3.1153 \(\int \frac{(a+a \sec (c+d x))^{5/2} (A+C \sec ^2(c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=332 \[ \frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a^{5/2} (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{512 d}+\frac{a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d \cos ^{\frac{7}{2}}(c+d x)} \]

[Out]

(a^(5/2)*(1304*A + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Se
c[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]])
+ (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 10
15*C)*Sin[c + d*x])/(512*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 23*C)*Sqrt[a + a*Sec[c
+ d*x]]*Sin[c + d*x])/(96*d*Cos[c + d*x]^(7/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Cos[c +
d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(7/2))

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Rubi [A]  time = 1.03235, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a^{5/2} (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{512 d}+\frac{a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d \cos ^{\frac{7}{2}}(c+d x)} \]

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]^(5/2),x]

[Out]

(a^(5/2)*(1304*A + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Se
c[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]])
+ (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 10
15*C)*Sin[c + d*x])/(512*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 23*C)*Sqrt[a + a*Sec[c
+ d*x]]*Sin[c + d*x])/(96*d*Cos[c + d*x]^(7/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Cos[c +
d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*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))^{5/2} \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))^{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)}{6 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))^{5/2} \left (\frac{1}{2} a (12 A+5 C)+\frac{5}{2} a C \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 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{15}{4} a^2 (8 A+5 C)+\frac{5}{4} a^2 (24 A+23 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac{a^2 (24 A+23 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 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}{8} a^3 (312 A+235 C)+\frac{15}{8} a^3 (136 A+109 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+23 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{384} \left (a^2 (1304 A+1015 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^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+23 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{1}{512} \left (a^2 (1304 A+1015 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 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+23 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{\left (a^2 (1304 A+1015 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}{1024}\\ &=\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+23 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{\left (a^2 (1304 A+1015 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 )}{512 d}\\ &=\frac{a^{5/2} (1304 A+1015 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)}}{512 d}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (24 A+23 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac{7}{2}}(c+d x)}\\ \end{align*}

Mathematica [A]  time = 4.76885, size = 200, normalized size = 0.6 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (14 (4056 A+4591 C) \cos (c+d x)+16 (1496 A+1711 C) \cos (2 (c+d x))+25448 A \cos (3 (c+d x))+5216 A \cos (4 (c+d x))+3912 A \cos (5 (c+d x))+18720 A+21721 C \cos (3 (c+d x))+4060 C \cos (4 (c+d x))+3045 C \cos (5 (c+d x))+27412 C)+24 \sqrt{2} (1304 A+1015 C) \cos ^6(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{24576 d \cos ^{\frac{11}{2}}(c+d x)} \]

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]^(5/2),x]

[Out]

(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(24*Sqrt[2]*(1304*A + 1015*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2
]]*Cos[c + d*x]^6 + (18720*A + 27412*C + 14*(4056*A + 4591*C)*Cos[c + d*x] + 16*(1496*A + 1711*C)*Cos[2*(c + d
*x)] + 25448*A*Cos[3*(c + d*x)] + 21721*C*Cos[3*(c + d*x)] + 5216*A*Cos[4*(c + d*x)] + 4060*C*Cos[4*(c + d*x)]
+ 3912*A*Cos[5*(c + d*x)] + 3045*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(24576*d*Cos[c + d*x]^(11/2))

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Maple [A]  time = 0.373, size = 564, normalized size = 1.7 \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)^(5/2),x)

[Out]

