3.165 \(\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \, dx\)

Optimal. Leaf size=32 \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{b \sec (c+d x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0066139, antiderivative size = 32, normalized size of antiderivative = 1., 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, 2637} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{b \sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[b*Sec[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 2637

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

Mathematica [A]  time = 0.0361462, size = 32, normalized size = 1. \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{b \sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.154, size = 41, normalized size = 1.3 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}}}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 2.09274, size = 18, normalized size = 0.56 \begin{align*} \frac{\sin \left (d x + c\right )}{\sqrt{b} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(d*x + c)/(sqrt(b)*d)

________________________________________________________________________________________

Fricas [A]  time = 1.7484, size = 81, normalized size = 2.53 \begin{align*} \frac{\sqrt{\frac{b}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 25.9602, size = 36, normalized size = 1.12 \begin{align*} \begin{cases} \frac{\tan{\left (c + d x \right )}}{\sqrt{b} d \sec{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{\sqrt{b \sec{\left (c \right )}} \sqrt{\sec{\left (c \right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((tan(c + d*x)/(sqrt(b)*d*sec(c + d*x)), Ne(d, 0)), (x/(sqrt(b*sec(c))*sqrt(sec(c))), True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (d x + c\right )} \sqrt{\sec \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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