3.2.95 \(\int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx\) [195]

Optimal. Leaf size=35 \[ \frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt {b \cos (c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 2717} \begin {gather*} \frac {\sin (c+d x) \sqrt {\cos (c+d x)}}{b^2 d \sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^(7/2)/(b*Cos[c + d*x])^(5/2),x]

[Out]

(Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b^2*d*Sqrt[b*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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \cos (c+d x) \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt {b \cos (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\cos (c+d x)} \sin (c+d x)}{b^2 d \sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^(7/2)/(b*Cos[c + d*x])^(5/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 29, normalized size = 0.83

method	result	size
default	\(\frac {\sin \left (d x +c \right ) \left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right )}{d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\)	\(29\)
risch	\(\frac {\sin \left (d x +c \right ) \left (\sqrt {\cos }\left (d x +c \right )\right )}{b^{2} d \sqrt {b \cos \left (d x +c \right )}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.58, size = 13, normalized size = 0.37 \begin {gather*} \frac {\sin \left (d x + c\right )}{b^{\frac {5}{2}} 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))^(5/2),x, algorithm="maxima")

[Out]

sin(d*x + c)/(b^(5/2)*d)

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 31, normalized size = 0.89 \begin {gather*} \frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{b^{3} 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))^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

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))^(5/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^(7/2)/(b*cos(d*x + c))^(5/2), x)

________________________________________________________________________________________

Mupad [B]
time = 0.41, size = 47, normalized size = 1.34 \begin {gather*} \frac {\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {b\,\cos \left (c+d\,x\right )}}{b^3\,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))^(5/2),x)

[Out]

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

________________________________________________________________________________________