3.2.65 \(\int \frac {(b \cos (c+d x))^{5/2}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\) [165]

Optimal. Leaf size=27 \[ \frac {b^2 x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 27, 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, 8} \begin {gather*} \frac {b^2 x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*x*Sqrt[b*Cos[c + d*x]])/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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.89 \begin {gather*} \frac {x (b \cos (c+d x))^{5/2}}{\cos ^{\frac {5}{2}}(c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 28, normalized size = 1.04

method	result	size
risch	\(\frac {b^{2} x \sqrt {b \cos \left (d x +c \right )}}{\sqrt {\cos \left (d x +c \right )}}\)	\(24\)
default	\(\frac {\left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (d x +c \right )}{d \cos \left (d x +c \right )^{\frac {5}{2}}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 26, normalized size = 0.96 \begin {gather*} \frac {2 \, b^{\frac {5}{2}} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.40, size = 97, normalized size = 3.59 \begin {gather*} \left [\frac {\sqrt {-b} b^{2} \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 \, d}, \frac {b^{\frac {5}{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 )}{d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(-b)*b^2*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
)/d, b^(5/2)*arctan(sqrt(b*cos(d*x + c))*sin(d*x + c)/(sqrt(b)*cos(d*x + c)^(3/2)))/d]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6189 deep

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

[Out]

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

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Mupad [B]
time = 0.09, size = 23, normalized size = 0.85 \begin {gather*} \frac {b^2\,x\,\sqrt {b\,\cos \left (c+d\,x\right )}}{\sqrt {\cos \left (c+d\,x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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