3.278 \(\int \frac{\sec ^2(e+f x)}{(a+b \sec ^2(e+f x))^{3/2}} \, dx\)

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

[Out]

Tan[e + f*x]/((a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])

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Rubi [A]  time = 0.0745116, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4146, 191} \[ \frac{\tan (e+f x)}{f (a+b) \sqrt{a+b \tan ^2(e+f x)+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]

[Out]

Tan[e + f*x]/((a + b)*f*Sqrt[a + b + b*Tan[e + f*x]^2])

Rule 4146

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fre
eFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/
2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{\sec ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x)}{(a+b) f \sqrt{a+b+b \tan ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.656438, size = 57, normalized size = 1.78 \[ \frac{\tan (e+f x) \sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b)}{2 f (a+b) \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^2/(a + b*Sec[e + f*x]^2)^(3/2),x]

[Out]

((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^2*Tan[e + f*x])/(2*(a + b)*f*(a + b*Sec[e + f*x]^2)^(3/2))

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Maple [A]  time = 0.23, size = 59, normalized size = 1.8 \begin{align*}{\frac{ \left ( b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2} \right ) \sin \left ( fx+e \right ) }{f \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ({\frac{b+a \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x)

[Out]

1/f/(a+b)*(b+a*cos(f*x+e)^2)*sin(f*x+e)/cos(f*x+e)^3/((b+a*cos(f*x+e)^2)/cos(f*x+e)^2)^(3/2)

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Maxima [A]  time = 1.07825, size = 41, normalized size = 1.28 \begin{align*} \frac{\tan \left (f x + e\right )}{\sqrt{b \tan \left (f x + e\right )^{2} + a + b}{\left (a + b\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="maxima")

[Out]

tan(f*x + e)/(sqrt(b*tan(f*x + e)^2 + a + b)*(a + b)*f)

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Fricas [B]  time = 0.560278, size = 159, normalized size = 4.97 \begin{align*} \frac{\sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{{\left (a^{2} + a b\right )} f \cos \left (f x + e\right )^{2} +{\left (a b + b^{2}\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**2/(a+b*sec(f*x+e)**2)**(3/2),x)

[Out]

Integral(sec(e + f*x)**2/(a + b*sec(e + f*x)**2)**(3/2), x)

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Giac [B]  time = 1.94885, size = 157, normalized size = 4.91 \begin{align*} -\frac{2 \, a^{2} b^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{{\left (a^{3} b^{2} + a^{2} b^{3}\right )} \sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} + b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a + b} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2/(a+b*sec(f*x+e)^2)^(3/2),x, algorithm="giac")

[Out]

-2*a^2*b^2*sgn(tan(1/2*f*x + 1/2*e)^2 - 1)*tan(1/2*f*x + 1/2*e)/((a^3*b^2 + a^2*b^3)*sqrt(a*tan(1/2*f*x + 1/2*
e)^4 + b*tan(1/2*f*x + 1/2*e)^4 - 2*a*tan(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e)^2 + a + b)*f)