1/3072/d*a^2*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)*(-1+cos(d*x+c))*(3912*A*2^(1/2)*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)^6-3912*A*2^(1/2)*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)^6+3045*C*2^(1/2)*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)^6-3045*C*2^(1/2)*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)^6-7824*A*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*cos(d*x+c)^5-6090*C*(-2/(cos(d*x+c)+1))^
(1/2)*sin(d*x+c)*cos(d*x+c)^5-5216*A*sin(d*x+c)*(-2/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)^4-4060*C*sin(d*x+c)*(-2/(
cos(d*x+c)+1))^(1/2)*cos(d*x+c)^4-2944*A*sin(d*x+c)*cos(d*x+c)^3*(-2/(cos(d*x+c)+1))^(1/2)-3248*C*cos(d*x+c)^3
*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-768*A*cos(d*x+c)^2*sin(d*x+c)*(-2/(cos(d*x+c)+1))^(1/2)-2784*C*cos(d*x+c
)^2*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-1792*C*(-2/(cos(d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)-512*C*(-2/(cos
(d*x+c)+1))^(1/2)*sin(d*x+c))/(-2/(cos(d*x+c)+1))^(1/2)/sin(d*x+c)^2/cos(d*x+c)^(11/2)

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Maxima [B]  time = 11.3096, size = 14959, normalized size = 45.06 \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)^(5/2),x, algorithm="maxima")

[Out]

