3.297 \(\int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=229 \[ -\frac {b (5 A+4 C) \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {b (5 A+4 C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d}+\frac {3 b B \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d}+\frac {b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{5 d} \]

[Out]

1/4*b*B*cos(d*x+c)^(5/2)*sin(d*x+c)*(b*cos(d*x+c))^(1/2)/d+1/5*b*C*cos(d*x+c)^(7/2)*sin(d*x+c)*(b*cos(d*x+c))^
(1/2)/d+3/8*b*B*x*(b*cos(d*x+c))^(1/2)/cos(d*x+c)^(1/2)+1/5*b*(5*A+4*C)*sin(d*x+c)*(b*cos(d*x+c))^(1/2)/d/cos(
d*x+c)^(1/2)-1/15*b*(5*A+4*C)*sin(d*x+c)^3*(b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)+3/8*b*B*sin(d*x+c)*cos(d*x+
c)^(1/2)*(b*cos(d*x+c))^(1/2)/d

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Rubi [A]  time = 0.13, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {17, 3023, 2748, 2633, 2635, 8} \[ -\frac {b (5 A+4 C) \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {b (5 A+4 C) \sin (c+d x) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b B \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{4 d}+\frac {3 b B \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}{8 d}+\frac {b C \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

(3*b*B*x*Sqrt[b*Cos[c + d*x]])/(8*Sqrt[Cos[c + d*x]]) + (b*(5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5*d
*Sqrt[Cos[c + d*x]]) + (3*b*B*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(8*d) + (b*B*Cos[c + d*x]^
(5/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(4*d) + (b*C*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(5
*d) - (b*(5*A + 4*C)*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(15*d*Sqrt[Cos[c + d*x]])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 17

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + 1/2)*b^(n - 1/2)*Sqrt[b*v])/Sqrt[a*v]
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] &&  !IntegerQ[m] && IGtQ[n + 1/2, 0] && IntegerQ[m + n]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 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 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]

Rubi steps

\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) (5 A+4 C+5 B \cos (c+d x)) \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {\left (b B \sqrt {b \cos (c+d x)}\right ) \int \cos ^4(c+d x) \, dx}{\sqrt {\cos (c+d x)}}+\frac {\left (b (5 A+4 C) \sqrt {b \cos (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=\frac {b B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}+\frac {\left (3 b B \sqrt {b \cos (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt {\cos (c+d x)}}-\frac {\left (b (5 A+4 C) \sqrt {b \cos (c+d x)}\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d \sqrt {\cos (c+d x)}}\\ &=\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {\left (3 b B \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{8 \sqrt {\cos (c+d x)}}\\ &=\frac {3 b B x \sqrt {b \cos (c+d x)}}{8 \sqrt {\cos (c+d x)}}+\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d \sqrt {\cos (c+d x)}}+\frac {3 b B \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)} \sin (c+d x)}{8 d}+\frac {b B \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{4 d}+\frac {b C \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)} \sin (c+d x)}{5 d}-\frac {b (5 A+4 C) \sqrt {b \cos (c+d x)} \sin ^3(c+d x)}{15 d \sqrt {\cos (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 109, normalized size = 0.48 \[ \frac {(b \cos (c+d x))^{3/2} (60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x)))}{480 d \cos ^{\frac {3}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]

[Out]

((b*Cos[c + d*x])^(3/2)*(180*B*c + 180*B*d*x + 60*(6*A + 5*C)*Sin[c + d*x] + 120*B*Sin[2*(c + d*x)] + 40*A*Sin
[3*(c + d*x)] + 50*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d*Cos[c + d*x]^(3/
2))

