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]

1/3*tan(x)/(a*sec(x)^2)^(3/2)+2/3*tan(x)/a/(a*sec(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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]

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 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
]

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 = 0.02, 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))

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 24, normalized size = 0.67 \[ \frac {{\left (\cos \relax (x)^{3} + 2 \, \cos \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{2}}} \sin \relax (x)}{3 \, a^{2}} \]

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

________________________________________________________________________________________

giac [B]  time = 0.60, size = 58, normalized size = 1.61 \[ \frac {2 \, {\left (3 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) - 4 \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )\right )}}{3 \, a^{\frac {3}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{3}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.30, size = 23, normalized size = 0.64 \[ \frac {\sin \relax (x ) \left (\cos ^{2}\relax (x )+2\right )}{3 \cos \relax (x )^{3} \left (\frac {a}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}}} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.00, size = 14, normalized size = 0.39 \[ \frac {\sin \left (3 \, x\right ) + 9 \, \sin \relax (x)}{12 \, a^{\frac {3}{2}}} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (\frac {a}{{\cos \relax (x)}^2}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/cos(x)^2)^(3/2),x)

[Out]

int(1/(a/cos(x)^2)^(3/2), x)

________________________________________________________________________________________

sympy [A]  time = 1.18, size = 37, normalized size = 1.03 \[ \frac {2 \tan ^{3}{\relax (x )}}{3 a^{\frac {3}{2}} \left (\sec ^{2}{\relax (x )}\right )^{\frac {3}{2}}} + \frac {\tan {\relax (x )}}{a^{\frac {3}{2}} \left (\sec ^{2}{\relax (x )}\right )^{\frac {3}{2}}} \]

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))

________________________________________________________________________________________