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

Optimal. Leaf size=214 \[ \frac{a (48 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a (48 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{64 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}} \]

[Out]

(Sqrt[a]*(48*A + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(48*A + 35*C)*Sec
[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*
x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) +
(C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)

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Rubi [A]  time = 0.45962, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4089, 4016, 3803, 3801, 215} \[ \frac{a (48 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a (48 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{64 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a]*(48*A + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(48*A + 35*C)*Sec
[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*
x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) +
(C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)

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 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 \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{1}{2} a (8 A+5 C)+\frac{1}{2} a C \sec (c+d x)\right ) \, dx}{4 a}\\ &=\frac{a C \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{C \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{48} (48 A+35 C) \int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a (48 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}+\frac{a C \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{C \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{64} (48 A+35 C) \int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a (48 A+35 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{a (48 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}+\frac{a C \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{C \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}+\frac{1}{128} (48 A+35 C) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a (48 A+35 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{a (48 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}+\frac{a C \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{C \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}-\frac{(48 A+35 C) \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 )}{64 d}\\ &=\frac{\sqrt{a} (48 A+35 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{64 d}+\frac{a (48 A+35 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{a (48 A+35 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}+\frac{a C \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{C \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 2.1597, size = 238, normalized size = 1.11 \[ \frac{\cos ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (A+C \sec ^2(c+d x)\right ) \left (\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{9}{2}}(c+d x) ((432 A+539 C) \cos (c+d x)+4 (48 A+35 C) \cos (2 (c+d x))+144 A \cos (3 (c+d x))+192 A+105 C \cos (3 (c+d x))+332 C)-\frac{12 (48 A+35 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left (\log (\sec (c+d x)+1)-\log \left (\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right )\right )}{\sqrt{\sec (c+d x)+1}}\right )}{384 d (A \cos (2 (c+d x))+A+2 C)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]

[Out]

(Cos[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(A + C*Sec[c + d*x]^2)*((192*A + 332*C + (432*A + 539*C)*Cos[c + d*
x] + 4*(48*A + 35*C)*Cos[2*(c + d*x)] + 144*A*Cos[3*(c + d*x)] + 105*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(9/2)*Ta
n[(c + d*x)/2] - (12*(48*A + 35*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]
^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2])/Sqrt[1 + Sec[c + d*x]]))/(384*d*(
A + 2*C + A*Cos[2*(c + d*x)]))

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Maple [B]  time = 0.387, size = 449, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^(5/2)*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x)

[Out]

1/768/d*(-144*A*cos(d*x+c)^4*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)))+1
44*A*cos(d*x+c)^4*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)-105*C*cos(d*
x+c)^4*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)))+105*C*cos(d*x+c)^4*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)))+288*A*sin(d*x+c)*cos(d*x+c)^3*(-2/
(cos(d*x+c)+1))^(1/2)+210*C*sin(d*x+c)*cos(d*x+c)^3*(-2/(cos(d*x+c)+1))^(1/2)+192*A*cos(d*x+c)^2*sin(d*x+c)*(-
2/(cos(d*x+c)+1))^(1/2)+140*C*sin(d*x+c)*cos(d*x+c)^2*(-2/(cos(d*x+c)+1))^(1/2)+112*C*sin(d*x+c)*cos(d*x+c)*(-
2/(cos(d*x+c)+1))^(1/2)+96*C*(-2/(cos(d*x+c)+1))^(1/2)*sin(d*x+c))*(1/cos(d*x+c))^(5/2)*(a*(cos(d*x+c)+1)/cos(
d*x+c))^(1/2)*(-2/(cos(d*x+c)+1))^(1/2)/sin(d*x+c)^2/cos(d*x+c)*(cos(d*x+c)^2-1)

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Maxima [B]  time = 3.37684, size = 5963, normalized size = 27.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(5/2)*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

