3.1270 \(\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=301 \[ \frac{a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (400 A+326 B+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a (2 B+C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]

[Out]

(a^(5/2)*(400*A + 326*B + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*S
qrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 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 + 326*B + 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 + 110*B + 79*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*(2*B + 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))

________________________________________________________________________________________

Rubi [A]  time = 1.00604, antiderivative size = 301, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {4265, 4088, 4018, 4016, 3803, 3801, 215} \[ \frac{a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (400 A+326 B+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{128 d}+\frac{a (2 B+C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]

[Out]

(a^(5/2)*(400*A + 326*B + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*S
qrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 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 + 326*B + 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 + 110*B + 79*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*(2*B + 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 4088

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + 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 + b*B*(m + n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A,
B, 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+B \sec (c+d x)+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+B \sec (c+d x)+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 (2 B+C) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac{a (2 B+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+30 B+39 C)+\frac{1}{4} a^2 (80 A+110 B+79 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac{a^2 (80 A+110 B+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a (2 B+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+170 B+157 C)+\frac{1}{8} a^3 (1040 A+950 B+787 C) \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac{a^3 (1040 A+950 B+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+110 B+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a (2 B+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+326 B+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+950 B+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+326 B+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+110 B+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a (2 B+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+326 B+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+950 B+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+326 B+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+110 B+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a (2 B+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+326 B+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+326 B+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+950 B+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+326 B+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+110 B+79 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a (2 B+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 = 6.32817, size = 212, normalized size = 0.7 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} \left (\sin \left (\frac{1}{2} (c+d x)\right ) (12 (1360 A+1950 B+2343 C) \cos (c+d x)+4 (6640 A+6730 B+6509 C) \cos (2 (c+d x))+5440 A \cos (3 (c+d x))+6000 A \cos (4 (c+d x))+20560 A+6520 B \cos (3 (c+d x))+4890 B \cos (4 (c+d x))+22030 B+5660 C \cos (3 (c+d x))+4245 C \cos (4 (c+d x))+24863 C)+60 \sqrt{2} (400 A+326 B+283 C) \cos ^5(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{15360 d \cos ^{\frac{9}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + 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 + 326*B + 283*C)*ArcTanh[Sqrt[2]*Sin[(c +
d*x)/2]]*Cos[c + d*x]^5 + (20560*A + 22030*B + 24863*C + 12*(1360*A + 1950*B + 2343*C)*Cos[c + d*x] + 4*(6640*
A + 6730*B + 6509*C)*Cos[2*(c + d*x)] + 5440*A*Cos[3*(c + d*x)] + 6520*B*Cos[3*(c + d*x)] + 5660*C*Cos[3*(c +
d*x)] + 6000*A*Cos[4*(c + d*x)] + 4890*B*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 = 0.375, size = 722, normalized size = 2.4 \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+B*sec(d*x+c)+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*cos(d*x+c)^5*arctan(1/4*2^(1/2)*(-2/(
cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1-sin(d*x+c)))*2^(1/2)-6000*A*cos(d*x+c)^5*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)
+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))*2^(1/2)+4890*B*cos(d*x+c)^5*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)
*(cos(d*x+c)+1-sin(d*x+c)))*2^(1/2)-4890*B*cos(d*x+c)^5*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+
c)+1+sin(d*x+c)))*2^(1/2)+4245*C*cos(d*x+c)^5*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1-sin(d
*x+c)))*2^(1/2)-4245*C*cos(d*x+c)^5*arctan(1/4*2^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))*2^
(1/2)-12000*A*cos(d*x+c)^4*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-9780*B*cos(d*x+c)^4*(-2/(cos(d*x+c)+1))^(1/2)*
sin(d*x+c)-8490*C*cos(d*x+c)^4*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-5440*A*sin(d*x+c)*cos(d*x+c)^3*(-2/(cos(d*
x+c)+1))^(1/2)-6520*B*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)-3680*B*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)-960*B*cos(d*x+c)*sin(d*x+c)*(-
2/(cos(d*x+c)+1))^(1/2)-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/cos(d*x+c)^(9/2)/(-2/(cos(d*x+c)+1))^(1/2)

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Maxima [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+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 1.67689, size = 1561, normalized size = 5.19 \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+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

[1/7680*(4*(15*(400*A + 326*B + 283*C)*a^2*cos(d*x + c)^4 + 10*(272*A + 326*B + 283*C)*a^2*cos(d*x + c)^3 + 8*
(80*A + 230*B + 283*C)*a^2*cos(d*x + c)^2 + 48*(10*B + 29*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 + 326*B + 283*C)*a^2*cos(d*x + c)^6 + (400*A
 + 326*B + 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))*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*(400*A + 326*B + 283*C)*a^2*cos(d*x + c)^4
 + 10*(272*A + 326*B + 283*C)*a^2*cos(d*x + c)^3 + 8*(80*A + 230*B + 283*C)*a^2*cos(d*x + c)^2 + 48*(10*B + 29
*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 + 326*B + 283*C)*a^2*cos(d*x + c)^6 + (400*A + 326*B + 283*C)*a^2*cos(d*x + c)^5)*sqrt(-a)*arctan(2*s
qrt(-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+B*sec(d*x+c)+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} + B \sec \left (d x + c\right ) + 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+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^(5/2)/cos(d*x + c)^(3/2), x)