3.1143 \(\int \frac{(a+a \sec (c+d x))^{3/2} (A+C \sec ^2(c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\)
Optimal. Leaf size=285 \[ \frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{3 a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)} \]
[Out]
(a^(3/2)*(176*A + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[
c + d*x]])/(128*d) + (a^2*(80*A + 67*C)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a
^2*(176*A + 133*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Si
n[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x]
)/(40*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))
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Rubi [A] time = 0.807993, antiderivative size = 285, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used =
{4265, 4089, 4018, 4016, 3803, 3801, 215} \[ \frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{3 a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]
[Out]
(a^(3/2)*(176*A + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[
c + d*x]])/(128*d) + (a^2*(80*A + 67*C)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a
^2*(176*A + 133*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Si
n[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x]
)/(40*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))
Rule 4265
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Dist[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]
Rule 4089
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*(m + n + 1)
), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n + a
*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1
)] && !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]
Rule 4018
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> -Simp[(b*B*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n)/(f*(m + n
)), x] + Dist[1/(d*(m + n)), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*d*(m + n) + B*(b*d*n
) + (A*b*d*(m + n) + a*B*d*(2*m + n - 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, n}, x] && Ne
Q[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1]
Rule 4016
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(
B_.) + (A_)), x_Symbol] :> Simp[(-2*b*B*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Csc[e + f*x]]
), x] + Dist[(A*b*(2*n + 1) + 2*a*B*n)/(b*(2*n + 1)), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^n, x], x]
/; FreeQ[{a, b, d, e, f, A, B, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n
, 0] && !LtQ[n, 0]
Rule 3803
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*b*d
*Cot[e + f*x]*(d*Csc[e + f*x])^(n - 1))/(f*(2*n - 1)*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(2*a*d*(n - 1))/(b*(
2*n - 1)), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a
^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
Rule 3801
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*a*Sq
rt[(a*d)/b])/(b*f), Subst[Int[1/Sqrt[1 + x^2/a], x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; Free
Q[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[(a*d)/b, 0]
Rule 215
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{5}{2} a (2 A+C)+\frac{3}{2} a C \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{3 a C \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{5}{4} a^2 (16 A+11 C)+\frac{1}{4} a^2 (80 A+67 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{3 a C \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{96} \left (a (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{3 a C \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{128} \left (a (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{3 a C \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{256} \left (a (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{3 a C \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{\left (a (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{128 d}\\ &=\frac{a^{3/2} (176 A+133 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{128 d}+\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{3 a C \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 4.62212, size = 176, normalized size = 0.62 \[ \frac{a \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (12 (880 A+1273 C) \cos (c+d x)+4 (3280 A+3059 C) \cos (2 (c+d x))+3520 A \cos (3 (c+d x))+2640 A \cos (4 (c+d x))+10480 A+2660 C \cos (3 (c+d x))+1995 C \cos (4 (c+d x))+13313 C)+60 \sqrt{2} (176 A+133 C) \cos ^5(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{15360 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]
[Out]
(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(176*A + 133*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*C
os[c + d*x]^5 + (10480*A + 13313*C + 12*(880*A + 1273*C)*Cos[c + d*x] + 4*(3280*A + 3059*C)*Cos[2*(c + d*x)] +
3520*A*Cos[3*(c + d*x)] + 2660*C*Cos[3*(c + d*x)] + 2640*A*Cos[4*(c + d*x)] + 1995*C*Cos[4*(c + d*x)])*Sin[(c
+ d*x)/2]))/(15360*d*Cos[c + d*x]^(9/2))
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Maple [B] time = 0.317, size = 500, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x)
[Out]
-1/3840/d*a*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))*(2640*A*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))
