3.60 \(\int \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=182 \[ -\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{163 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 d}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]

[Out]

(-163*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) - (163*a^3*Cot[c + d*x])/(64*d*
Sqrt[a + a*Sin[c + d*x]]) - (163*a^3*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (17*a^3*Cot[
c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c +
d*x]])/(4*d)

________________________________________________________________________________________

Rubi [A]  time = 0.335321, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2762, 2980, 2772, 2773, 206} \[ -\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a \sin (c+d x)+a}}-\frac{163 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{64 d}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a \sin (c+d x)+a}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a \sin (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

(-163*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/(64*d) - (163*a^3*Cot[c + d*x])/(64*d*
Sqrt[a + a*Sin[c + d*x]]) - (163*a^3*Cot[c + d*x]*Csc[c + d*x])/(96*d*Sqrt[a + a*Sin[c + d*x]]) - (17*a^3*Cot[
c + d*x]*Csc[c + d*x]^2)/(24*d*Sqrt[a + a*Sin[c + d*x]]) - (a^2*Cot[c + d*x]*Csc[c + d*x]^3*Sqrt[a + a*Sin[c +
d*x]])/(4*d)

Rule 2762

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(b^2*(b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(b*c
+ a*d)), x] + Dist[b^2/(d*(n + 1)*(b*c + a*d)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*
Simp[a*c*(m - 2) - b*d*(m - 2*n - 4) - (b*c*(m - 1) - a*d*(m + 2*n + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
&& (IntegersQ[2*m, 2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2980

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^2*(B*c - A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n
+ 1)*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(2*d*(n + 1)
*(b*c + a*d)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, -1]

Rule 2772

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[((b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]]), x]
+ Dist[((2*n + 3)*(b*c - a*d))/(2*b*(n + 1)*(c^2 - d^2)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &
& LtQ[n, -1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]

Rule 2773

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(-2*
b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^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 \csc ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{1}{4} a \int \csc ^4(c+d x) \left (-\frac{17 a}{2}-\frac{13}{2} a \sin (c+d x)\right ) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}+\frac{1}{48} \left (163 a^2\right ) \int \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}+\frac{1}{64} \left (163 a^2\right ) \int \csc ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}+\frac{1}{128} \left (163 a^2\right ) \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}-\frac{\left (163 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 d}\\ &=-\frac{163 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{64 d}-\frac{163 a^3 \cot (c+d x)}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{163 a^3 \cot (c+d x) \csc (c+d x)}{96 d \sqrt{a+a \sin (c+d x)}}-\frac{17 a^3 \cot (c+d x) \csc ^2(c+d x)}{24 d \sqrt{a+a \sin (c+d x)}}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x) \sqrt{a+a \sin (c+d x)}}{4 d}\\ \end{align*}

Mathematica [B]  time = 1.60193, size = 370, normalized size = 2.03 \[ -\frac{a^2 \csc ^{13}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sin (c+d x)+1)} \left (1030 \sin \left (\frac{1}{2} (c+d x)\right )+3102 \sin \left (\frac{3}{2} (c+d x)\right )+326 \sin \left (\frac{5}{2} (c+d x)\right )-978 \sin \left (\frac{7}{2} (c+d x)\right )-1030 \cos \left (\frac{1}{2} (c+d x)\right )+3102 \cos \left (\frac{3}{2} (c+d x)\right )-326 \cos \left (\frac{5}{2} (c+d x)\right )-978 \cos \left (\frac{7}{2} (c+d x)\right )-1956 \cos (2 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+489 \cos (4 (c+d x)) \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+1467 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+1956 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-489 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )-1467 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{192 d \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac{1}{4} (c+d x)\right )-\sec ^2\left (\frac{1}{4} (c+d x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]

[Out]

-(a^2*Csc[(c + d*x)/2]^13*Sqrt[a*(1 + Sin[c + d*x])]*(-1030*Cos[(c + d*x)/2] + 3102*Cos[(3*(c + d*x))/2] - 326
*Cos[(5*(c + d*x))/2] - 978*Cos[(7*(c + d*x))/2] + 1467*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 1956*Co
s[2*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 489*Cos[4*(c + d*x)]*Log[1 + Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]] - 1467*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1956*Cos[2*(c + d*x)]*Log[1 - Cos[(c +
d*x)/2] + Sin[(c + d*x)/2]] - 489*Cos[4*(c + d*x)]*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1030*Sin[(c
+ d*x)/2] + 3102*Sin[(3*(c + d*x))/2] + 326*Sin[(5*(c + d*x))/2] - 978*Sin[(7*(c + d*x))/2]))/(192*d*(1 + Cot
[(c + d*x)/2])*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^4)

________________________________________________________________________________________

Maple [A]  time = 0.727, size = 162, normalized size = 0.9 \begin{align*} -{\frac{1+\sin \left ( dx+c \right ) }{192\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 1047\,\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{a}^{11/2}-2303\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{3/2}{a}^{9/2}+1793\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{5/2}{a}^{7/2}-489\, \left ( -a \left ( \sin \left ( dx+c \right ) -1 \right ) \right ) ^{7/2}{a}^{5/2}+489\,{\it Artanh} \left ({\frac{\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{\sqrt{a}}} \right ){a}^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{4} \right ){a}^{-{\frac{7}{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(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x)

