3.695 \(\int \frac{(a+b \cos (c+d x))^3 (A+C \cos ^2(c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx\)
Optimal. Leaf size=293 \[ \frac{2 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)} \]
[Out]
(-2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*b*(7*b^2*(A + 3*C) + 3*a^2*
(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Cos[
c + d*x]^(5/2)) + (2*b*(8*A*b^2 + 9*a^2*(5*A + 7*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*a*(9*b^2*(3*
A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*
x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))
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Rubi [A] time = 0.788261, antiderivative size = 293, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used =
{3048, 3047, 3031, 3021, 2748, 2636, 2639, 2641} \[ \frac{2 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]
[Out]
(-2*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*b*(7*b^2*(A + 3*C) + 3*a^2*
(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(24*A*b^2 + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Cos[
c + d*x]^(5/2)) + (2*b*(8*A*b^2 + 9*a^2*(5*A + 7*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*a*(9*b^2*(3*
A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (4*A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*
x])/(21*d*Cos[c + d*x]^(7/2)) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))
Rule 3048
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m
- 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(
m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] &
& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Rule 3047
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Rule 3031
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e
_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((b*c - a*d)*(A*b^2 - a*b*B + a^2*C)*
Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b^2*f*(m + 1)*(a^2 - b^2)), x] - Dist[1/(b^2*(m + 1)*(a^2 - b^2)),
Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b
^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1))))*Sin[e + f*x] - b*C*d*(m +
1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && Ne
Q[a^2 - b^2, 0] && LtQ[m, -1]
Rule 3021
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m +
1)*(a^2 - b^2)), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B + a*
C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e,
f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Rule 2748
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Rule 2636
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]
Rule 2639
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]
Rule 2641
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{11}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2}{9} \int \frac{(a+b \cos (c+d x))^2 \left (3 A b+\frac{1}{2} a (7 A+9 C) \cos (c+d x)+\frac{1}{2} b (A+9 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{4 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4}{63} \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} \left (24 A b^2+7 a^2 (7 A+9 C)\right )+\frac{1}{2} a b (43 A+63 C) \cos (c+d x)+\frac{1}{4} b^2 (13 A+63 C) \cos ^2(c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (24 A b^2+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{8}{315} \int \frac{-\frac{15}{8} b \left (8 A b^2+9 a^2 (5 A+7 C)\right )-\frac{21}{8} a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \cos (c+d x)-\frac{5}{8} b^3 (13 A+63 C) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (24 A b^2+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (8 A b^2+9 a^2 (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{16}{945} \int \frac{-\frac{63}{16} a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )-\frac{45}{16} b \left (7 b^2 (A+3 C)+3 a^2 (5 A+7 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a \left (24 A b^2+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (8 A b^2+9 a^2 (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{1}{21} \left (b \left (7 b^2 (A+3 C)+3 a^2 (5 A+7 C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{15} \left (a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 b \left (7 b^2 (A+3 C)+3 a^2 (5 A+7 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a \left (24 A b^2+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (8 A b^2+9 a^2 (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{4 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{1}{15} \left (a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right )\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{2 b \left (7 b^2 (A+3 C)+3 a^2 (5 A+7 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 a \left (24 A b^2+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left (8 A b^2+9 a^2 (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{4 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac{9}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 5.2872, size = 250, normalized size = 0.85 \[ \frac{10 \left (3 a^2 b (5 A+7 C)+7 b^3 (A+3 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-14 \left (a^3 (7 A+9 C)+9 a b^2 (3 A+5 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{14 a \left (a^2 (7 A+9 C)+27 A b^2\right ) \sin (c+d x)}{3 \cos ^{\frac{5}{2}}(c+d x)}+\frac{10 b \left (3 a^2 (5 A+7 C)+7 A b^2\right ) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{14 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{90 a^2 A b \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{70 a^3 A \sin (c+d x)}{3 \cos ^{\frac{9}{2}}(c+d x)}}{105 d} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]
[Out]
(-14*(9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(7*b^3*(A + 3*C) + 3*a^2*b*(5*A +
7*C))*EllipticF[(c + d*x)/2, 2] + (70*a^3*A*Sin[c + d*x])/(3*Cos[c + d*x]^(9/2)) + (90*a^2*A*b*Sin[c + d*x])/C
os[c + d*x]^(7/2) + (14*a*(27*A*b^2 + a^2*(7*A + 9*C))*Sin[c + d*x])/(3*Cos[c + d*x]^(5/2)) + (10*b*(7*A*b^2 +
3*a^2*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (14*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x
])/Sqrt[Cos[c + d*x]])/(105*d)
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Maple [B] time = 2.273, size = 1270, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x)
[Out]
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*C*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d
*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)
)-2/5*a*(3*A*b^2+C*a^2)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+
1/2*c)^2*(12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/
2)*sin(1/2*d*x+1/2*c)^4-24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(s
in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*sin(1/2*d*x+1/2*c)^2+24*sin(1/2*d*x+1/2*c)^4*cos(1
/2*d*x+1/2*c)+3*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^
(1/2)-8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*b*(A*b
^2+3*C*a^2)*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/
2*c)^2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*
d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+6*A*a^2*b*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x
+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^4-5/42*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1/2*c)^2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2
*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/
2)))+2*A*a^3*(-1/144*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x
+1/2*c)^2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(-1/2+cos(1/2*d*x+1
/2*c)^2)^3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(
1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+
1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)
^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-Ellipt
icE(cos(1/2*d*x+1/2*c),2^(1/2))))+6*C*a*b^2*(-(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(-
2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+2*(-2*sin(1/2*d*x+1/2
*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x
+1/2*c)^2-1))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algorithm="maxima")
[Out]
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/cos(d*x + c)^(11/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C b^{3} \cos \left (d x + c\right )^{5} + 3 \, C a b^{2} \cos \left (d x + c\right )^{4} + 3 \, A a^{2} b \cos \left (d x + c\right ) + A a^{3} +{\left (3 \, C a^{2} b + A b^{3}\right )} \cos \left (d x + c\right )^{3} +{\left (C a^{3} + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}}{\cos \left (d x + c\right )^{\frac{11}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algorithm="fricas")
[Out]
integral((C*b^3*cos(d*x + c)^5 + 3*C*a*b^2*cos(d*x + c)^4 + 3*A*a^2*b*cos(d*x + c) + A*a^3 + (3*C*a^2*b + A*b^
3)*cos(d*x + c)^3 + (C*a^3 + 3*A*a*b^2)*cos(d*x + c)^2)/cos(d*x + c)^(11/2), x)
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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+b*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(11/2),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 (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac{11}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/cos(d*x + c)^(11/2), x)