3.361 \(\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx\)

Optimal. Leaf size=205 \[ \frac{2 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d} \]

[Out]

(2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 +
21*a^2*b*B + 5*b^3*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d
*x]]*Sin[c + d*x])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sqrt[Cos[
c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 0.477762, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2990, 3033, 3023, 2748, 2641, 2639} \[ \frac{2 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 b \left (18 a^2 B+21 a A b+5 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]

[Out]

(2*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^3*A + 21*a*A*b^2 +
21*a^2*b*B + 5*b^3*B)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(21*a*A*b + 18*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d
*x]]*Sin[c + d*x])/(21*d) + (2*b^2*(7*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Sqrt[Cos[
c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)

Rule 2990

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 - 2)*(c + d*Sin[e + f*x
])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*(m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c -
b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x]^2, x], x], x] /; F
reeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m,
1] &&  !(IGtQ[n, 1] && ( !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 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx &=\frac{2 b B \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{2}{7} \int \frac{(a+b \cos (c+d x)) \left (\frac{1}{2} a (7 a A+b B)+\frac{1}{2} \left (5 b^2 B+7 a (2 A b+a B)\right ) \cos (c+d x)+\frac{1}{2} b (7 A b+11 a B) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b^2 (7 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{4}{35} \int \frac{\frac{5}{4} a^2 (7 a A+b B)+\frac{7}{4} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \cos (c+d x)+\frac{5}{4} b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b^2 (7 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{8}{105} \int \frac{\frac{5}{8} \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right )+\frac{21}{8} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b^2 (7 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}+\frac{1}{5} \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{21} \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (15 a^2 A b+3 A b^3+5 a^3 B+9 a b^2 B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (21 a^3 A+21 a A b^2+21 a^2 b B+5 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 b \left (21 a A b+18 a^2 B+5 b^2 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b^2 (7 A b+11 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 1.26794, size = 158, normalized size = 0.77 \[ \frac{10 \left (21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+42 \left (15 a^2 A b+5 a^3 B+9 a b^2 B+3 A b^3\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+b \sin (c+d x) \sqrt{\cos (c+d x)} \left (5 \left (42 a^2 B+42 a A b+3 b^2 B \cos (2 (c+d x))+13 b^2 B\right )+42 b (3 a B+A b) \cos (c+d x)\right )}{105 d} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]

[Out]

(42*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE[(c + d*x)/2, 2] + 10*(21*a^3*A + 21*a*A*b^2 + 21*a^
2*b*B + 5*b^3*B)*EllipticF[(c + d*x)/2, 2] + b*Sqrt[Cos[c + d*x]]*(42*b*(A*b + 3*a*B)*Cos[c + d*x] + 5*(42*a*A
*b + 42*a^2*B + 13*b^2*B + 3*b^2*B*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)

________________________________________________________________________________________

Maple [B]  time = 3.418, size = 664, normalized size = 3.2 \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+B*cos(d*x+c))/cos(d*x+c)^(1/2),x)

[Out]

-2/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*B*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^8+(-168*A*b^3-504*B*a*b^2-360*B*b^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(420*A*a*b^2+168*A*b^3+420*B*a^
2*b+504*B*a*b^2+280*B*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-210*A*a*b^2-42*A*b^3-210*B*a^2*b-126*B*a*
b^2-80*B*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-315*A*(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^2*b-63*A*(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))*b^3+105*A*a^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))+105*A*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-105*B*(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*B*(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*B*(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))+25*B*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)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sqrt(cos(d*x + c)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B b^{3} \cos \left (d x + c\right )^{4} + A a^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (B a^{2} b + A a b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

integral((B*b^3*cos(d*x + c)^4 + A*a^3 + (3*B*a*b^2 + A*b^3)*cos(d*x + c)^3 + 3*(B*a^2*b + A*a*b^2)*cos(d*x +
c)^2 + (B*a^3 + 3*A*a^2*b)*cos(d*x + c))/sqrt(cos(d*x + c)), x)

________________________________________________________________________________________

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+B*cos(d*x+c))/cos(d*x+c)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c))/cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/sqrt(cos(d*x + c)), x)