3.2.76 \(\int \frac {\sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}} \, dx\) [176]

Optimal. Leaf size=24 \[ \frac {x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, 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 {x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[c + d*x]]/Sqrt[b*Cos[c + d*x]],x]

[Out]

(x*Sqrt[Cos[c + d*x]])/Sqrt[b*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 {\sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int 1 \, dx}{\sqrt {b \cos (c+d x)}}\\ &=\frac {x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[c + d*x]]/Sqrt[b*Cos[c + d*x]],x]

[Out]

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

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Maple [A]
time = 0.10, size = 28, normalized size = 1.17

method	result	size
risch	\(\frac {x \left (\sqrt {\cos }\left (d x +c \right )\right )}{\sqrt {b \cos \left (d x +c \right )}}\)	\(21\)
default	\(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \left (d x +c \right )}{d \sqrt {b \cos \left (d x +c \right )}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.52, size = 26, normalized size = 1.08 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{\sqrt {b} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.42, size = 97, normalized size = 4.04 \begin {gather*} \left [-\frac {\sqrt {-b} \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 \, b d}, \frac {\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 )}{\sqrt {b} d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 1.09, size = 22, normalized size = 0.92 \begin {gather*} \frac {x \sqrt {\cos {\left (c + d x \right )}}}{\sqrt {b \cos {\left (c + d x \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(cos(c + d*x))/sqrt(b*cos(c + d*x))

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

[Out]

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

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Mupad [B]
time = 0.26, size = 37, normalized size = 1.54 \begin {gather*} \frac {2\,x\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {b\,\cos \left (c+d\,x\right )}}{b\,\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)^(1/2)/(b*cos(c + d*x))^(1/2),x)

[Out]

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

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