\(\int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx\) [144]

Optimal result

Integrand size = 23, antiderivative size = 24 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 8} \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.92 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\left [\frac {\sqrt {-b} \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, d}, \frac {\sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{d}\right ] \]

[In]

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

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x \sqrt {b \cos {\left (c + d x \right )}}}{\sqrt {\cos {\left (c + d x \right )}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \, \sqrt {b} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {\sqrt {b \cos \left (d x + c\right )}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx=\frac {x\,\sqrt {b\,\cos \left (c+d\,x\right )}}{\sqrt {\cos \left (c+d\,x\right )}} \]

[In]

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

[Out]

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