3.2.73 \(\int \frac {\cos ^{\frac {7}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx\) [173]

Optimal. Leaf size=70 \[ \frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}-\frac {\sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {b \cos (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 70, 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, 2713} \begin {gather*} \frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {b \cos (c+d x)}}-\frac {\sin ^3(c+d x) \sqrt {\cos (c+d x)}}{3 d \sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(7/2)/Sqrt[b*Cos[c + d*x]],x]

[Out]

(Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Cos[c + d*x]]) - (Sqrt[Cos[c + d*x]]*Sin[c + d*x]^3)/(3*d*Sqrt[b*C
os[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 2713

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 \frac {\cos ^{\frac {7}{2}}(c+d x)}{\sqrt {b \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos ^3(c+d x) \, dx}{\sqrt {b \cos (c+d x)}}\\ &=-\frac {\sqrt {\cos (c+d x)} \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt {b \cos (c+d x)}}\\ &=\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{d \sqrt {b \cos (c+d x)}}-\frac {\sqrt {\cos (c+d x)} \sin ^3(c+d x)}{3 d \sqrt {b \cos (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 45, normalized size = 0.64 \begin {gather*} \frac {\sqrt {\cos (c+d x)} (5+\cos (2 (c+d x))) \sin (c+d x)}{6 d \sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(7/2)/Sqrt[b*Cos[c + d*x]],x]

[Out]

(Sqrt[Cos[c + d*x]]*(5 + Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d*Sqrt[b*Cos[c + d*x]])

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Maple [A]
time = 0.13, size = 40, normalized size = 0.57

method	result	size
default	\(\frac {\left (\cos ^{2}\left (d x +c \right )+2\right ) \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{3 d \sqrt {b \cos \left (d x +c \right )}}\)	\(40\)
risch	\(\frac {3 \sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{4 d \sqrt {b \cos \left (d x +c \right )}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \sin \left (3 d x +3 c \right )}{12 \sqrt {b \cos \left (d x +c \right )}\, d}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.57, size = 42, normalized size = 0.60 \begin {gather*} \frac {\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \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 )}{12 \, \sqrt {b} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

1/12*(sin(3*d*x + 3*c) + 9*sin(1/3*arctan2(sin(3*d*x + 3*c), cos(3*d*x + 3*c))))/(sqrt(b)*d)

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Fricas [A]
time = 0.39, size = 42, normalized size = 0.60 \begin {gather*} \frac {\sqrt {b \cos \left (d x + c\right )} {\left (\cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right )}{3 \, b d \sqrt {\cos \left (d x + c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**(7/2)/(b*cos(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(7/2)/sqrt(b*cos(d*x + c)), x)

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Mupad [B]
time = 0.64, size = 60, normalized size = 0.86 \begin {gather*} \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (10\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (4\,c+4\,d\,x\right )\right )}{12\,b\,d\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^(7/2)/(b*cos(c + d*x))^(1/2),x)

[Out]

(cos(c + d*x)^(1/2)*(b*cos(c + d*x))^(1/2)*(10*sin(2*c + 2*d*x) + sin(4*c + 4*d*x)))/(12*b*d*(cos(2*c + 2*d*x)
+ 1))

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