3.160 \(\int \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=116 \[ \frac {b^2 \sin ^5(c+d x) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}} \]

[Out]

b^2*sin(d*x+c)*(b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)-2/3*b^2*sin(d*x+c)^3*(b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^
(1/2)+1/5*b^2*sin(d*x+c)^5*(b*cos(d*x+c))^(1/2)/d/cos(d*x+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 2633} \[ \frac {b^2 \sin ^5(c+d x) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b^2 \sin ^3(c+d x) \sqrt {b \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}+\frac {b^2 \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(5/2),x]

[Out]

(b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) - (2*b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^3)/(
3*d*Sqrt[Cos[c + d*x]]) + (b^2*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]^5)/(5*d*Sqrt[Cos[c + d*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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 57, normalized size = 0.49 \[ \frac {\sin (c+d x) \left (3 \sin ^4(c+d x)-10 \sin ^2(c+d x)+15\right ) (b \cos (c+d x))^{5/2}}{15 d \cos ^{\frac {5}{2}}(c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(5/2),x]

[Out]

((b*Cos[c + d*x])^(5/2)*Sin[c + d*x]*(15 - 10*Sin[c + d*x]^2 + 3*Sin[c + d*x]^4))/(15*d*Cos[c + d*x]^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 61, normalized size = 0.53 \[ \frac {{\left (3 \, b^{2} \cos \left (d x + c\right )^{4} + 4 \, b^{2} \cos \left (d x + c\right )^{2} + 8 \, b^{2}\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{15 \, d \sqrt {\cos \left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/2)*(b*cos(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/15*(3*b^2*cos(d*x + c)^4 + 4*b^2*cos(d*x + c)^2 + 8*b^2)*sqrt(b*cos(d*x + c))*sin(d*x + c)/(d*sqrt(cos(d*x +
c)))

________________________________________________________________________________________

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)^(5/2)*(b*cos(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.12, size = 52, normalized size = 0.45 \[ \frac {\left (3 \left (\cos ^{4}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right )+8\right ) \sin \left (d x +c \right ) \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}{15 d \cos \left (d x +c \right )^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(5/2)*(b*cos(d*x+c))^(5/2),x)

[Out]

1/15/d*(3*cos(d*x+c)^4+4*cos(d*x+c)^2+8)*sin(d*x+c)*(b*cos(d*x+c))^(5/2)/cos(d*x+c)^(5/2)

________________________________________________________________________________________

maxima [A]  time = 1.02, size = 77, normalized size = 0.66 \[ \frac {{\left (3 \, b^{2} \sin \left (5 \, d x + 5 \, c\right ) + 25 \, b^{2} \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right ) + 150 \, b^{2} \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (5 \, d x + 5 \, c\right ), \cos \left (5 \, d x + 5 \, c\right )\right )\right )\right )} \sqrt {b}}{240 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(5/2)*(b*cos(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

1/240*(3*b^2*sin(5*d*x + 5*c) + 25*b^2*sin(3/5*arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))) + 150*b^2*sin(1/5*
arctan2(sin(5*d*x + 5*c), cos(5*d*x + 5*c))))*sqrt(b)/d

________________________________________________________________________________________

mupad [B]  time = 1.35, size = 73, normalized size = 0.63 \[ \frac {b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (175\,\sin \left (2\,c+2\,d\,x\right )+28\,\sin \left (4\,c+4\,d\,x\right )+3\,\sin \left (6\,c+6\,d\,x\right )\right )}{240\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^(5/2)*(b*cos(c + d*x))^(5/2),x)

[Out]

(b^2*cos(c + d*x)^(1/2)*(b*cos(c + d*x))^(1/2)*(175*sin(2*c + 2*d*x) + 28*sin(4*c + 4*d*x) + 3*sin(6*c + 6*d*x
)))/(240*d*(cos(2*c + 2*d*x) + 1))

________________________________________________________________________________________

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)**(5/2)*(b*cos(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________