[Out]

-1/192*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(1047*(-a*(sin(d*x+c)-1))^(1/2)*a^(11/2)-2303*(-a*(sin(d*x+c)-
1))^(3/2)*a^(9/2)+1793*(-a*(sin(d*x+c)-1))^(5/2)*a^(7/2)-489*(-a*(sin(d*x+c)-1))^(7/2)*a^(5/2)+489*arctanh((-a
*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^6*sin(d*x+c)^4)/a^(7/2)/sin(d*x+c)^4/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \csc \left (d x + c\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(5/2)*csc(d*x + c)^5, x)

________________________________________________________________________________________

Fricas [B]  time = 1.91775, size = 1238, normalized size = 6.8 \begin{align*} \frac{489 \,{\left (a^{2} \cos \left (d x + c\right )^{5} + a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) + a^{2} +{\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} - 9 \, a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \,{\left (489 \, a^{2} \cos \left (d x + c\right )^{4} + 326 \, a^{2} \cos \left (d x + c\right )^{3} - 836 \, a^{2} \cos \left (d x + c\right )^{2} - 374 \, a^{2} \cos \left (d x + c\right ) + 299 \, a^{2} +{\left (489 \, a^{2} \cos \left (d x + c\right )^{3} + 163 \, a^{2} \cos \left (d x + c\right )^{2} - 673 \, a^{2} \cos \left (d x + c\right ) - 299 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{768 \,{\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/768*(489*(a^2*cos(d*x + c)^5 + a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^3 - 2*a^2*cos(d*x + c)^2 + a^2*cos(d*
x + c) + a^2 + (a^2*cos(d*x + c)^4 - 2*a^2*cos(d*x + c)^2 + a^2)*sin(d*x + c))*sqrt(a)*log((a*cos(d*x + c)^3 -
7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x
+ c) + a)*sqrt(a) - 9*a*cos(d*x + c) + (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x +
c)^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1)) + 4*(489*a^2*cos(d*x + c)^4 +
326*a^2*cos(d*x + c)^3 - 836*a^2*cos(d*x + c)^2 - 374*a^2*cos(d*x + c) + 299*a^2 + (489*a^2*cos(d*x + c)^3 + 1
63*a^2*cos(d*x + c)^2 - 673*a^2*cos(d*x + c) - 299*a^2)*sin(d*x + c))*sqrt(a*sin(d*x + c) + a))/(d*cos(d*x + c
)^5 + d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 - 2*d*cos(d*x + c)^2 + d*cos(d*x + c) + (d*cos(d*x + c)^4 - 2*d*co
s(d*x + c)^2 + d)*sin(d*x + c) + d)

________________________________________________________________________________________

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(csc(d*x+c)**5*(a+a*sin(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 3.28643, size = 1102, normalized size = 6.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+a*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

1/384*(978*a^3*arctan(-(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))/sqrt(-a))*sgn(tan(1
/2*d*x + 1/2*c) + 1)/sqrt(-a) - 489*a^(5/2)*log(abs(-sqrt(a)*tan(1/2*d*x + 1/2*c) + sqrt(a*tan(1/2*d*x + 1/2*c
)^2 + a)))*sgn(tan(1/2*d*x + 1/2*c) + 1) + (400*a^2*sgn(tan(1/2*d*x + 1/2*c) + 1) + (135*a^2*sgn(tan(1/2*d*x +
1/2*c) + 1) + 2*(3*a^2*sgn(tan(1/2*d*x + 1/2*c) + 1)*tan(1/2*d*x + 1/2*c) + 20*a^2*sgn(tan(1/2*d*x + 1/2*c) +
1))*tan(1/2*d*x + 1/2*c))*tan(1/2*d*x + 1/2*c))*sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a) - (11736*sqrt(2)*a^3*arcta
n((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 5868*sqrt(2)*sqrt(-a)*a^(5/2)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 16626
*a^3*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - 8313*sqrt(-a)*a^(5/2)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 536
6*sqrt(2)*sqrt(-a)*a^(5/2) + 7552*sqrt(-a)*a^(5/2))*sgn(tan(1/2*d*x + 1/2*c) + 1)/(12*sqrt(2)*sqrt(-a) + 17*sq
rt(-a)) + 2*(135*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^7*a^3*sgn(tan(1/2*d*x + 1
/2*c) + 1) + 480*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^6*a^(7/2)*sgn(tan(1/2*d*x
+ 1/2*c) + 1) - 111*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^5*a^4*sgn(tan(1/2*d*x
+ 1/2*c) + 1) - 1200*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^4*a^(9/2)*sgn(tan(1/
2*d*x + 1/2*c) + 1) - 111*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^3*a^5*sgn(tan(1/
2*d*x + 1/2*c) + 1) + 1120*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a^(11/2)*sgn(
tan(1/2*d*x + 1/2*c) + 1) + 135*(sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*tan(1/2*d*x + 1/2*c)^2 + a))*a^6*sgn(ta
n(1/2*d*x + 1/2*c) + 1) - 400*a^(13/2)*sgn(tan(1/2*d*x + 1/2*c) + 1))/((sqrt(a)*tan(1/2*d*x + 1/2*c) - sqrt(a*
tan(1/2*d*x + 1/2*c)^2 + a))^2 - a)^4)/d