3.1152 \(\int \frac{(a+a \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\)
Optimal. Leaf size=285 \[ \frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1040 A+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (400 A+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]
[Out]
(a^(5/2)*(400*A + 283*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^3*(1040*A + 787*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) +
(a^3*(400*A + 283*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 79*C)*S
qrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x
])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))
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Rubi [A] time = 0.932856, 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} \[ \frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1040 A+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (400 A+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]
[Out]
(a^(5/2)*(400*A + 283*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^3*(1040*A + 787*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) +
(a^3*(400*A + 283*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 79*C)*S
qrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x
])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))
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))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (10 A+3 C)+\frac{5}{2} a C \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{1}{4} a^2 (80 A+39 C)+\frac{1}{4} a^2 (80 A+79 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (80 A+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left (\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)} \left (\frac{3}{8} a^3 (240 A+157 C)+\frac{1}{8} a^3 (1040 A+787 C) \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac{a^3 (1040 A+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{1}{128} \left (a^2 (400 A+283 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^3 (1040 A+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{1}{256} \left (a^2 (400 A+283 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^3 (1040 A+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{\left (a^2 (400 A+283 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^{5/2} (400 A+283 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^3 (1040 A+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (80 A+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 3.57572, size = 178, normalized size = 0.62 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (12 (1360 A+2343 C) \cos (c+d x)+4 (6640 A+6509 C) \cos (2 (c+d x))+5440 A \cos (3 (c+d x))+6000 A \cos (4 (c+d x))+20560 A+5660 C \cos (3 (c+d x))+4245 C \cos (4 (c+d x))+24863 C)+60 \sqrt{2} (400 A+283 C) \cos ^5(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{15360 d \cos ^{\frac{9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]
[Out]
(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(400*A + 283*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]
*Cos[c + d*x]^5 + (20560*A + 24863*C + 12*(1360*A + 2343*C)*Cos[c + d*x] + 4*(6640*A + 6509*C)*Cos[2*(c + d*x)
] + 5440*A*Cos[3*(c + d*x)] + 5660*C*Cos[3*(c + d*x)] + 6000*A*Cos[4*(c + d*x)] + 4245*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 = 0.346, size = 502, 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))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x)
[Out]
-1/3840/d*a^2*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))*(-6000*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)+6000*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)-4245*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)+4245*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)+12000*A*sin(d*x+c)*(-2/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^4+8490*C*sin(d*x+c)*(-2/(c
os(d*x+c)+1))^(1/2)*cos(d*x+c)^4+5440*A*sin(d*x+c)*cos(d*x+c)^3*(-2/(cos(d*x+c)+1))^(1/2)+5660*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)+4528*C*cos(d*x+c
)^2*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)+2784*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/(-2/(cos(d*x+c)+1))^(1/2)/cos(d*x+c)^(9/2)
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Maxima [B] time = 6.46128, size = 11950, normalized size = 41.93 \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))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="maxima")
[Out]
1/7680*(80*(300*sqrt(2)*a^2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))*sin(6*d*x + 6*c) - 28
*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) + 28*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) - 28*(sqrt(2)*a^2*sin(9/2*d*x + 9/2*c)
- sqrt(2)*a^2*sin(3/2*d*x + 3/2*c))*cos(6*d*x + 6*c) - 300*(sqrt(2)*a^2*sin(6*d*x + 6*c) + 3*sqrt(2)*a^2*sin(
8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 3*sqrt(2)*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c),
cos(3/2*d*x + 3/2*c))))*cos(11/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 12*(7*sqrt(2)*a^2*sin
(9/2*d*x + 9/2*c) - 7*sqrt(2)*a^2*sin(3/2*d*x + 3/2*c) - 114*sqrt(2)*a^2*sin(7/3*arctan2(sin(3/2*d*x + 3/2*c),
