3.2.77 \(\int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}} \, dx\) [177]

Optimal. Leaf size=33 \[ \frac {\tanh ^{-1}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{d \sqrt {b \cos (c+d x)}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 33, 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 = {18, 3855} \begin {gather*} \frac {\sqrt {\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]),x]

[Out]

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

Rule 18

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[a^(m - 1/2)*b^(n + 1/2)*(Sqrt[a*v]/Sqrt[b*v])
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] &&  !IntegerQ[m] && ILtQ[n - 1/2, 0] && IntegerQ[m + n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \sec (c+d x) \, dx}{\sqrt {b \cos (c+d x)}}\\ &=\frac {\tanh ^{-1}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{d \sqrt {b \cos (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{d \sqrt {b \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]),x]

[Out]

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

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Maple [A]
time = 0.13, size = 42, normalized size = 1.27

method	result	size
default	\(-\frac {2 \left (\sqrt {\cos }\left (d x +c \right )\right ) \arctanh \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )}{d \sqrt {b \cos \left (d x +c \right )}}\)	\(42\)
risch	\(\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{\sqrt {b \cos \left (d x +c \right )}\, d}-\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{\sqrt {b \cos \left (d x +c \right )}\, d}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
time = 0.56, size = 65, normalized size = 1.97 \begin {gather*} \frac {\log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}{2 \, \sqrt {b} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.44, size = 116, normalized size = 3.52 \begin {gather*} \left [\frac {\log \left (-\frac {b \cos \left (d x + c\right )^{3} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{3}}\right )}{2 \, \sqrt {b} d}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sin \left (d x + c\right )}{b \sqrt {\cos \left (d x + c\right )}}\right )}{b d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b \cos {\left (c + d x \right )}} \sqrt {\cos {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {b\,\cos \left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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