3.1.52 \(\int \frac {1}{(a \sec ^2(x))^{3/2}} \, dx\) [52]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {4207, 198, 197} \begin {gather*} \frac {2 \tan (x)}{3 a \sqrt {a \sec ^2(x)}}+\frac {\tan (x)}{3 \left (a \sec ^2(x)\right )^{3/2}} \end {gather*}

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 197

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 198

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 4207

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 \text {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} \text {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*}

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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.75 \begin {gather*} \frac {\sec ^3(x) (9 \sin (x)+\sin (3 x))}{12 \left (a \sec ^2(x)\right )^{3/2}} \end {gather*}

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 = 0.18, size = 23, normalized size = 0.64

method	result	size
default	\(\frac {\sin \left (x \right ) \left (\cos ^{2}\left (x \right )+2\right )}{3 \cos \left (x \right )^{3} \left (\frac {a}{\cos \left (x \right )^{2}}\right )^{\frac {3}{2}}}\)	\(23\)
risch	\(-\frac {i {\mathrm e}^{4 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}-\frac {3 i {\mathrm e}^{2 i x}}{8 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}+\frac {3 i}{8 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}+\frac {i {\mathrm e}^{-2 i x}}{24 a \left ({\mathrm e}^{2 i x}+1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.60, size = 14, normalized size = 0.39 \begin {gather*} \frac {\sin \left (3 \, x\right ) + 9 \, \sin \left (x\right )}{12 \, a^{\frac {3}{2}}} \end {gather*}

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 = 3.06, size = 24, normalized size = 0.67 \begin {gather*} \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 {gather*}

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 = 0.36, size = 31, normalized size = 0.86 \begin {gather*} \frac {2 \tan ^{3}{\left (x \right )}}{3 \left (a \sec ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} + \frac {\tan {\left (x \right )}}{\left (a \sec ^{2}{\left (x \right )}\right )^{\frac {3}{2}}} \end {gather*}

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

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Giac [A]
time = 0.44, size = 26, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {a} \sin \left (x\right )^{3} - 3 \, \sqrt {a} \sin \left (x\right )}{3 \, a^{2} \mathrm {sgn}\left (\cos \left (x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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