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

Optimal. Leaf size=29 \[ \frac{2 \tan (x)}{3 \sqrt{\sec ^2(x)}}+\frac{\tan (x)}{3 \sec ^2(x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0105769, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4122, 192, 191} \[ \frac{2 \tan (x)}{3 \sqrt{\sec ^2(x)}}+\frac{\tan (x)}{3 \sec ^2(x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0149289, size = 23, normalized size = 0.79 \[ \frac{(9 \sin (x)+\sin (3 x)) \sec (x)}{12 \sqrt{\sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.059, size = 21, normalized size = 0.7 \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 ( \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/(sec(x)^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.08934, size = 34, normalized size = 1.17 \begin{align*} \frac{2 \, \tan \left (x\right )}{3 \, \sqrt{\tan \left (x\right )^{2} + 1}} + \frac{\tan \left (x\right )}{3 \,{\left (\tan \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*tan(x)/sqrt(tan(x)^2 + 1) + 1/3*tan(x)/(tan(x)^2 + 1)^(3/2)

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Fricas [A]  time = 1.31181, size = 38, normalized size = 1.31 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(cos(x)^2 + 2)*sin(x)

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Sympy [A]  time = 1.5059, size = 27, normalized size = 0.93 \begin{align*} \frac{2 \tan ^{3}{\left (x \right )}}{3 \left (\sec ^{2}{\left (x \right )}\right )^{\frac{3}{2}}} + \frac{\tan{\left (x \right )}}{\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/(sec(x)**2)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.35019, size = 22, normalized size = 0.76 \begin{align*} -\frac{1}{3} \, \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right )^{3} + \mathrm{sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sgn(cos(x))*sin(x)^3 + sgn(cos(x))*sin(x)