3.168 \(\int \frac{\sec ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\)

Optimal. Leaf size=221 \[ -\frac{231 a \cos (c+d x)}{512 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a \sec ^3(c+d x)}{60 d (a \sin (c+d x)+a)^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{77 a \sec (c+d x)}{320 d (a \sin (c+d x)+a)^{3/2}}-\frac{231 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{512 \sqrt{2} \sqrt{a} d} \]

[Out]

(-231*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(512*Sqrt[2]*Sqrt[a]*d) - (231*a*Cos
[c + d*x])/(512*d*(a + a*Sin[c + d*x])^(3/2)) - (77*a*Sec[c + d*x])/(320*d*(a + a*Sin[c + d*x])^(3/2)) - (11*a
*Sec[c + d*x]^3)/(60*d*(a + a*Sin[c + d*x])^(3/2)) + (77*Sec[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (11*
Sec[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^5/(5*d*Sqrt[a + a*Sin[c + d*x]])

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Rubi [A]  time = 0.35688, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2687, 2681, 2650, 2649, 206} \[ -\frac{231 a \cos (c+d x)}{512 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a \sec ^3(c+d x)}{60 d (a \sin (c+d x)+a)^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{77 a \sec (c+d x)}{320 d (a \sin (c+d x)+a)^{3/2}}-\frac{231 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{512 \sqrt{2} \sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

(-231*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(512*Sqrt[2]*Sqrt[a]*d) - (231*a*Cos
[c + d*x])/(512*d*(a + a*Sin[c + d*x])^(3/2)) - (77*a*Sec[c + d*x])/(320*d*(a + a*Sin[c + d*x])^(3/2)) - (11*a
*Sec[c + d*x]^3)/(60*d*(a + a*Sin[c + d*x])^(3/2)) + (77*Sec[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (11*
Sec[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^5/(5*d*Sqrt[a + a*Sin[c + d*x]])

Rule 2687

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> -Simp[(b*(g*
Cos[e + f*x])^(p + 1))/(a*f*g*(p + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(a*(2*p + 1))/(2*g^2*(p + 1)), Int[
(g*Cos[e + f*x])^(p + 2)/(a + b*Sin[e + f*x])^(3/2), x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
&& LtQ[p, -1] && IntegerQ[2*p]

Rule 2681

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(2*m + p + 1)), x] + Dist[(m + p + 1)/(a*(2*m + p + 1)),
Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && IntegersQ[2*m, 2*p]

Rule 2650

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

Rule 2649

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2/d, Subst[Int[1/(2*a - x^2), x], x, (b*C
os[c + d*x])/Sqrt[a + b*Sin[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{\sec ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{10} (11 a) \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{33}{40} \int \frac{\sec ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{80} (77 a) \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac{11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{77}{128} \int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac{11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{256} (231 a) \int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac{77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac{11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{231 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{1024}\\ &=-\frac{231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac{77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac{11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}-\frac{231 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{512 d}\\ &=-\frac{231 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{512 \sqrt{2} \sqrt{a} d}-\frac{231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac{77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac{11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 0.661624, size = 140, normalized size = 0.63 \[ \frac{\frac{1}{16} \sec ^5(c+d x) (36850 \sin (c+d x)+17787 \sin (3 (c+d x))+3465 \sin (5 (c+d x))+11352 \cos (2 (c+d x))+2310 \cos (4 (c+d x))+11090)+(3465+3465 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )}{7680 d \sqrt{a (\sin (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]

[Out]

((3465 + 3465*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]) + (Sec[c + d*x]^5*(11090 + 11352*Cos[2*(c + d*x)] + 2310*Cos[4*(c + d*x)] + 36850*Sin[c + d*x] + 1
7787*Sin[3*(c + d*x)] + 3465*Sin[5*(c + d*x)]))/16)/(7680*d*Sqrt[a*(1 + Sin[c + d*x])])

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Maple [A]  time = 0.184, size = 308, normalized size = 1.4 \begin{align*} -{\frac{1}{15360\, \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}\cos \left ( dx+c \right ) d} \left ( -6930\,{a}^{11/2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+ \left ( -3696\,{a}^{11/2}-3465\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) + \left ( -2816\,{a}^{11/2}+13860\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2} \right ) \sin \left ( dx+c \right ) -2310\,{a}^{11/2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+ \left ( -528\,{a}^{11/2}-10395\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-256\,{a}^{11/2}+13860\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{3} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2} \right ){a}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^6/(a+a*sin(d*x+c))^(1/2),x)

