3.63 \(\int (a \sec ^4(x))^{3/2} \, dx\)

Optimal. Leaf size=61 \[ a \sin (x) \cos (x) \sqrt{a \sec ^4(x)}+\frac{1}{5} a \sin ^2(x) \tan ^3(x) \sqrt{a \sec ^4(x)}+\frac{2}{3} a \sin ^2(x) \tan (x) \sqrt{a \sec ^4(x)} \]

[Out]

a*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (2*a*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x])/3 + (a*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan
[x]^3)/5

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Rubi [A]  time = 0.0220653, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4123, 3767} \[ a \sin (x) \cos (x) \sqrt{a \sec ^4(x)}+\frac{1}{5} a \sin ^2(x) \tan ^3(x) \sqrt{a \sec ^4(x)}+\frac{2}{3} a \sin ^2(x) \tan (x) \sqrt{a \sec ^4(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sec[x]^4)^(3/2),x]

[Out]

a*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (2*a*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan[x])/3 + (a*Sqrt[a*Sec[x]^4]*Sin[x]^2*Tan
[x]^3)/5

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps

\begin{align*} \int \left (a \sec ^4(x)\right )^{3/2} \, dx &=\left (a \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \int \sec ^6(x) \, dx\\ &=-\left (\left (a \cos ^2(x) \sqrt{a \sec ^4(x)}\right ) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )\right )\\ &=a \cos (x) \sqrt{a \sec ^4(x)} \sin (x)+\frac{2}{3} a \sqrt{a \sec ^4(x)} \sin ^2(x) \tan (x)+\frac{1}{5} a \sqrt{a \sec ^4(x)} \sin ^2(x) \tan ^3(x)\\ \end{align*}

Mathematica [A]  time = 0.0571263, size = 30, normalized size = 0.49 \[ \frac{1}{15} \sin (x) \cos (x) (6 \cos (2 x)+\cos (4 x)+8) \left (a \sec ^4(x)\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sec[x]^4)^(3/2),x]

[Out]

(Cos[x]*(8 + 6*Cos[2*x] + Cos[4*x])*(a*Sec[x]^4)^(3/2)*Sin[x])/15

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Maple [A]  time = 0.067, size = 29, normalized size = 0.5 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( x \right ) \right ) ^{4}+4\, \left ( \cos \left ( x \right ) \right ) ^{2}+3 \right ) \cos \left ( x \right ) \sin \left ( x \right ) }{15} \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{4}}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sec(x)^4)^(3/2),x)

[Out]

1/15*(8*cos(x)^4+4*cos(x)^2+3)*cos(x)*sin(x)*(a/cos(x)^4)^(3/2)

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Maxima [A]  time = 1.69223, size = 34, normalized size = 0.56 \begin{align*} \frac{1}{5} \, a^{\frac{3}{2}} \tan \left (x\right )^{5} + \frac{2}{3} \, a^{\frac{3}{2}} \tan \left (x\right )^{3} + a^{\frac{3}{2}} \tan \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)^4)^(3/2),x, algorithm="maxima")

[Out]

1/5*a^(3/2)*tan(x)^5 + 2/3*a^(3/2)*tan(x)^3 + a^(3/2)*tan(x)

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Fricas [A]  time = 1.4485, size = 101, normalized size = 1.66 \begin{align*} \frac{{\left (8 \, a \cos \left (x\right )^{4} + 4 \, a \cos \left (x\right )^{2} + 3 \, a\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4}}} \sin \left (x\right )}{15 \, \cos \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)^4)^(3/2),x, algorithm="fricas")

[Out]

1/15*(8*a*cos(x)^4 + 4*a*cos(x)^2 + 3*a)*sqrt(a/cos(x)^4)*sin(x)/cos(x)^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sec ^{4}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)**4)**(3/2),x)

[Out]

Integral((a*sec(x)**4)**(3/2), x)

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Giac [A]  time = 1.30526, size = 30, normalized size = 0.49 \begin{align*} \frac{1}{15} \,{\left (3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )\right )} a^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sec(x)^4)^(3/2),x, algorithm="giac")

[Out]

1/15*(3*tan(x)^5 + 10*tan(x)^3 + 15*tan(x))*a^(3/2)