3.319 \(\int \frac {1}{\sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)}} \, dx\)

Optimal. Leaf size=32 \[ \frac {2 \sqrt {b \tan (e+f x)}}{b f \sqrt {d \sec (e+f x)}} \]

[Out]

2*(b*tan(f*x+e))^(1/2)/b/f/(d*sec(f*x+e))^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2605} \[ \frac {2 \sqrt {b \tan (e+f x)}}{b f \sqrt {d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]]),x]

[Out]

(2*Sqrt[b*Tan[e + f*x]])/(b*f*Sqrt[d*Sec[e + f*x]])

Rule 2605

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> -Simp[((a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*m), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 1, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d \sec (e+f x)} \sqrt {b \tan (e+f x)}} \, dx &=\frac {2 \sqrt {b \tan (e+f x)}}{b f \sqrt {d \sec (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 32, normalized size = 1.00 \[ \frac {2 \sqrt {b \tan (e+f x)}}{b f \sqrt {d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d*Sec[e + f*x]]*Sqrt[b*Tan[e + f*x]]),x]

[Out]

(2*Sqrt[b*Tan[e + f*x]])/(b*f*Sqrt[d*Sec[e + f*x]])

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fricas [A]  time = 0.59, size = 47, normalized size = 1.47 \[ \frac {2 \, \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{b d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*sin(f*x + e)/cos(f*x + e))*sqrt(d/cos(f*x + e))*cos(f*x + e)/(b*d*f)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d \sec \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*sec(f*x + e))*sqrt(b*tan(f*x + e))), x)

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maple [A]  time = 0.59, size = 50, normalized size = 1.56 \[ \frac {2 \sin \left (f x +e \right )}{f \sqrt {\frac {d}{\cos \left (f x +e \right )}}\, \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}\, \cos \left (f x +e \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*sec(f*x+e))^(1/2)/(b*tan(f*x+e))^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d \sec \left (f x + e\right )} \sqrt {b \tan \left (f x + e\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(d*sec(f*x + e))*sqrt(b*tan(f*x + e))), x)

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mupad [B]  time = 2.90, size = 52, normalized size = 1.62 \[ \frac {2\,\sin \left (e+f\,x\right )\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}}{d\,f\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*tan(e + f*x))^(1/2)*(d/cos(e + f*x))^(1/2)),x)

[Out]

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

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sympy [A]  time = 19.00, size = 51, normalized size = 1.59 \[ \begin {cases} \frac {2 \sqrt {\tan {\left (e + f x \right )}}}{\sqrt {b} \sqrt {d} f \sqrt {\sec {\left (e + f x \right )}}} & \text {for}\: f \neq 0 \\\frac {x}{\sqrt {b \tan {\relax (e )}} \sqrt {d \sec {\relax (e )}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*sec(f*x+e))**(1/2)/(b*tan(f*x+e))**(1/2),x)

[Out]

Piecewise((2*sqrt(tan(e + f*x))/(sqrt(b)*sqrt(d)*f*sqrt(sec(e + f*x))), Ne(f, 0)), (x/(sqrt(b*tan(e))*sqrt(d*s
ec(e))), True))

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