^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))*cos(d*x+c)^5*2^(1/2)-2640*A*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(co
s(d*x+c)+1-sin(d*x+c)))*cos(d*x+c)^5*2^(1/2)+1995*C*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1
+sin(d*x+c)))*cos(d*x+c)^5*2^(1/2)-1995*C*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1-sin(d*x+c
)))*cos(d*x+c)^5*2^(1/2)+5280*A*sin(d*x+c)*(-2/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^4+3990*C*sin(d*x+c)*(-2/(cos(d
*x+c)+1))^(1/2)*cos(d*x+c)^4+3520*A*sin(d*x+c)*cos(d*x+c)^3*(-2/(cos(d*x+c)+1))^(1/2)+2660*C*cos(d*x+c)^3*(-2/
(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+1280*A*cos(d*x+c)^2*sin(d*x+c)*(-2/(cos(d*x+c)+1))^(1/2)+2128*C*cos(d*x+c)^2*
(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+1824*C*(-2/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)+768*C*(-2/(cos(d*x
+c)+1))^(1/2)*sin(d*x+c))/sin(d*x+c)^2/cos(d*x+c)^(9/2)/(-2/(cos(d*x+c)+1))^(1/2)
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Maxima [B] time = 5.97014, size = 9767, normalized size = 34.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x, algorithm="maxima")
[Out]
-1/7680*(80*(132*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*co
s(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 44*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x +
4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 216*(sqrt(2)*a*si
n(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c))) - 216*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x
+ 2*c))*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(sqrt(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*si
n(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 132*(sqr
t(2)*a*sin(6*d*x + 6*c) + 3*sqrt(2)*a*sin(4*d*x + 4*c) + 3*sqrt(2)*a*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))) - 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a
*sin(6*d*x + 6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2
+ 2*(3*a*cos(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d
*x + 4*c) + 6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos
(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2
+ 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c))) + 2) + 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a*sin
(6*d*x + 6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2 + 2
*(3*a*cos(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d*x +
4*c) + 6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2
*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), co
s(2*d*x + 2*c))) + 2) - 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a*sin(6*d
*x + 6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2 + 2*(3*
a*cos(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c
) + 6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqr
t(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*
d*x + 2*c))) + 2) + 33*(a*cos(6*d*x + 6*c)^2 + 9*a*cos(4*d*x + 4*c)^2 + 9*a*cos(2*d*x + 2*c)^2 + a*sin(6*d*x +
6*c)^2 + 9*a*sin(4*d*x + 4*c)^2 + 18*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*a*sin(2*d*x + 2*c)^2 + 2*(3*a*co
s(4*d*x + 4*c) + 3*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 6*(3*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) +
6*a*cos(2*d*x + 2*c) + 6*(a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arctan2
(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)
*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))) + 2) - 132*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c)
+ sqrt(2)*a)*sin(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2
)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) - 216*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt
(2)*a)*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 216*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*co
s(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))
) + 44*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x + 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*
sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 132*(sqrt(2)*a*cos(6*d*x + 6*c) + 3*sqrt(2)*a*cos(4*d*x
+ 4*c) + 3*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*A*sq
rt(a)/(2*(3*cos(4*d*x + 4*c) + 3*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + cos(6*d*x + 6*c)^2 + 6*(3*cos(2*d*x
+ 2*c) + 1)*cos(4*d*x + 4*c) + 9*cos(4*d*x + 4*c)^2 + 9*cos(2*d*x + 2*c)^2 + 6*(sin(4*d*x + 4*c) + sin(2*d*x +
2*c))*sin(6*d*x + 6*c) + sin(6*d*x + 6*c)^2 + 9*sin(4*d*x + 4*c)^2 + 18*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9
*sin(2*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c) + 1) + (7980*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x +
8*c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(19/4*
arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2660*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*
c) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(17/4*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 38304*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c
) + 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(15/4*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 12160*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c)