-1/6144*(8*(1956*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*
c) + 4*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))) + 652*(sqrt(2)*a^2*
sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x +
2*c))*cos(13/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 6204*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*
a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(11/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) - 2060*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt
(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))
) + 2060*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*s
qrt(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))) - 6204*(sqrt(2)*a^2*sin(8*d*
x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*sqrt(2)*a^2*sin(2*d*x + 2*c))*c
os(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 652*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*
d*x + 6*c) + 6*sqrt(2)*a^2*sin(4*d*x + 4*c) + 4*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))) - 1956*(sqrt(2)*a^2*sin(8*d*x + 8*c) + 4*sqrt(2)*a^2*sin(6*d*x + 6*c) + 6*sqrt(2)*a^2*sin
(4*d*x + 4*c) + 4*sqrt(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))) - 489*(a^
2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2
*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*
x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*
d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*
c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) +
3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d
*x + 2*c))*sin(6*d*x + 6*c))*log(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*sq
rt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 489*(a^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d
*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*
x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2
+ 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) +
a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^
2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*
d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(2*co
s(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) - 489*(a^2*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)
^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^
2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*(4
*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*cos
(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x +
4*c) + 4*(2*a^2*sin(6*d*x + 6*c) + 3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*
a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*log(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) + 489*(a^2
*cos(8*d*x + 8*c)^2 + 16*a^2*cos(6*d*x + 6*c)^2 + 36*a^2*cos(4*d*x + 4*c)^2 + 16*a^2*cos(2*d*x + 2*c)^2 + a^2*
sin(8*d*x + 8*c)^2 + 16*a^2*sin(6*d*x + 6*c)^2 + 36*a^2*sin(4*d*x + 4*c)^2 + 48*a^2*sin(4*d*x + 4*c)*sin(2*d*x
+ 2*c) + 16*a^2*sin(2*d*x + 2*c)^2 + 8*a^2*cos(2*d*x + 2*c) + a^2 + 2*(4*a^2*cos(6*d*x + 6*c) + 6*a^2*cos(4*d
*x + 4*c) + 4*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 8*(6*a^2*cos(4*d*x + 4*c) + 4*a^2*cos(2*d*x + 2*c
) + a^2)*cos(6*d*x + 6*c) + 12*(4*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 4*(2*a^2*sin(6*d*x + 6*c) + 3
*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*a^2*sin(4*d*x + 4*c) + 2*a^2*sin(2*d*
x + 2*c))*sin(6*d*x + 6*c))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(s
in(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*sqr
t(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) - 1956*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2
)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*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))) - 652*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x
+ 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*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))) - 6204*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt
(2)*a^2*cos(4*d*x + 4*c) + 4*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(11/4*arctan2(sin(2*d*x + 2*c), co
s(2*d*x + 2*c))) + 2060*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d
*x + 4*c) + 4*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)))
- 2060*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*sq
rt(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))) + 6204*(sqrt(2)
*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*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))) + 652*(sqrt(2)*a^2*cos(8*d*x +
8*c) + 4*sqrt(2)*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*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))) + 1956*(sqrt(2)*a^2*cos(8*d*x + 8*c) + 4*sqrt(2)
*a^2*cos(6*d*x + 6*c) + 6*sqrt(2)*a^2*cos(4*d*x + 4*c) + 4*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))))*A*sqrt(a)/(2*(4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2
*d*x + 2*c) + 1)*cos(8*d*x + 8*c) + cos(8*d*x + 8*c)^2 + 8*(6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(6
*d*x + 6*c) + 16*cos(6*d*x + 6*c)^2 + 12*(4*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 36*cos(4*d*x + 4*c)^2 + 1
6*cos(2*d*x + 2*c)^2 + 4*(2*sin(6*d*x + 6*c) + 3*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin
(8*d*x + 8*c)^2 + 16*(3*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 16*sin(6*d*x + 6*c)^2 + 36*s
in(4*d*x + 4*c)^2 + 48*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*sin(2*d*x + 2*c)^2 + 8*cos(2*d*x + 2*c) + 1) + (
12180*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 2
0*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(23/4*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 4060*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x +
10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) +
6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(21/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 70644*(sqrt(2)*a^2*si
n(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d
*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*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))) + 22620*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*
a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2
*d*x + 2*c))*cos(17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 147592*(sqrt(2)*a^2*sin(12*d*x + 12*c) +
6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt
(2)*a^2*sin(4*d*x + 4*c) + 6*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)
)) - 37800*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c
) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(13
/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 37800*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10
*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x +
4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(11/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 147592*(sqrt(2)
*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*
sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*sqrt(2)*a^2*sin(2*d*x + 2*c))*cos(9/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c))) - 22620*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sq
rt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*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))) - 70644*(sqrt(2)*a^2*sin(12*d*x + 12*c
) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*
sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*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))) - 4060*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(10*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8
*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x + 4*c) + 6*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))) - 12180*(sqrt(2)*a^2*sin(12*d*x + 12*c) + 6*sqrt(2)*a^2*sin(1
0*d*x + 10*c) + 15*sqrt(2)*a^2*sin(8*d*x + 8*c) + 20*sqrt(2)*a^2*sin(6*d*x + 6*c) + 15*sqrt(2)*a^2*sin(4*d*x +
4*c) + 6*sqrt(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))) - 3045*(a^2*cos(1
2*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 + 225*a^2*cos(8*d*x + 8*c)^2 + 400*a^2*cos(6*d*x + 6*c)^2 + 225*
a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(12*d*x + 12*c)^2 + 36*a^2*sin(10*d*x + 10*c)^2 +
225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x + 6*c)^2 + 225*a^2*sin(4*d*x + 4*c)^2 + 180*a^2*sin(4*d*x + 4*c
