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

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

[Out]

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

________________________________________________________________________________________

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} \[ \frac {x \sqrt {b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 24, normalized size = 1.00 \[ \frac {x \sqrt {b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 98, normalized size = 4.08 \[ \left [\frac {\sqrt {-b} \log \left (-2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{2 \, d}, \frac {\sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \sec \left (d x + c\right )}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 1.03, size = 32, normalized size = 1.33 \[ \frac {\sqrt {\frac {b}{\cos \left (d x +c \right )}}\, \left (d x +c \right )}{d \sqrt {\frac {1}{\cos \left (d x +c \right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 26, normalized size = 1.08 \[ \frac {2 \, \sqrt {b} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 24, normalized size = 1.00 \[ \frac {x\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.58, size = 5, normalized size = 0.21 \[ \sqrt {b} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*x

________________________________________________________________________________________