+ 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(13/4*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 45400*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c)
+ 10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(11/4*arcta
n2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 45400*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) +
10*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(9/4*arctan2
(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 12160*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 1
0*sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(7/4*arctan2(s
in(2*d*x + 2*c), cos(2*d*x + 2*c))) - 38304*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*
sqrt(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(5/4*arctan2(sin
(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2660*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqr
t(2)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*
d*x + 2*c), cos(2*d*x + 2*c))) - 7980*(sqrt(2)*a*sin(10*d*x + 10*c) + 5*sqrt(2)*a*sin(8*d*x + 8*c) + 10*sqrt(2
)*a*sin(6*d*x + 6*c) + 10*sqrt(2)*a*sin(4*d*x + 4*c) + 5*sqrt(2)*a*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) - 1995*(a*cos(10*d*x + 10*c)^2 + 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^
2 + 100*a*cos(4*d*x + 4*c)^2 + 25*a*cos(2*d*x + 2*c)^2 + a*sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 10
0*a*sin(6*d*x + 6*c)^2 + 100*a*sin(4*d*x + 4*c)^2 + 100*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x +
2*c)^2 + 2*(5*a*cos(8*d*x + 8*c) + 10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*
cos(10*d*x + 10*c) + 10*(10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x +
8*c) + 20*(10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)
*cos(4*d*x + 4*c) + 10*a*cos(2*d*x + 2*c) + 10*(a*sin(8*d*x + 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*
c) + a*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + 100*(2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*s
qrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c))) + 2) + 1995*(a*cos(10*d*x + 10*c)^2 + 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^2 + 100*
a*cos(4*d*x + 4*c)^2 + 25*a*cos(2*d*x + 2*c)^2 + a*sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 100*a*sin(
6*d*x + 6*c)^2 + 100*a*sin(4*d*x + 4*c)^2 + 100*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x + 2*c)^2
+ 2*(5*a*cos(8*d*x + 8*c) + 10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(10*d
*x + 10*c) + 10*(10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) +
20*(10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)*cos(4*d
*x + 4*c) + 10*a*cos(2*d*x + 2*c) + 10*(a*sin(8*d*x + 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*s
in(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*si
n(8*d*x + 8*c) + 100*(2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arctan2(s
in(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*c
os(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c))) + 2) - 1995*(a*cos(10*d*x + 10*c)^2 + 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^2 + 100*a*cos(4*
d*x + 4*c)^2 + 25*a*cos(2*d*x + 2*c)^2 + a*sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 100*a*sin(6*d*x +
6*c)^2 + 100*a*sin(4*d*x + 4*c)^2 + 100*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x + 2*c)^2 + 2*(5*a
*cos(8*d*x + 8*c) + 10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(10*d*x + 10*
c) + 10*(10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 20*(10*a
*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c
) + 10*a*cos(2*d*x + 2*c) + 10*(a*sin(8*d*x + 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x
+ 2*c))*sin(10*d*x + 10*c) + 50*(2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x
+ 8*c) + 100*(2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) +
2) + 1995*(a*cos(10*d*x + 10*c)^2 + 25*a*cos(8*d*x + 8*c)^2 + 100*a*cos(6*d*x + 6*c)^2 + 100*a*cos(4*d*x + 4*
c)^2 + 25*a*cos(2*d*x + 2*c)^2 + a*sin(10*d*x + 10*c)^2 + 25*a*sin(8*d*x + 8*c)^2 + 100*a*sin(6*d*x + 6*c)^2 +
100*a*sin(4*d*x + 4*c)^2 + 100*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a*sin(2*d*x + 2*c)^2 + 2*(5*a*cos(8*d
*x + 8*c) + 10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(10*d*x + 10*c) + 10*
(10*a*cos(6*d*x + 6*c) + 10*a*cos(4*d*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(8*d*x + 8*c) + 20*(10*a*cos(4*d
*x + 4*c) + 5*a*cos(2*d*x + 2*c) + a)*cos(6*d*x + 6*c) + 20*(5*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 10*a
*cos(2*d*x + 2*c) + 10*(a*sin(8*d*x + 8*c) + 2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))
*sin(10*d*x + 10*c) + 50*(2*a*sin(6*d*x + 6*c) + 2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) +
100*(2*a*sin(4*d*x + 4*c) + a*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + a)*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(s
in(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) - 79
80*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a
*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(19/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2
*c))) - 2660*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10
*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(17/4*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c))) - 38304*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x