)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2 + 12*a^2*cos(2*d*x + 2*c) + a^2 + 2*(6*a^2*cos(10*d*x + 10*c) +
15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*c
os(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*co
s(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2
*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x +
6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 2*(6*a^2*sin(10*d*x + 10*c) + 15*a^2*sin(8*d*x + 8
*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 12*(15*
a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(10*d*x
+ 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 1
20*(5*a^2*sin(4*d*x + 4*c) + 2*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(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) + 30
45*(a^2*cos(12*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 + 225*a^2*cos(8*d*x + 8*c)^2 + 400*a^2*cos(6*d*x +
6*c)^2 + 225*a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(12*d*x + 12*c)^2 + 36*a^2*sin(10*d*x
+ 10*c)^2 + 225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x + 6*c)^2 + 225*a^2*sin(4*d*x + 4*c)^2 + 180*a^2*si
n(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2 + 12*a^2*cos(2*d*x + 2*c) + a^2 + 2*(6*a^2*cos(10*
d*x + 10*c) + 15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x +
2*c) + a^2)*cos(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*
c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c)
+ 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)
*cos(6*d*x + 6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 2*(6*a^2*sin(10*d*x + 10*c) + 15*a^2*
sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(12*d*x + 12
*c) + 12*(15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c)
)*sin(10*d*x + 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*sin(8*d
*x + 8*c) + 120*(5*a^2*sin(4*d*x + 4*c) + 2*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) - 3045*(a^2*cos(12*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 + 225*a^2*cos(8*d*x + 8*c)^2 + 400*a^2
*cos(6*d*x + 6*c)^2 + 225*a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(12*d*x + 12*c)^2 + 36*a
^2*sin(10*d*x + 10*c)^2 + 225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x + 6*c)^2 + 225*a^2*sin(4*d*x + 4*c)^2
+ 180*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2 + 12*a^2*cos(2*d*x + 2*c) + a^2 + 2*(
6*a^2*cos(10*d*x + 10*c) + 15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2
*cos(2*d*x + 2*c) + a^2)*cos(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*c
os(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(
4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x
+ 2*c) + a^2)*cos(6*d*x + 6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 2*(6*a^2*sin(10*d*x + 10
*c) + 15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x + 2*c))*si
n(12*d*x + 12*c) + 12*(15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin
(2*d*x + 2*c))*sin(10*d*x + 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d*x +
2*c))*sin(8*d*x + 8*c) + 120*(5*a^2*sin(4*d*x + 4*c) + 2*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), co
s(2*d*x + 2*c))) + 2) + 3045*(a^2*cos(12*d*x + 12*c)^2 + 36*a^2*cos(10*d*x + 10*c)^2 + 225*a^2*cos(8*d*x + 8*c
)^2 + 400*a^2*cos(6*d*x + 6*c)^2 + 225*a^2*cos(4*d*x + 4*c)^2 + 36*a^2*cos(2*d*x + 2*c)^2 + a^2*sin(12*d*x + 1
2*c)^2 + 36*a^2*sin(10*d*x + 10*c)^2 + 225*a^2*sin(8*d*x + 8*c)^2 + 400*a^2*sin(6*d*x + 6*c)^2 + 225*a^2*sin(4
*d*x + 4*c)^2 + 180*a^2*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*a^2*sin(2*d*x + 2*c)^2 + 12*a^2*cos(2*d*x + 2*c
) + a^2 + 2*(6*a^2*cos(10*d*x + 10*c) + 15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*c) + 15*a^2*cos(4*d*x +
4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(12*d*x + 12*c) + 12*(15*a^2*cos(8*d*x + 8*c) + 20*a^2*cos(6*d*x + 6*
c) + 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(10*d*x + 10*c) + 30*(20*a^2*cos(6*d*x + 6*c)
+ 15*a^2*cos(4*d*x + 4*c) + 6*a^2*cos(2*d*x + 2*c) + a^2)*cos(8*d*x + 8*c) + 40*(15*a^2*cos(4*d*x + 4*c) + 6*a
^2*cos(2*d*x + 2*c) + a^2)*cos(6*d*x + 6*c) + 30*(6*a^2*cos(2*d*x + 2*c) + a^2)*cos(4*d*x + 4*c) + 2*(6*a^2*si
n(10*d*x + 10*c) + 15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2*sin(2*d
*x + 2*c))*sin(12*d*x + 12*c) + 12*(15*a^2*sin(8*d*x + 8*c) + 20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c
) + 6*a^2*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 30*(20*a^2*sin(6*d*x + 6*c) + 15*a^2*sin(4*d*x + 4*c) + 6*a^2
*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 120*(5*a^2*sin(4*d*x + 4*c) + 2*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) - 12180*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c)
+ 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt
(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(23/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 4060*(sqrt(2)*
a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*c
os(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(21/4*arc
tan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 70644*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x +
10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) +
6*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))) - 22620*(
sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(
2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*sqrt(2)*a^2*cos(2*d*x + 2*c) + sqrt(2)*a^2)*sin(
17/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 147592*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos
(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x
+ 4*c) + 6*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)))
+ 37800*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) +
20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*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))) - 37800*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)
*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*c
os(4*d*x + 4*c) + 6*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))) + 147592*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x
+ 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*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))) + 22620*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6
*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(
2)*a^2*cos(4*d*x + 4*c) + 6*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))) + 70644*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos
(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*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))) + 4060*(sqrt(2)*a^2*cos(12*d*x + 12*c
) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*
sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*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))) + 12180*(sqrt(2)*a^2*cos(12*d*x + 12*c) + 6*sqrt(2)*a^2*cos(10*d*x + 10*c) + 15*sqrt(2)*a^
2*cos(8*d*x + 8*c) + 20*sqrt(2)*a^2*cos(6*d*x + 6*c) + 15*sqrt(2)*a^2*cos(4*d*x + 4*c) + 6*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*(6*cos(10*d*x + 10
*c) + 15*cos(8*d*x + 8*c) + 20*cos(6*d*x + 6*c) + 15*cos(4*d*x + 4*c) + 6*cos(2*d*x + 2*c) + 1)*cos(12*d*x + 1
2*c) + cos(12*d*x + 12*c)^2 + 12*(15*cos(8*d*x + 8*c) + 20*cos(6*d*x + 6*c) + 15*cos(4*d*x + 4*c) + 6*cos(2*d*
x + 2*c) + 1)*cos(10*d*x + 10*c) + 36*cos(10*d*x + 10*c)^2 + 30*(20*cos(6*d*x + 6*c) + 15*cos(4*d*x + 4*c) + 6
*cos(2*d*x + 2*c) + 1)*cos(8*d*x + 8*c) + 225*cos(8*d*x + 8*c)^2 + 40*(15*cos(4*d*x + 4*c) + 6*cos(2*d*x + 2*c
) + 1)*cos(6*d*x + 6*c) + 400*cos(6*d*x + 6*c)^2 + 30*(6*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 225*cos(4*d*
x + 4*c)^2 + 36*cos(2*d*x + 2*c)^2 + 2*(6*sin(10*d*x + 10*c) + 15*sin(8*d*x + 8*c) + 20*sin(6*d*x + 6*c) + 15*
sin(4*d*x + 4*c) + 6*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + sin(12*d*x + 12*c)^2 + 12*(15*sin(8*d*x + 8*c) + 2
0*sin(6*d*x + 6*c) + 15*sin(4*d*x + 4*c) + 6*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 36*sin(10*d*x + 10*c)^2 +
30*(20*sin(6*d*x + 6*c) + 15*sin(4*d*x + 4*c) + 6*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 225*sin(8*d*x + 8*c)^2
+ 120*(5*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 400*sin(6*d*x + 6*c)^2 + 225*sin(4*d*x + 4*
c)^2 + 180*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 36*sin(2*d*x + 2*c)^2 + 12*cos(2*d*x + 2*c) + 1))/d

