3.1335 \(\int \frac {(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)+C \cos ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\)

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

[Out]

1/8*a*(2*B+C)*(a+a*cos(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+1/5*C*(a+a*cos(d*x+c))^(5/2)*sin(d*x+c)/d/s
ec(d*x+c)^(3/2)+1/960*a^3*(1040*A+950*B+787*C)*sin(d*x+c)/d/sec(d*x+c)^(3/2)/(a+a*cos(d*x+c))^(1/2)+1/240*a^2*
(80*A+110*B+79*C)*sin(d*x+c)*(a+a*cos(d*x+c))^(1/2)/d/sec(d*x+c)^(3/2)+1/128*a^3*(400*A+326*B+283*C)*sin(d*x+c
)/d/(a+a*cos(d*x+c))^(1/2)/sec(d*x+c)^(1/2)+1/128*a^(5/2)*(400*A+326*B+283*C)*arcsin(sin(d*x+c)*a^(1/2)/(a+a*c
os(d*x+c))^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d

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Rubi [A]  time = 1.04, antiderivative size = 301, normalized size of antiderivative = 1.00, 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 = {4221, 3045, 2976, 2981, 2770, 2774, 216} \[ \frac {a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (80 A+110 B+79 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{240 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^{5/2} (400 A+326 B+283 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{128 d}+\frac {a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {a (2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \sec ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^(5/2)*(400*A + 326*B + 283*C)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sqrt[a + a*Cos[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sq
rt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 787*C)*Sin[c + d*x])/(960*d*Sqrt[a + a*Cos[c + d*x]]*Sec[c
+ d*x]^(3/2)) + (a^2*(80*A + 110*B + 79*C)*Sqrt[a + a*Cos[c + d*x]]*Sin[c + d*x])/(240*d*Sec[c + d*x]^(3/2)) +
(a*(2*B + C)*(a + a*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Sec[c + d*x]^(3/2)) + (C*(a + a*Cos[c + d*x])^(5/2
)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (a^3*(400*A + 326*B + 283*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Cos[c
+ d*x]]*Sqrt[Sec[c + d*x]])

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]

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 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 3045

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*
sin[(e_.) + (f_.)*(x_)] + (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)) + b*B*
d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, 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 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]

Rubi steps

\begin {align*} \int \frac {(a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \left (\frac {1}{2} a (10 A+3 C)+\frac {5}{2} a (2 B+C) \cos (c+d x)\right ) \, dx}{5 a}\\ &=\frac {a (2 B+C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+a \cos (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) \cos (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a^2 (80 A+110 B+79 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{240 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a (2 B+C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {a+a \cos (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) \cos (c+d x)\right ) \, dx}{60 a}\\ &=\frac {a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^2 (80 A+110 B+79 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{240 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a (2 B+C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{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 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)} \, dx\\ &=\frac {a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^2 (80 A+110 B+79 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{240 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a (2 B+C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (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 \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^2 (80 A+110 B+79 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{240 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a (2 B+C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (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 \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{128 d}\\ &=\frac {a^{5/2} (400 A+326 B+283 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)}}{128 d}+\frac {a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sqrt {a+a \cos (c+d x)} \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^2 (80 A+110 B+79 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{240 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a (2 B+C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {C (a+a \cos (c+d x))^{5/2} \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 1.34, size = 193, normalized size = 0.64 \[ \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \sqrt {a (\cos (c+d x)+1)} \left (15 \sqrt {2} (400 A+326 B+283 C) \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cos (c+d x)}+\left (\sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) ((2720 A+3620 B+3874 C) \cos (c+d x)+4 (80 A+230 B+331 C) \cos (2 (c+d x))+6320 A+120 B \cos (3 (c+d x))+5810 B+348 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+5521 C)\right )}{3840 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(15*Sqrt[2]*(400*A + 326*B + 283*C)*ArcSin
[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (6320*A + 5810*B + 5521*C + (2720*A + 3620*B + 3874*C)*Cos[c +
d*x] + 4*(80*A + 230*B + 331*C)*Cos[2*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 348*C*Cos[3*(c + d*x)] + 48*C*Cos
[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)

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fricas [A]  time = 1.02, size = 218, normalized size = 0.72 \[ -\frac {15 \, {\left ({\left (400 \, A + 326 \, B + 283 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (400 \, A + 326 \, B + 283 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {{\left (384 \, C a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (10 \, B + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (80 \, A + 230 \, B + 283 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (272 \, A + 326 \, B + 283 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (400 \, A + 326 \, B + 283 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1920*(15*((400*A + 326*B + 283*C)*a^2*cos(d*x + c) + (400*A + 326*B + 283*C)*a^2)*sqrt(a)*arctan(sqrt(a*cos
(d*x + c) + a)*sqrt(cos(d*x + c))/(sqrt(a)*sin(d*x + c))) - (384*C*a^2*cos(d*x + c)^5 + 48*(10*B + 29*C)*a^2*c
os(d*x + c)^4 + 8*(80*A + 230*B + 283*C)*a^2*cos(d*x + c)^3 + 10*(272*A + 326*B + 283*C)*a^2*cos(d*x + c)^2 +
15*(400*A + 326*B + 283*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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.58, size = 591, normalized size = 1.96 \[ \frac {\left (-1+\cos \left (d x +c \right )\right )^{2} \left (384 C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+480 B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1392 C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+640 A \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+1840 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2264 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2720 A \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3260 B \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2830 C \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+6000 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+4890 B \sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+4245 C \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+6000 A \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )+4890 B \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )+4245 C \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )\right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, a^{2}}{1920 d \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920/d*(-1+cos(d*x+c))^2*(384*C*sin(d*x+c)*cos(d*x+c)^4*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+480*B*sin(d*x+c)*c
os(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+1392*C*sin(d*x+c)*cos(d*x+c)^3*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)
+640*A*cos(d*x+c)^2*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+1840*B*sin(d*x+c)*cos(d*x+c)^2*(cos(d*x+c)/(1
+cos(d*x+c)))^(1/2)+2264*C*sin(d*x+c)*cos(d*x+c)^2*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+2720*A*sin(d*x+c)*cos(d*x
+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+3260*B*sin(d*x+c)*cos(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+2830*C*(c
os(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)*cos(d*x+c)+6000*A*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+4890
*B*sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)+4245*C*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)+6000*A*arc
tan(sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)/cos(d*x+c))+4890*B*arctan(sin(d*x+c)*(cos(d*x+c)/(1+cos(d*x+c
)))^(1/2)/cos(d*x+c))+4245*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)))^(3/2)/(1/cos(d*x+c))^(1/2)/sin(d*x+c)^4*a^2

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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