3.52 \(\int \frac{1}{(a \sec ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 \tan (x)}{3 a \sqrt{a \sec ^2(x)}}+\frac{\tan (x)}{3 \left (a \sec ^2(x)\right )^{3/2}} \]

[Out]

Tan[x]/(3*(a*Sec[x]^2)^(3/2)) + (2*Tan[x])/(3*a*Sqrt[a*Sec[x]^2])

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Rubi [A]  time = 0.0182093, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ \frac{2 \tan (x)}{3 a \sqrt{a \sec ^2(x)}}+\frac{\tan (x)}{3 \left (a \sec ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sec[x]^2)^(-3/2),x]

[Out]

Tan[x]/(3*(a*Sec[x]^2)^(3/2)) + (2*Tan[x])/(3*a*Sqrt[a*Sec[x]^2])

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 192

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

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{1}{\left (a \sec ^2(x)\right )^{3/2}} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\tan (x)\right )\\ &=\frac{\tan (x)}{3 \left (a \sec ^2(x)\right )^{3/2}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac{\tan (x)}{3 \left (a \sec ^2(x)\right )^{3/2}}+\frac{2 \tan (x)}{3 a \sqrt{a \sec ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0178713, size = 27, normalized size = 0.75 \[ \frac{(9 \sin (x)+\sin (3 x)) \sec ^3(x)}{12 \left (a \sec ^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sec[x]^2)^(-3/2),x]

[Out]

(Sec[x]^3*(9*Sin[x] + Sin[3*x]))/(12*(a*Sec[x]^2)^(3/2))

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Maple [A]  time = 0.057, size = 23, normalized size = 0.6 \begin{align*}{\frac{\sin \left ( x \right ) \left ( \left ( \cos \left ( x \right ) \right ) ^{2}+2 \right ) }{3\, \left ( \cos \left ( x \right ) \right ) ^{3}} \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sec(x)^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.94539, size = 19, normalized size = 0.53 \begin{align*} \frac{\sin \left (3 \, x\right ) + 9 \, \sin \left (x\right )}{12 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^2)^(3/2),x, algorithm="maxima")

[Out]

1/12*(sin(3*x) + 9*sin(x))/a^(3/2)

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Fricas [A]  time = 1.4321, size = 74, normalized size = 2.06 \begin{align*} \frac{{\left (\cos \left (x\right )^{3} + 2 \, \cos \left (x\right )\right )} \sqrt{\frac{a}{\cos \left (x\right )^{2}}} \sin \left (x\right )}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(cos(x)^3 + 2*cos(x))*sqrt(a/cos(x)^2)*sin(x)/a^2

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Sympy [A]  time = 1.48685, size = 37, normalized size = 1.03 \begin{align*} \frac{2 \tan ^{3}{\left (x \right )}}{3 a^{\frac{3}{2}} \left (\sec ^{2}{\left (x \right )}\right )^{\frac{3}{2}}} + \frac{\tan{\left (x \right )}}{a^{\frac{3}{2}} \left (\sec ^{2}{\left (x \right )}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)**2)**(3/2),x)

[Out]

2*tan(x)**3/(3*a**(3/2)*(sec(x)**2)**(3/2)) + tan(x)/(a**(3/2)*(sec(x)**2)**(3/2))

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Giac [B]  time = 1.37879, size = 78, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (3 \,{\left (\frac{1}{\tan \left (\frac{1}{2} \, x\right )} + \tan \left (\frac{1}{2} \, x\right )\right )}^{2} \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) - 4 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )\right )}}{3 \, a^{\frac{3}{2}}{\left (\frac{1}{\tan \left (\frac{1}{2} \, x\right )} + \tan \left (\frac{1}{2} \, x\right )\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)^2)^(3/2),x, algorithm="giac")

[Out]

2/3*(3*(1/tan(1/2*x) + tan(1/2*x))^2*sgn(-tan(1/2*x)^2 + 1) - 4*sgn(-tan(1/2*x)^2 + 1))/(a^(3/2)*(1/tan(1/2*x)
+ tan(1/2*x))^3)