cos(3/2*d*x + 3/2*c))) + 114*sqrt(2)*a^2*sin(5/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 75*sq
rt(2)*a^2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*cos(8/3*arctan2(sin(3/2*d*x + 3/2*c),
cos(3/2*d*x + 3/2*c))) - 456*(sqrt(2)*a^2*sin(6*d*x + 6*c) + 3*sqrt(2)*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c
), cos(3/2*d*x + 3/2*c))))*cos(7/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 456*(sqrt(2)*a^2*sin
(6*d*x + 6*c) + 3*sqrt(2)*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*cos(5/3*arctan2(si
n(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 12*(7*sqrt(2)*a^2*sin(9/2*d*x + 9/2*c) - 7*sqrt(2)*a^2*sin(3/2*d*
x + 3/2*c) + 75*sqrt(2)*a^2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*cos(4/3*arctan2(sin(
3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 75*(a^2*cos(6*d*x + 6*c)^2 + 9*a^2*cos(8/3*arctan2(sin(3/2*d*x + 3/
2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*a^2*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + a^2*si
n(6*d*x + 6*c)^2 + 9*a^2*sin(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 6*a^2*sin(6*d*x + 6*
c)*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 9*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c),
cos(3/2*d*x + 3/2*c)))^2 + 2*a^2*cos(6*d*x + 6*c) + a^2 + 6*(a^2*cos(6*d*x + 6*c) + 3*a^2*cos(4/3*arctan2(sin(
3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + a^2)*cos(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) +
6*(a^2*cos(6*d*x + 6*c) + a^2)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(a^2*sin(6*d*
x + 6*c) + 3*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*sin(8/3*arctan2(sin(3/2*d*x + 3
/2*c), cos(3/2*d*x + 3/2*c))))*log(2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/
3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos
(3/2*d*x + 3/2*c))) + 2*sqrt(2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) - 75*(a^2*co
s(6*d*x + 6*c)^2 + 9*a^2*cos(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*a^2*cos(4/3*arctan
2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + a^2*sin(6*d*x + 6*c)^2 + 9*a^2*sin(8/3*arctan2(sin(3/2*d*x
+ 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 6*a^2*sin(6*d*x + 6*c)*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x
+ 3/2*c))) + 9*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*a^2*cos(6*d*x + 6*c) + a
^2 + 6*(a^2*cos(6*d*x + 6*c) + 3*a^2*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + a^2)*cos(8
/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(a^2*cos(6*d*x + 6*c) + a^2)*cos(4/3*arctan2(sin(3
/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c)
, cos(3/2*d*x + 3/2*c))))*sin(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*log(2*cos(1/3*arctan2(
sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))
)^2 + 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 2*sqrt(2)*sin(1/3*arctan2(sin(3
/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) + 75*(a^2*cos(6*d*x + 6*c)^2 + 9*a^2*cos(8/3*arctan2(sin(3/2*d*x
+ 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*a^2*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + a^
2*sin(6*d*x + 6*c)^2 + 9*a^2*sin(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 6*a^2*sin(6*d*x
+ 6*c)*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 9*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*
c), cos(3/2*d*x + 3/2*c)))^2 + 2*a^2*cos(6*d*x + 6*c) + a^2 + 6*(a^2*cos(6*d*x + 6*c) + 3*a^2*cos(4/3*arctan2(
sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + a^2)*cos(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)
)) + 6*(a^2*cos(6*d*x + 6*c) + a^2)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(a^2*sin(
6*d*x + 6*c) + 3*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*sin(8/3*arctan2(sin(3/2*d*x
+ 3/2*c), cos(3/2*d*x + 3/2*c))))*log(2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*si
n(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 - 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c),
cos(3/2*d*x + 3/2*c))) + 2*sqrt(2)*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) - 75*(a^
2*cos(6*d*x + 6*c)^2 + 9*a^2*cos(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 9*a^2*cos(4/3*ar
ctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + a^2*sin(6*d*x + 6*c)^2 + 9*a^2*sin(8/3*arctan2(sin(3/2*
d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 6*a^2*sin(6*d*x + 6*c)*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*
d*x + 3/2*c))) + 9*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*a^2*cos(6*d*x + 6*c)
+ a^2 + 6*(a^2*cos(6*d*x + 6*c) + 3*a^2*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + a^2)*c
os(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(a^2*cos(6*d*x + 6*c) + a^2)*cos(4/3*arctan2(s
in(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 6*(a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4/3*arctan2(sin(3/2*d*x + 3/
2*c), cos(3/2*d*x + 3/2*c))))*sin(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*log(2*cos(1/3*arct
an2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2
*c)))^2 - 2*sqrt(2)*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 2*sqrt(2)*sin(1/3*arctan2(s
in(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 2) + 28*(sqrt(2)*a^2*cos(9/2*d*x + 9/2*c) - sqrt(2)*a^2*cos(3/2*