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fricas [A]  time = 0.61, size = 309, normalized size = 1.35 \[ \left [\frac {45 \, B \sqrt {-b} b \cos \left (d x + c\right ) \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) + 2 \, {\left (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, B b \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 45 \, B b \cos \left (d x + c\right ) + 16 \, {\left (5 \, A + 4 \, C\right )} b\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )}, \frac {45 \, B b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right ) + {\left (24 \, C b \cos \left (d x + c\right )^{4} + 30 \, B b \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 4 \, C\right )} b \cos \left (d x + c\right )^{2} + 45 \, B b \cos \left (d x + c\right ) + 16 \, {\left (5 \, A + 4 \, C\right )} b\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{120 \, d \cos \left (d x + c\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/240*(45*B*sqrt(-b)*b*cos(d*x + c)*log(2*b*cos(d*x + c)^2 - 2*sqrt(b*cos(d*x + c))*sqrt(-b)*sqrt(cos(d*x + c
))*sin(d*x + c) - b) + 2*(24*C*b*cos(d*x + c)^4 + 30*B*b*cos(d*x + c)^3 + 8*(5*A + 4*C)*b*cos(d*x + c)^2 + 45*
B*b*cos(d*x + c) + 16*(5*A + 4*C)*b)*sqrt(b*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)), 1
/120*(45*B*b^(3/2)*arctan(sqrt(b*cos(d*x + c))*sin(d*x + c)/(sqrt(b)*cos(d*x + c)^(3/2)))*cos(d*x + c) + (24*C
*b*cos(d*x + c)^4 + 30*B*b*cos(d*x + c)^3 + 8*(5*A + 4*C)*b*cos(d*x + c)^2 + 45*B*b*cos(d*x + c) + 16*(5*A + 4
*C)*b)*sqrt(b*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))]

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

[Out]

Timed out

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maple [A]  time = 0.29, size = 134, normalized size = 0.59 \[ \frac {\left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (24 C \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+30 B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+40 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+32 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+45 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+80 A \sin \left (d x +c \right )+45 B \left (d x +c \right )+64 C \sin \left (d x +c \right )\right )}{120 d \cos \left (d x +c \right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(3/2)*(b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)

[Out]

1/120/d*(b*cos(d*x+c))^(3/2)*(24*C*sin(d*x+c)*cos(d*x+c)^4+30*B*sin(d*x+c)*cos(d*x+c)^3+40*A*cos(d*x+c)^2*sin(
d*x+c)+32*C*sin(d*x+c)*cos(d*x+c)^2+45*B*cos(d*x+c)*sin(d*x+c)+80*A*sin(d*x+c)+45*B*(d*x+c)+64*C*sin(d*x+c))/c
os(d*x+c)^(3/2)

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maxima [A]  time = 0.73, size = 169, normalized size = 0.74 \[ \frac {40 \, {\left (b \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} A \sqrt {b} + 15 \, {\left (12 \, {\left (d x + c\right )} b + b \sin \left (4 \, d x + 4 \, c\right ) + 8 \, b \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )\right )} B \sqrt {b} + 2 \, {\left (3 \, b \sin \left (5 \, d x + 5 \, c\right ) + 25 \, b \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right ) + 150 \, b \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right )\right )} C \sqrt {b}}{480 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(40*(b*sin(3*d*x + 3*c) + 9*b*sin(1/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))))*A*sqrt(b) + 15*(12*(
d*x + c)*b + b*sin(4*d*x + 4*c) + 8*b*sin(1/2*arctan2(sin(4*d*x + 4*c), cos(4*d*x + 4*c))))*B*sqrt(b) + 2*(3*b
*sin(5*d*x + 5*c) + 25*b*sin(3/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) + 150*b*sin(1/5*arctan2(sin(5*d*
x + 5*c), cos(5*d*x + 5*c))))*C*sqrt(b))/d

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mupad [B]  time = 3.15, size = 142, normalized size = 0.62 \[ \frac {b\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (120\,B\,\sin \left (c+d\,x\right )+400\,A\,\sin \left (2\,c+2\,d\,x\right )+40\,A\,\sin \left (4\,c+4\,d\,x\right )+135\,B\,\sin \left (3\,c+3\,d\,x\right )+15\,B\,\sin \left (5\,c+5\,d\,x\right )+350\,C\,\sin \left (2\,c+2\,d\,x\right )+56\,C\,\sin \left (4\,c+4\,d\,x\right )+6\,C\,\sin \left (6\,c+6\,d\,x\right )+360\,B\,d\,x\,\cos \left (c+d\,x\right )\right )}{480\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^(3/2)*(b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)

[Out]

(b*cos(c + d*x)^(1/2)*(b*cos(c + d*x))^(1/2)*(120*B*sin(c + d*x) + 400*A*sin(2*c + 2*d*x) + 40*A*sin(4*c + 4*d
*x) + 135*B*sin(3*c + 3*d*x) + 15*B*sin(5*c + 5*d*x) + 350*C*sin(2*c + 2*d*x) + 56*C*sin(4*c + 4*d*x) + 6*C*si
n(6*c + 6*d*x) + 360*B*d*x*cos(c + d*x)))/(480*d*(cos(2*c + 2*d*x) + 1))

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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(cos(d*x+c)**(3/2)*(b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)

[Out]

Timed out

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