-1/768*(48*(12*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(d*x + c), cos(d*x +
c))) + 4*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(5/2*arctan2(sin(d*x + c), cos(d*x + c)))
- 4*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin(d*x + c), cos(d*x + c))) - 12
*(sqrt(2)*sin(4*d*x + 4*c) + 2*sqrt(2)*sin(2*d*x + 2*c))*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) - 3*(2*(
2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*
sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d
*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(d
*x + c), cos(d*x + c))) + 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) + 3*(2*(2*cos(2*d*x + 2*
c) + 1)*cos(4*d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)
*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x
+ c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x
+ c))) - 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - 3*(2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d*
x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c
) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*si
n(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2*sqrt
(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) + 3*(2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + cos(4
*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x
+ 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(si
n(d*x + c), cos(d*x + c)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) - 2*sqrt(2)*sin(1/2*arct
an2(sin(d*x + c), cos(d*x + c))) + 2) - 12*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*s
in(7/2*arctan2(sin(d*x + c), cos(d*x + c))) - 4*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) + sqrt(
2))*sin(5/2*arctan2(sin(d*x + c), cos(d*x + c))) + 4*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2*c) +
sqrt(2))*sin(3/2*arctan2(sin(d*x + c), cos(d*x + c))) + 12*(sqrt(2)*cos(4*d*x + 4*c) + 2*sqrt(2)*cos(2*d*x + 2
*c) + sqrt(2))*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))))*A*sqrt(a)/(2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d*x +
4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) +
4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1) + (420*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x + 6*c)
+ 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x + 2*c))*cos(15/2*arctan2(sin(d*x + c), cos(d*x + c))) + 140
*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x + 6*c) + 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x + 2
*c))*cos(13/2*arctan2(sin(d*x + c), cos(d*x + c))) + 1596*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x + 6*
c) + 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x + 2*c))*cos(11/2*arctan2(sin(d*x + c), cos(d*x + c))) +
500*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x + 6*c) + 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x
+ 2*c))*cos(9/2*arctan2(sin(d*x + c), cos(d*x + c))) - 500*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x + 6
*c) + 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x + 2*c))*cos(7/2*arctan2(sin(d*x + c), cos(d*x + c))) -
1596*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x + 6*c) + 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x
+ 2*c))*cos(5/2*arctan2(sin(d*x + c), cos(d*x + c))) - 140*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x +
6*c) + 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin(d*x + c), cos(d*x + c))) -
420*(sqrt(2)*sin(8*d*x + 8*c) + 4*sqrt(2)*sin(6*d*x + 6*c) + 6*sqrt(2)*sin(4*d*x + 4*c) + 4*sqrt(2)*sin(2*d*x
+ 2*c))*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) - 105*(2*(4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*co
s(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)*co
s(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
+ 16*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 + 3
6*sin(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)*
log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sq
rt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) +
2) + 105*(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 + 16*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*sin(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)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^
2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))
- 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - 105*(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 + 16*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*sin(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)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^
2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x +
c))) + 2) + 105*(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 + 16*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*sin(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)*log(2*cos(1/2*arctan2(sin(d*x + c), cos(d*x
+ c)))^2 + 2*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c)))^2 - 2*sqrt(2)*cos(1/2*arctan2(sin(d*x + c), cos(d*x
+ c))) - 2*sqrt(2)*sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))) + 2) - 420*(sqrt(2)*cos(8*d*x + 8*c) + 4*sqrt
(2)*cos(6*d*x + 6*c) + 6*sqrt(2)*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(15/2*arctan2(sin
(d*x + c), cos(d*x + c))) - 140*(sqrt(2)*cos(8*d*x + 8*c) + 4*sqrt(2)*cos(6*d*x + 6*c) + 6*sqrt(2)*cos(4*d*x +
4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(13/2*arctan2(sin(d*x + c), cos(d*x + c))) - 1596*(sqrt(2)*co
s(8*d*x + 8*c) + 4*sqrt(2)*cos(6*d*x + 6*c) + 6*sqrt(2)*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2
))*sin(11/2*arctan2(sin(d*x + c), cos(d*x + c))) - 500*(sqrt(2)*cos(8*d*x + 8*c) + 4*sqrt(2)*cos(6*d*x + 6*c)
+ 6*sqrt(2)*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(9/2*arctan2(sin(d*x + c), cos(d*x + c
))) + 500*(sqrt(2)*cos(8*d*x + 8*c) + 4*sqrt(2)*cos(6*d*x + 6*c) + 6*sqrt(2)*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(
2*d*x + 2*c) + sqrt(2))*sin(7/2*arctan2(sin(d*x + c), cos(d*x + c))) + 1596*(sqrt(2)*cos(8*d*x + 8*c) + 4*sqrt
(2)*cos(6*d*x + 6*c) + 6*sqrt(2)*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(5/2*arctan2(sin(
d*x + c), cos(d*x + c))) + 140*(sqrt(2)*cos(8*d*x + 8*c) + 4*sqrt(2)*cos(6*d*x + 6*c) + 6*sqrt(2)*cos(4*d*x +
4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*sin(3/2*arctan2(sin(d*x + c), cos(d*x + c))) + 420*(sqrt(2)*cos(8
*d*x + 8*c) + 4*sqrt(2)*cos(6*d*x + 6*c) + 6*sqrt(2)*cos(4*d*x + 4*c) + 4*sqrt(2)*cos(2*d*x + 2*c) + sqrt(2))*
sin(1/2*arctan2(sin(d*x + c), cos(d*x + c))))*C*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 + 16
*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*si
n(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))/d

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Fricas [A]  time = 1.02463, size = 1238, normalized size = 5.79 \begin{align*} \left [\frac{3 \,{\left ({\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - \frac{4 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac{4 \,{\left (3 \,{\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{2} + 56 \, C \cos \left (d x + c\right ) + 48 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{768 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}, \frac{3 \,{\left ({\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{3}\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 ) + \frac{2 \,{\left (3 \,{\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{3} + 2 \,{\left (48 \, A + 35 \, C\right )} \cos \left (d x + c\right )^{2} + 56 \, C \cos \left (d x + c\right ) + 48 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{384 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(5/2)*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[1/768*(3*((48*A + 35*C)*cos(d*x + c)^4 + (48*A + 35*C)*cos(d*x + c)^3)*sqrt(a)*log((a*cos(d*x + c)^3 - 7*a*co
s(d*x + c)^2 - 4*(cos(d*x + c)^2 - 2*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c
)/sqrt(cos(d*x + c)) + 8*a)/(cos(d*x + c)^3 + cos(d*x + c)^2)) + 4*(3*(48*A + 35*C)*cos(d*x + c)^3 + 2*(48*A +
35*C)*cos(d*x + c)^2 + 56*C*cos(d*x + c) + 48*C)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/sqrt(co
s(d*x + c)))/(d*cos(d*x + c)^4 + d*cos(d*x + c)^3), 1/384*(3*((48*A + 35*C)*cos(d*x + c)^4 + (48*A + 35*C)*cos
(d*x + c)^3)*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)) + 2*(3*(48*A + 35*C)*cos(d*x + c)^3 + 2*(48*A + 35*C)*cos(d*x + c
)^2 + 56*C*cos(d*x + c) + 48*C)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*co
s(d*x + c)^4 + d*cos(d*x + c)^3)]

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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(sec(d*x+c)**(5/2)*(A+C*sec(d*x+c)**2)*(a+a*sec(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt{a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(5/2)*(A+C*sec(d*x+c)^2)*(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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