3.1230 \(\int \frac{(a+a \cos (c+d x))^{5/2} (A+C \cos ^2(c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\)

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

[Out]

(a^(5/2)*(1304*A + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec
[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) +
(a^2*(24*A + 23*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d*Sec[c + d*x]^(5/2)) + (a*C*(a + a*Cos[c + d*x
])^(3/2)*Sin[c + d*x])/(12*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*
x]^(5/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1
304*A + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])

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Rubi [A]  time = 1.04587, 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 = {4221, 3046, 2976, 2981, 2770, 2774, 216} \[ \frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^{5/2} (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac{5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(5/2)*(1304*A + 1015*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec
[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2)) +
(a^2*(24*A + 23*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(96*d*Sec[c + d*x]^(5/2)) + (a*C*(a + a*Cos[c + d*x
])^(3/2)*Sin[c + d*x])/(12*d*Sec[c + d*x]^(5/2)) + (C*(a + a*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Sec[c + d*
x]^(5/2)) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)) + (a^3*(1
304*A + 1015*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])

Rule 4221

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*
sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n
+ 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp
[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b
, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-
1)] && NeQ[m + n + 2, 0]

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rule 2770

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp
[(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] + Dist[(2*n*(b*c + a*d)
)/(b*(2*n + 1)), Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && IntegerQ[2*n]

Rule 2774

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2/f, Su
bst[Int[1/Sqrt[1 - x^2/a], x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, d, e, f}, x]
&& EqQ[a^2 - b^2, 0] && EqQ[d, a/b]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{(a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (12 A+5 C)+\frac{5}{2} a C \cos (c+d x)\right ) \, dx}{6 a}\\ &=\frac{a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left (\frac{15}{4} a^2 (8 A+5 C)+\frac{5}{4} a^2 (24 A+23 C) \cos (c+d x)\right ) \, dx}{30 a}\\ &=\frac{a^2 (24 A+23 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left (\frac{5}{8} a^3 (312 A+235 C)+\frac{15}{8} a^3 (136 A+109 C) \cos (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (24 A+23 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{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 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (24 A+23 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{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 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (24 A+23 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (a^2 (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx}{1024}\\ &=\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (24 A+23 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (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 \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{512 d}\\ &=\frac{a^{5/2} (1304 A+1015 C) \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (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 \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^2 (24 A+23 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 1.25709, size = 192, normalized size = 0.58 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left (3 \sqrt{2} (1304 A+1015 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}+\left (\sin \left (\frac{3}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) ((2896 A+3234 C) \cos (c+d x)+4 (184 A+315 C) \cos (2 (c+d x))+96 A \cos (3 (c+d x))+4648 A+428 C \cos (3 (c+d x))+112 C \cos (4 (c+d x))+16 C \cos (5 (c+d x))+4193 C)\right )}{3072 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(1304*A + 1015*C)*ArcSin[Sqrt[2
]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (4648*A + 4193*C + (2896*A + 3234*C)*Cos[c + d*x] + 4*(184*A + 315*C)
*Cos[2*(c + d*x)] + 96*A*Cos[3*(c + d*x)] + 428*C*Cos[3*(c + d*x)] + 112*C*Cos[4*(c + d*x)] + 16*C*Cos[5*(c +
d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3072*d)

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Maple [A]  time = 0.203, size = 491, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1536/d*a^2*(-1+cos(d*x+c))^3*(256*C*sin(d*x+c)*cos(d*x+c)^5*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+896*C*sin(d*x
+c)*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+384*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^3*sin(d*
x+c)+1392*C*sin(d*x+c)*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1472*A*cos(d*x+c)^2*sin(d*x+c)*(cos(d*x+
c)/(1+cos(d*x+c)))^(1/2)+1624*C*sin(d*x+c)*cos(d*x+c)^2*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+2608*A*sin(d*x+c)*co
s(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+2030*C*cos(d*x+c)*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+3912
*A*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+3045*C*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+3912*A*arc
tan(sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)/cos(d*x+c))+3045*C*arctan(sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c
)))^(1/2)/cos(d*x+c)))*cos(d*x+c)*(a*(1+cos(d*x+c)))^(1/2)/(cos(d*x+c)/(1+cos(d*x+c)))^(5/2)/(1/cos(d*x+c))^(3
/2)/sin(d*x+c)^6

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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*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 2.62512, size = 603, normalized size = 1.82 \begin{align*} -\frac{3 \,{\left ({\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (1304 \, A + 1015 \, C\right )} a^{2}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right ) - \frac{{\left (256 \, C a^{2} \cos \left (d x + c\right )^{6} + 896 \, C a^{2} \cos \left (d x + c\right )^{5} + 48 \,{\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \,{\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \,{\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{1536 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

-1/1536*(3*((1304*A + 1015*C)*a^2*cos(d*x + c) + (1304*A + 1015*C)*a^2)*sqrt(a)*arctan(sqrt(a*cos(d*x + c) + a
)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - (256*C*a^2*cos(d*x + c)^6 + 896*C*a^2*cos(d*x + c)^5 + 48*(8*A
+ 29*C)*a^2*cos(d*x + c)^4 + 8*(184*A + 203*C)*a^2*cos(d*x + c)^3 + 2*(1304*A + 1015*C)*a^2*cos(d*x + c)^2 + 3
*(1304*A + 1015*C)*a^2*cos(d*x + c))*sqrt(a*cos(d*x + c) + a)*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)
+ d)

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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*cos(d*x+c))**(5/2)*(A+C*cos(d*x+c)**2)/sec(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 \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sec \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*cos(d*x+c))^(5/2)*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algorithm="giac")

[Out]

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