3.360 \(\int \sec ^4(x) \tan ^{\frac{3}{2}}(x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{9} \tan ^{\frac{9}{2}}(x)+\frac{2}{5} \tan ^{\frac{5}{2}}(x) \]

[Out]

(2*Tan[x]^(5/2))/5 + (2*Tan[x]^(9/2))/9

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Rubi [A]  time = 0.0241418, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2607, 14} \[ \frac{2}{9} \tan ^{\frac{9}{2}}(x)+\frac{2}{5} \tan ^{\frac{5}{2}}(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Tan[x]^(5/2))/5 + (2*Tan[x]^(9/2))/9

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \sec ^4(x) \tan ^{\frac{3}{2}}(x) \, dx &=\operatorname{Subst}\left (\int x^{3/2} \left (1+x^2\right ) \, dx,x,\tan (x)\right )\\ &=\operatorname{Subst}\left (\int \left (x^{3/2}+x^{7/2}\right ) \, dx,x,\tan (x)\right )\\ &=\frac{2}{5} \tan ^{\frac{5}{2}}(x)+\frac{2}{9} \tan ^{\frac{9}{2}}(x)\\ \end{align*}

Mathematica [A]  time = 0.0365843, size = 22, normalized size = 1.05 \[ \frac{2}{45} (2 \cos (2 x)+7) \tan ^{\frac{5}{2}}(x) \sec ^2(x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(7 + 2*Cos[2*x])*Sec[x]^2*Tan[x]^(5/2))/45

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Maple [A]  time = 0.259, size = 26, normalized size = 1.2 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( x \right ) \right ) ^{2}+10 \right ) \sin \left ( x \right ) }{45\, \left ( \cos \left ( x \right ) \right ) ^{3}} \left ({\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*(4*cos(x)^2+5)*sin(x)*(sin(x)/cos(x))^(3/2)/cos(x)^3

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Maxima [A]  time = 0.935958, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{9} \, \tan \left (x\right )^{\frac{9}{2}} + \frac{2}{5} \, \tan \left (x\right )^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*tan(x)^(9/2) + 2/5*tan(x)^(5/2)

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Fricas [A]  time = 2.46133, size = 86, normalized size = 4.1 \begin{align*} -\frac{2 \,{\left (4 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 5\right )} \sqrt{\frac{\sin \left (x\right )}{\cos \left (x\right )}}}{45 \, \cos \left (x\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/45*(4*cos(x)^4 + cos(x)^2 - 5)*sqrt(sin(x)/cos(x))/cos(x)^4

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.0933, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{9} \, \tan \left (x\right )^{\frac{9}{2}} + \frac{2}{5} \, \tan \left (x\right )^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*tan(x)^(9/2) + 2/5*tan(x)^(5/2)