*cos(4*d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(
8*d*x + 8*c) + 16*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(cos(d*x + c)^2 + sin
(d*x + c)^2 + 2*sin(d*x + c) + 1) - 3*(a^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x +
4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x +
4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*x + 2*c) + a^2 +
2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 8*(6*a^
2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*
d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 1
6*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*
sin(d*x + c) + 1) - 4*(3*a^2*cos(7*d*x + 7*c) + 11*a^2*cos(5*d*x + 5*c) - 11*a^2*cos(3*d*x + 3*c) - 3*a^2*cos(
d*x + c))*sin(8*d*x + 8*c) + 12*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^
2)*sin(7*d*x + 7*c) - 16*(11*a^2*cos(5*d*x + 5*c) - 11*a^2*cos(3*d*x + 3*c) - 3*a^2*cos(d*x + c))*sin(6*d*x +
6*c) + 44*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*sin(5*d*x + 5*c) + 24*(11*a^2*cos(3*d*x + 3*
c) + 3*a^2*cos(d*x + c))*sin(4*d*x + 4*c) - 44*(4*a^2*cos(2*d*x + 2*c) + a^2)*sin(3*d*x + 3*c))*sqrt(a)/((2*(4
*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(8*d*x + 8*c) + cos(8*d*x + 8*c)^2 + 8*(6*
cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + 16*cos(6*d*x + 6*c)^2 + 12*(4*cos(2*d*x + 2*c) +
1)*cos(4*d*x + 4*c) + 36*cos(4*d*x + 4*c)^2 + 16*cos(2*d*x + 2*c)^2 + 4*(2*sin(6*d*x + 6*c) + 3*sin(4*d*x + 4
*c) + 2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin(8*d*x + 8*c)^2 + 16*(3*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))
*sin(6*d*x + 6*c) + 16*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 48*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*
sin(2*d*x + 2*c)^2 + 8*cos(2*d*x + 2*c) + 1)*d)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 7084 vs. \(2 (82) = 164\).
Time = 3.58 (sec) , antiderivative size = 7084, normalized size of antiderivative = 72.29
\[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\text {Too large to display}
\]
[In]
integrate((a+a*tan(d*x+c)^2)^(5/2),x, algorithm="giac")
[Out]
1/8*(3*(a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d
*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c) - a^
(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(
1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1))*log(abs(-tan(1/2*d*x)*ta
n(1/2*c) + tan(1/2*d*x) + tan(1/2*c) + 1))/(tan(1/2*c) - 1) - 3*(a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*t
an(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(
1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c) + a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x
)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 -
4*tan(1/2*d*x)*tan(1/2*c) + 1))*log(abs(-tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) + 1))/(tan(1/2*c
) + 1) - 2*(5*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan
(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d
*x)^7*tan(1/2*c)^16 + 34*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x
)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1
)*tan(1/2*d*x)^7*tan(1/2*c)^14 - 6*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - t
an(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(
1/2*c) + 1)*tan(1/2*d*x)^6*tan(1/2*c)^15 + 3*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/
2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2
*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^5*tan(1/2*c)^16 - 58*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*
x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4
- 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^7*tan(1/2*c)^12 - 354*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 -
4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - t
an(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^6*tan(1/2*c)^13 - 162*a^(5/2)*sgn(tan(1/2*d*x)^4*tan
(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(
1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^5*tan(1/2*c)^14 - 30*a^(5/2)*sgn(tan(1/2
*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/
2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^4*tan(1/2*c)^15 + 3*a^(5/2)*s
gn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)
- 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^16 - 6
*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*t
an(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^7*tan(1/2
*c)^10 + 330*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(
1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*
x)^6*tan(1/2*c)^11 + 1514*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*
x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) +
1)*tan(1/2*d*x)^5*tan(1/2*c)^12 + 1062*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3
- tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*
tan(1/2*c) + 1)*tan(1/2*d*x)^4*tan(1/2*c)^13 + 318*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*
tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*t
an(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^14 + 46*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(
1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2
*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^15 + 5*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^
4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3
- tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^16 + 30*a^(5/2)*sgn(tan(1/2*d*x)^4*ta
n(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan
(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^6*tan(1/2*c)^9 - 506*a^(5/2)*sgn(tan(1/
2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1
/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^5*tan(1/2*c)^10 - 2750*a^(5/
2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2
*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^4*tan(1/2*c)^11
- 2198*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d
*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*
tan(1/2*c)^12 - 742*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 -
4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan
(1/2*d*x)^2*tan(1/2*c)^13 - 126*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(
1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2
*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^14 - 10*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)
^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x
)*tan(1/2*c) + 1)*tan(1/2*c)^15 + 6*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 -
tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan
(1/2*c) + 1)*tan(1/2*d*x)^7*tan(1/2*c)^6 + 30*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1
/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/
2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^6*tan(1/2*c)^7 - 330*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d
*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4
- 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^4*tan(1/2*c)^9 + 1382*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 -
4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 -
tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^10 + 1182*a^(5/2)*sgn(tan(1/2*d*x)^4*t
an(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*ta
n(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^11 + 326*a^(5/2)*sgn(tan(
1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan
(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^12 + 34*a^(5/2)
*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c
) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^13 + 58*a^(5/2)*sgn
(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) -
4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^7*tan(1/2*c)^4 + 330*
a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*ta
n(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^6*tan(1/2*
c)^5 + 506*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/
2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)
^5*tan(1/2*c)^6 - 330*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4
- 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*t
an(1/2*d*x)^4*tan(1/2*c)^7 - 486*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan
(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/
2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^9 - 294*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2
*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*
d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^10 - 42*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^
3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4
*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^11 - 34*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*ta
n(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan
(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^7*tan(1/2*c)^2 - 354*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/
2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c
)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^6*tan(1/2*c)^3 - 1514*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^
4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3
- tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^5*tan(1/2*c)^4 - 2750*a^(5/2)*sgn(tan(1/2*d*x)^4
*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*
tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^4*tan(1/2*c)^5 - 1382*a^(5/2)*sgn(ta
n(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*t
an(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^6 - 486*a^(
5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1
/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^
7 + 18*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*
x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^9 - 5
*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*t
an(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^7 - 6*a^(
5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1
/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^6*tan(1/2*c)
+ 162*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x
)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^5*ta
n(1/2*c)^2 + 1062*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4
*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1
/2*d*x)^4*tan(1/2*c)^3 + 2198*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/
2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c
) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^4 + 1182*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c
)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*
x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^5 + 294*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^
3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4
*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^6 + 18*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1
/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*
c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^7 - 3*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x
)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 -
4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^5 - 30*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3
*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*
tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^4*tan(1/2*c) - 318*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1
/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*
c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^2 - 742*a^(5/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^
4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3
- tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^3 - 326*a^(5/2)*sgn(tan(1/2*d*x)^4*
tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*t
an(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^4 - 42*a^(5/2)*sgn(tan(1/2
*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/
2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^5 - 3*a^(5/2)*sgn(tan(1/2*d*x)^
4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)
*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3 + 46*a^(5/2)*sgn(tan(1/2*d*x)^4*t
an(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*ta
n(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c) + 126*a^(5/2)*sgn(tan(1/2
*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/
2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^2 + 34*a^(5/2)*sgn
(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) -
4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^3 - 5*a^(5/2)*sgn(tan(1
/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(
1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x) - 10*a^(5/2)*sgn(tan(1/2*d*
x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d
*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c))/((tan(1/2*c)^8 - 4*tan(1/2*c)^6 +
6*tan(1/2*c)^4 - 4*tan(1/2*c)^2 + 1)*(tan(1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*x)*tan(1/2*c
) - tan(1/2*c)^2 + 1)^4))/d
Mupad [F(-1)]
Timed out. \[
\int \left (a+a \tan ^2(c+d x)\right )^{5/2} \, dx=\int {\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x
\]
[In]
int((a + a*tan(c + d*x)^2)^(5/2),x)
[Out]
int((a + a*tan(c + d*x)^2)^(5/2), x)