d*x + 3/2*c))*sin(6*d*x + 6*c) + 300*(sqrt(2)*a^2*cos(6*d*x + 6*c) + 3*sqrt(2)*a^2*cos(8/3*arctan2(sin(3/2*d*x
+ 3/2*c), cos(3/2*d*x + 3/2*c))) + 3*sqrt(2)*a^2*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))
+ sqrt(2)*a^2)*sin(11/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 12*(7*sqrt(2)*a^2*cos(9/2*d*x
+ 9/2*c) - 7*sqrt(2)*a^2*cos(3/2*d*x + 3/2*c) - 114*sqrt(2)*a^2*cos(7/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*
d*x + 3/2*c))) + 114*sqrt(2)*a^2*cos(5/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 75*sqrt(2)*a^2
*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*sin(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d
*x + 3/2*c))) + 456*(sqrt(2)*a^2*cos(6*d*x + 6*c) + 3*sqrt(2)*a^2*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/
2*d*x + 3/2*c))) + sqrt(2)*a^2)*sin(7/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 456*(sqrt(2)*a^
2*cos(6*d*x + 6*c) + 3*sqrt(2)*a^2*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + sqrt(2)*a^2)
*sin(5/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 12*(7*sqrt(2)*a^2*cos(9/2*d*x + 9/2*c) - 7*sqr
t(2)*a^2*cos(3/2*d*x + 3/2*c) + 75*sqrt(2)*a^2*cos(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*s
in(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) - 300*(sqrt(2)*a^2*cos(6*d*x + 6*c) + sqrt(2)*a^2)
*sin(1/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))))*A*sqrt(a)/(cos(6*d*x + 6*c)^2 + 6*(cos(6*d*x +
6*c) + 3*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 1)*cos(8/3*arctan2(sin(3/2*d*x + 3/2*c
), cos(3/2*d*x + 3/2*c))) + 9*cos(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 6*(cos(6*d*x +
6*c) + 1)*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 9*cos(4/3*arctan2(sin(3/2*d*x + 3/2*c
), cos(3/2*d*x + 3/2*c)))^2 + sin(6*d*x + 6*c)^2 + 6*(sin(6*d*x + 6*c) + 3*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c
), cos(3/2*d*x + 3/2*c))))*sin(8/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c))) + 9*sin(8/3*arctan2(si
n(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 6*sin(6*d*x + 6*c)*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/
2*d*x + 3/2*c))) + 9*sin(4/3*arctan2(sin(3/2*d*x + 3/2*c), cos(3/2*d*x + 3/2*c)))^2 + 2*cos(6*d*x + 6*c) + 1)
- (16980*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) +
10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(19/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))) + 5660*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x +
6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(17/4*arctan2(sin(2*d*x + 2*c), c
os(2*d*x + 2*c))) + 81504*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*si
n(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(15/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) + 8320*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2
)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(13/4*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c))) + 86440*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) +
10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(11/4*a
rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 86440*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin(8*d*x
+ 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x + 2*c))*c
os(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 8320*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)*a^2*sin
(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(2*d*x +
2*c))*cos(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 81504*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*sqrt(2)
*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a^2*sin(
2*d*x + 2*c))*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 5660*(sqrt(2)*a^2*sin(10*d*x + 10*c) + 5*
sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*sqrt(2)*a
^2*sin(2*d*x + 2*c))*cos(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 16980*(sqrt(2)*a^2*sin(10*d*x + 10
*c) + 5*sqrt(2)*a^2*sin(8*d*x + 8*c) + 10*sqrt(2)*a^2*sin(6*d*x + 6*c) + 10*sqrt(2)*a^2*sin(4*d*x + 4*c) + 5*s
qrt(2)*a^2*sin(2*d*x + 2*c))*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 4245*(a^2*cos(10*d*x + 10*
c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2*cos(2*d*x
+ 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*a^2*sin(4*d
*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*d*x + 2*c)
+ a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c)
+ a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) +
a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 20*(5*a
^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*
d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) +
a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))
*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) + 4245*(a^2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos
(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*s
in(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d
*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos
(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*
d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x +
4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) +