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Fricas [A]  time = 0.864351, size = 1539, normalized size = 4.64 \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)^(5/2),x, algorithm="fricas")

[Out]

[1/6144*(4*(3*(1304*A + 1015*C)*a^2*cos(d*x + c)^5 + 2*(1304*A + 1015*C)*a^2*cos(d*x + c)^4 + 8*(184*A + 203*C
)*a^2*cos(d*x + c)^3 + 48*(8*A + 29*C)*a^2*cos(d*x + c)^2 + 896*C*a^2*cos(d*x + c) + 256*C*a^2)*sqrt((a*cos(d*
x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 3*((1304*A + 1015*C)*a^2*cos(d*x + c)^7 + (1304*A
+ 1015*C)*a^2*cos(d*x + c)^6)*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)^7 + d*cos(d*x + c)^6), 1/3072*(2*(3*(1304*A + 1015*C)*a^2*cos(d*x + c)^5 + 2*(1304*A +
1015*C)*a^2*cos(d*x + c)^4 + 8*(184*A + 203*C)*a^2*cos(d*x + c)^3 + 48*(8*A + 29*C)*a^2*cos(d*x + c)^2 + 896*C
*a^2*cos(d*x + c) + 256*C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) + 3*((1
304*A + 1015*C)*a^2*cos(d*x + c)^7 + (1304*A + 1015*C)*a^2*cos(d*x + c)^6)*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)^7 + d*cos(d*x + c)^6)]

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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)**(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{5}{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))^(5/2)*(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)^(5/2)/cos(d*x + c)^(5/2), x)