3.691 \(\int \frac{(a+b \cos (c+d x))^3 (A+C \cos ^2(c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\)
Optimal. Leaf size=244 \[ \frac{2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}-\frac{2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}-\frac{2 a b^2 (35 A-11 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}} \]
[Out]
(-2*a*(5*a^2*(A - C) - 3*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(21*a^2*(3*A + C) + b^2*(7*A
+ 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) - (2*b*(6*a^2*(7*A - 3*C) - b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin
[c + d*x])/(21*d) - (2*a*b^2*(35*A - 11*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(7*A - C)*Sqrt[Cos[c
+ d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c
+ d*x]])
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Rubi [A] time = 0.737913, antiderivative size = 244, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used =
{3048, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}-\frac{2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}-\frac{2 a b^2 (35 A-11 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]
[Out]
(-2*a*(5*a^2*(A - C) - 3*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*b*(21*a^2*(3*A + C) + b^2*(7*A
+ 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) - (2*b*(6*a^2*(7*A - 3*C) - b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin
[c + d*x])/(21*d) - (2*a*b^2*(35*A - 11*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) - (2*b*(7*A - C)*Sqrt[Cos[c
+ d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c
+ d*x]])
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 3049
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/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Rule 3033
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[(C*d*Cos[e + f*x]*Sin[e + f*x]*(a + b
*Sin[e + f*x])^(m + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[(a + b*Sin[e + f*x])^m*Simp[a*C*d + A*b*c*
(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e +
f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] &&
!LtQ[m, -1]
Rule 3023
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((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 + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
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 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]
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]
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+2 \int \frac{(a+b \cos (c+d x))^2 \left (3 A b-\frac{1}{2} a (A-C) \cos (c+d x)-\frac{1}{2} b (7 A-C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b (7 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{4}{7} \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{4} a b (35 A+C)-\frac{1}{4} \left (7 a^2 (A-C)-b^2 (7 A+5 C)\right ) \cos (c+d x)-\frac{1}{4} a b (35 A-11 C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 a b^2 (35 A-11 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac{2 b (7 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{8}{35} \int \frac{\frac{5}{8} a^2 b (35 A+C)-\frac{7}{8} a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \cos (c+d x)-\frac{5}{8} b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}-\frac{2 a b^2 (35 A-11 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac{2 b (7 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{16}{105} \int \frac{\frac{5}{16} b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right )-\frac{21}{16} a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}-\frac{2 a b^2 (35 A-11 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac{2 b (7 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{1}{5} \left (a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right )\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 a \left (5 a^2 (A-C)-3 b^2 (5 A+3 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}-\frac{2 b \left (6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}-\frac{2 a b^2 (35 A-11 C) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}-\frac{2 b (7 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.77046, size = 172, normalized size = 0.7 \[ \frac{20 b \left (21 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )-84 \left (5 a^3 (A-C)-3 a b^2 (5 A+3 C)\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{\sin (c+d x) \left (5 b \left (84 a^2 C+28 A b^2+29 b^2 C\right ) \cos (c+d x)+3 \left (140 a^3 A+42 a b^2 C \cos (2 (c+d x))+42 a b^2 C+5 b^3 C \cos (3 (c+d x))\right )\right )}{\sqrt{\cos (c+d x)}}}{210 d} \]
Antiderivative was successfully verified.
[In]
Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]
[Out]
(-84*(5*a^3*(A - C) - 3*a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 20*b*(21*a^2*(3*A + C) + b^2*(7*A + 5*C
))*EllipticF[(c + d*x)/2, 2] + ((5*b*(28*A*b^2 + 84*a^2*C + 29*b^2*C)*Cos[c + d*x] + 3*(140*a^3*A + 42*a*b^2*C
+ 42*a*b^2*C*Cos[2*(c + d*x)] + 5*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(210*d)
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Maple [B] time = 0.624, size = 943, normalized size = 3.9 \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)^(3/2),x)
[Out]
-2/105*(240*C*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8
-72*C*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b^2*(7*a+5*b)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*
c)+28*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*b*(5*A*b^2+15*C*a^2+18*C*a*b+10*C*b^2)*sin(1/2*d*x+
1/2*c)^4*cos(1/2*d*x+1/2*c)-2*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(105*A*a^3+35*A*b^3+105*C*a
^2*b+63*C*a*b^2+40*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+105*A*(-2*sin(1/2*d*x+1/2*c)^4+sin(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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1
/2))*a^3-315*A*(-2*sin(1/2*d*x+1/2*c)^4+sin(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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+315*A*a^2*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si
n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^
2)^(1/2)+35*A*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2
^(1/2))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-105*C*(-2*sin(1/2*d*x+1/2*c)^4+sin(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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*
a^3-189*C*(-2*sin(1/2*d*x+1/2*c)^4+sin(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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+105*a^2*b*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2
*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1
/2)+25*C*b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2
))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/
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{3}{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)^(3/2),x, algorithm="maxima")
[Out]
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/cos(d*x + c)^(3/2), 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{3}{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)^(3/2),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)^(3/2), 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)**(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 (b \cos \left (d x + c\right ) + a\right )}^{3}}{\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+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algorithm="giac")
[Out]
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/cos(d*x + c)^(3/2), x)