10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x
+ 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(
2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*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) - 4245*(a^
2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 +
25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x + 10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2
+ 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2
*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x + 8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^
2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*c
os(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*
x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6
*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*s
in(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))
*sin(6*d*x + 6*c))*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) + 4245*(a^2*cos(10*d*x + 10*c)^2 + 25*a^2*cos(8*d*x + 8*
c)^2 + 100*a^2*cos(6*d*x + 6*c)^2 + 100*a^2*cos(4*d*x + 4*c)^2 + 25*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(10*d*x +
10*c)^2 + 25*a^2*sin(8*d*x + 8*c)^2 + 100*a^2*sin(6*d*x + 6*c)^2 + 100*a^2*sin(4*d*x + 4*c)^2 + 100*a^2*sin(4*
d*x + 4*c)*sin(2*d*x + 2*c) + 25*a^2*sin(2*d*x + 2*c)^2 + 10*a^2*cos(2*d*x + 2*c) + a^2 + 2*(5*a^2*cos(8*d*x +
8*c) + 10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) +
10*(10*a^2*cos(6*d*x + 6*c) + 10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 20*(
10*a^2*cos(4*d*x + 4*c) + 5*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 20*(5*a^2*cos(2*d*x + 2*c) + a^2)*c
os(4*d*x + 4*c) + 10*(a^2*sin(8*d*x + 8*c) + 2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x +
2*c))*sin(10*d*x + 10*c) + 50*(2*a^2*sin(6*d*x + 6*c) + 2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(8*
d*x + 8*c) + 100*(2*a^2*sin(4*d*x + 4*c) + a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*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) - 16980*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x +
6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(19/4*arctan2(sin(2*
d*x + 2*c), cos(2*d*x + 2*c))) - 5660*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sq
rt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*s
in(17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 81504*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*c
os(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x
+ 2*c) + sqrt(2)*a^2)*sin(15/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 8320*(sqrt(2)*a^2*cos(10*d*x + 1
0*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*
sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 86440*(sqr
t(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^
2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*
x + 2*c))) + 86440*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x
+ 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(9/4*arctan2(sin(
2*d*x + 2*c), cos(2*d*x + 2*c))) + 8320*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*
sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)
*sin(7/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 81504*(sqrt(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*
cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x
+ 2*c) + sqrt(2)*a^2)*sin(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 5660*(sqrt(2)*a^2*cos(10*d*x + 1
0*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2*cos(4*d*x + 4*c) + 5*
sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(3/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 16980*(sqrt
(2)*a^2*cos(10*d*x + 10*c) + 5*sqrt(2)*a^2*cos(8*d*x + 8*c) + 10*sqrt(2)*a^2*cos(6*d*x + 6*c) + 10*sqrt(2)*a^2
*cos(4*d*x + 4*c) + 5*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*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(1
0*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 = 0.856404, size = 1426, normalized size = 5. \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))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="fricas")
[Out]
[1/7680*(4*(15*(400*A + 283*C)*a^2*cos(d*x + c)^4 + 10*(272*A + 283*C)*a^2*cos(d*x + c)^3 + 8*(80*A + 283*C)*a
^2*cos(d*x + c)^2 + 1392*C*a^2*cos(d*x + c) + 384*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x
+ c))*sin(d*x + c) + 15*((400*A + 283*C)*a^2*cos(d*x + c)^6 + (400*A + 283*C)*a^2*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))*si
n(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*(400*A + 283*C)*a^2*cos(d*x + c)^4 + 10*(272*A + 283*C)*a^2*cos(d*x + c)^3 + 8*(80*A + 283*C
)*a^2*cos(d*x + c)^2 + 1392*C*a^2*cos(d*x + c) + 384*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d
*x + c))*sin(d*x + c) + 15*((400*A + 283*C)*a^2*cos(d*x + c)^6 + (400*A + 283*C)*a^2*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))**(5/2)*(A+C*sec(d*x+c)**2)/cos(d*x+c)**(3/2),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{5}{2}}}{\cos \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sec(d*x+c))^(5/2)*(A+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="giac")
[Out]
integrate((C*sec(d*x + c)^2 + A)*(a*sec(d*x + c) + a)^(5/2)/cos(d*x + c)^(3/2), x)