[Out]

-1/15360*(-6930*a^(11/2)*sin(d*x+c)*cos(d*x+c)^4+(-3696*a^(11/2)-3465*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/
2)*2^(1/2)/a^(1/2))*a^3*(a-a*sin(d*x+c))^(5/2))*cos(d*x+c)^2*sin(d*x+c)+(-2816*a^(11/2)+13860*2^(1/2)*arctanh(
1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^3*(a-a*sin(d*x+c))^(5/2))*sin(d*x+c)-2310*a^(11/2)*cos(d*x+c)^4+
(-528*a^(11/2)-10395*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^3*(a-a*sin(d*x+c))^(5/2))*c
os(d*x+c)^2-256*a^(11/2)+13860*2^(1/2)*arctanh(1/2*(a-a*sin(d*x+c))^(1/2)*2^(1/2)/a^(1/2))*a^3*(a-a*sin(d*x+c)
)^(5/2))/a^(11/2)/(sin(d*x+c)-1)^2/(1+sin(d*x+c))^2/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{6}}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(sec(d*x + c)^6/sqrt(a*sin(d*x + c) + a), x)

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Fricas [A]  time = 2.72939, size = 695, normalized size = 3.14 \begin{align*} \frac{3465 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{5}\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \,{\left (1155 \, \cos \left (d x + c\right )^{4} + 264 \, \cos \left (d x + c\right )^{2} + 11 \,{\left (315 \, \cos \left (d x + c\right )^{4} + 168 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 128\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{30720 \,{\left (a d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

1/30720*(3465*sqrt(2)*(cos(d*x + c)^5*sin(d*x + c) + cos(d*x + c)^5)*sqrt(a)*log(-(a*cos(d*x + c)^2 - 2*sqrt(2
)*sqrt(a*sin(d*x + c) + a)*sqrt(a)*(cos(d*x + c) - sin(d*x + c) + 1) + 3*a*cos(d*x + c) - (a*cos(d*x + c) - 2*
a)*sin(d*x + c) + 2*a)/(cos(d*x + c)^2 - (cos(d*x + c) + 2)*sin(d*x + c) - cos(d*x + c) - 2)) + 4*(1155*cos(d*
x + c)^4 + 264*cos(d*x + c)^2 + 11*(315*cos(d*x + c)^4 + 168*cos(d*x + c)^2 + 128)*sin(d*x + c) + 128)*sqrt(a*
sin(d*x + c) + a))/(a*d*cos(d*x + c)^5*sin(d*x + c) + a*d*cos(d*x + c)^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**6/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 4.42085, size = 1438, normalized size = 6.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^6/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/7680*(3465*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a) +
sqrt(a))/sqrt(-a))/(sqrt(-a)*sgn(tan(1/2*d*x + 1/2*c) + 1)) + 64*(165*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*t
an(1/2*d*x + 1/2*c)^2 + a))^9 - 915*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^8*sqrt
(a) + 1340*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^7*a + 1420*(sqrt(a)*tan(1/2*d*x
+ 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*a^(3/2) - 3434*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/
2*d*x + 1/2*c)^2 + a))^5*a^2 - 1610*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^4*a^(5
/2) + 2700*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^3*a^3 + 2620*(sqrt(a)*tan(1/2*d
*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^(7/2) + 845*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1
/2*d*x + 1/2*c)^2 + a))*a^4 + 101*a^(9/2))/(((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a
))^2 - 2*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))*sqrt(a) - a)^5*sgn(tan(1/2*d*x +
1/2*c) + 1)) + 10*(1323*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^11 + 8793*(sqrt(a)
*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^10*sqrt(a) + 20907*(sqrt(a)*tan(1/2*d*x + 1/2*c) -
sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^9*a + 9237*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2
+ a))^8*a^(3/2) - 26274*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^7*a^2 - 12806*(sqr
t(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*a^(5/2) + 30342*(sqrt(a)*tan(1/2*d*x + 1/2*c
) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^5*a^3 - 4182*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*
c)^2 + a))^4*a^(7/2) - 12793*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^3*a^4 + 9405*
(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^(9/2) - 2721*(sqrt(a)*tan(1/2*d*x + 1/
2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))*a^5 + 337*a^(11/2))/(((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/
2*d*x + 1/2*c)^2 + a))^2 + 2*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))*sqrt(a) - a)^
6*sgn(tan(1/2*d*x + 1/2*c) + 1)))/d