+ 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(15/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) - 12160*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a
*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(13/4*arctan2
(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 45400*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 1
0*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(1
1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 45400*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x
+ 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(
2)*a)*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 12160*(sqrt(2)*a*cos(10*d*x + 10*c) + 5*sqrt(2)*a
*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(2*d*x + 2*
c) + sqrt(2)*a)*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 38304*(sqrt(2)*a*cos(10*d*x + 10*c) + 5
*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(2)*a*cos(
2*d*x + 2*c) + sqrt(2)*a)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2660*(sqrt(2)*a*cos(10*d*x +
10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c) + 5*sqrt(
2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 7980*(sqrt(2)*a*cos(
10*d*x + 10*c) + 5*sqrt(2)*a*cos(8*d*x + 8*c) + 10*sqrt(2)*a*cos(6*d*x + 6*c) + 10*sqrt(2)*a*cos(4*d*x + 4*c)
+ 5*sqrt(2)*a*cos(2*d*x + 2*c) + sqrt(2)*a)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*C*sqrt(a)/(2
*(5*cos(8*d*x + 8*c) + 10*cos(6*d*x + 6*c) + 10*cos(4*d*x + 4*c) + 5*cos(2*d*x + 2*c) + 1)*cos(10*d*x + 10*c)
+ cos(10*d*x + 10*c)^2 + 10*(10*cos(6*d*x + 6*c) + 10*cos(4*d*x + 4*c) + 5*cos(2*d*x + 2*c) + 1)*cos(8*d*x + 8
*c) + 25*cos(8*d*x + 8*c)^2 + 20*(10*cos(4*d*x + 4*c) + 5*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + 100*cos(6*d
*x + 6*c)^2 + 20*(5*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 100*cos(4*d*x + 4*c)^2 + 25*cos(2*d*x + 2*c)^2 +
10*(sin(8*d*x + 8*c) + 2*sin(6*d*x + 6*c) + 2*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + sin(10
*d*x + 10*c)^2 + 50*(2*sin(6*d*x + 6*c) + 2*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 25*sin(8*d
*x + 8*c)^2 + 100*(2*sin(4*d*x + 4*c) + sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 100*sin(6*d*x + 6*c)^2 + 100*sin(
4*d*x + 4*c)^2 + 100*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*sin(2*d*x + 2*c)^2 + 10*cos(2*d*x + 2*c) + 1))/d
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Fricas [A] time = 0.863369, size = 1385, normalized size = 4.86 \begin{align*} \left [\frac{4 \,{\left (15 \,{\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{4} + 10 \,{\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{3} + 8 \,{\left (80 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{2} + 912 \, C a \cos \left (d x + c\right ) + 384 \, C a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 15 \,{\left ({\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{6} +{\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{5}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 4 \, \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}{\left (\cos \left (d x + c\right ) - 2\right )} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{7680 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}, \frac{2 \,{\left (15 \,{\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{4} + 10 \,{\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{3} + 8 \,{\left (80 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{2} + 912 \, C a \cos \left (d x + c\right ) + 384 \, C a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 15 \,{\left ({\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{6} +{\left (176 \, A + 133 \, C\right )} a \cos \left (d x + c\right )^{5}\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right )}{3840 \,{\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x, algorithm="fricas")
[Out]
[1/7680*(4*(15*(176*A + 133*C)*a*cos(d*x + c)^4 + 10*(176*A + 133*C)*a*cos(d*x + c)^3 + 8*(80*A + 133*C)*a*cos
(d*x + c)^2 + 912*C*a*cos(d*x + c) + 384*C*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d
*x + c) + 15*((176*A + 133*C)*a*cos(d*x + c)^6 + (176*A + 133*C)*a*cos(d*x + c)^5)*sqrt(a)*log((a*cos(d*x + c)
^3 - 4*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*(cos(d*x + c) - 2)*sqrt(cos(d*x + c))*sin(d*x + c) - 7*
a*cos(d*x + c)^2 + 8*a)/(cos(d*x + c)^3 + cos(d*x + c)^2)))/(d*cos(d*x + c)^6 + d*cos(d*x + c)^5), 1/3840*(2*(
15*(176*A + 133*C)*a*cos(d*x + c)^4 + 10*(176*A + 133*C)*a*cos(d*x + c)^3 + 8*(80*A + 133*C)*a*cos(d*x + c)^2
+ 912*C*a*cos(d*x + c) + 384*C*a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 15
*((176*A + 133*C)*a*cos(d*x + c)^6 + (176*A + 133*C)*a*cos(d*x + c)^5)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(
d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 - a*cos(d*x + c) - 2*a)))/(d*cos
(d*x + c)^6 + 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((a+a*sec(d*x+c))**(3/2)*(A+C*sec(d*x+c)**2)/cos(d*x+c)**(5/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\cos \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2),x, algorithm="giac")
[Out]
integrate((C*sec(d*x + c)^2 + A)*(a*sec(d*x + c) + a)^(3/2)/cos(d*x + c)^(5/2), x)