3.354 \(\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx\)

Optimal. Leaf size=182 \[ \frac{2 \left (5 a^2 A+6 a b B+3 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (7 a (a B+2 A b)+5 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (7 a (a B+2 A b)+5 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d} \]

[Out]

(2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*EllipticF
[(c + d*x)/2, 2])/(21*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*
A*b + 9*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*Sin[c +
d*x])/(7*d)

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Rubi [A]  time = 0.31587, antiderivative size = 182, 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, 3023, 2748, 2639, 2635, 2641} \[ \frac{2 \left (5 a^2 A+6 a b B+3 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (7 a (a B+2 A b)+5 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (7 a (a B+2 A b)+5 b^2 B\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]

[Out]

(2*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*EllipticF
[(c + d*x)/2, 2])/(21*d) + (2*(5*b^2*B + 7*a*(2*A*b + a*B))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*b*(7*
A*b + 9*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*b*B*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*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 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 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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

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 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx &=\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{2}{7} \int \sqrt{\cos (c+d x)} \left (\frac{1}{2} a (7 a A+3 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+9 a B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{2 b (7 A b+9 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{4}{35} \int \sqrt{\cos (c+d x)} \left (\frac{7}{4} \left (5 a^2 A+3 A b^2+6 a b B\right )+\frac{5}{4} \left (5 b^2 B+7 a (2 A b+a B)\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{2 b (7 A b+9 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{1}{5} \left (5 a^2 A+3 A b^2+6 a b B\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{7} \left (5 b^2 B+7 a (2 A b+a B)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx\\ &=\frac{2 \left (5 a^2 A+3 A b^2+6 a b B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b (7 A b+9 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 b^2 B+7 a (2 A b+a B)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 \left (5 a^2 A+3 A b^2+6 a b B\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 \left (5 b^2 B+7 a (2 A b+a B)\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{2 \left (5 b^2 B+7 a (2 A b+a B)\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{2 b (7 A b+9 a B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 b B \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 1.06837, size = 139, normalized size = 0.76 \[ \frac{10 \left (7 a^2 B+14 a A b+5 b^2 B\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+42 \left (5 a^2 A+6 a b B+3 A b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt{\cos (c+d x)} \left (5 \left (14 a^2 B+28 a A b+3 b^2 B \cos (2 (c+d x))+13 b^2 B\right )+42 b (2 a B+A b) \cos (c+d x)\right )}{105 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]

[Out]

(42*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2] + 10*(14*a*A*b + 7*a^2*B + 5*b^2*B)*EllipticF[(c +
d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(42*b*(A*b + 2*a*B)*Cos[c + d*x] + 5*(28*a*A*b + 14*a^2*B + 13*b^2*B + 3*b^2*
B*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)

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Maple [B]  time = 3.138, size = 548, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^2*(A+B*cos(d*x+c)),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^2*B*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^8+(-168*A*b^2-336*B*a*b-360*B*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(280*A*a*b+168*A*b^2+140*B*a^2+33
6*B*a*b+280*B*b^2)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-140*A*a*b-42*A*b^2-70*B*a^2-84*B*a*b-80*B*b^2)*si
n(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+70*A*a*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)*E
llipticF(cos(1/2*d*x+1/2*c),2^(1/2))-105*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)*Ellip
ticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2-63*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)*Ellipt
icE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2+35*B*a^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)*Ell
ipticF(cos(1/2*d*x+1/2*c),2^(1/2))+25*b^2*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)*Elli
pticF(cos(1/2*d*x+1/2*c),2^(1/2))-126*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)*Elliptic
E(cos(1/2*d*x+1/2*c),2^(1/2))*a*b)/(-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{\left (B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sqrt{\cos \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^2*sqrt(cos(d*x + c)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*b^2*cos(d*x + c)^3 + A*a^2 + (2*B*a*b + A*b^2)*cos(d*x + c)^2 + (B*a^2 + 2*A*a*b)*cos(d*x + c))*sq
rt(cos(d*x + c)), 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(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**2*(A+B*cos(d*x+c)),x)

[Out]

Timed out

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

[Out]

Timed out