3.288 \(\int \frac{\sin ^{\frac{3}{2}}(x)}{\cos ^{\frac{7}{2}}(x)} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 \sin ^{\frac{5}{2}}(x)}{5 \cos ^{\frac{5}{2}}(x)} \]

[Out]

(2*Sin[x]^(5/2))/(5*Cos[x]^(5/2))

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Rubi [A]  time = 0.0226398, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2563} \[ \frac{2 \sin ^{\frac{5}{2}}(x)}{5 \cos ^{\frac{5}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^(3/2)/Cos[x]^(7/2),x]

[Out]

(2*Sin[x]^(5/2))/(5*Cos[x]^(5/2))

Rule 2563

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[((a*Sin[e +
f*x])^(m + 1)*(b*Cos[e + f*x])^(n + 1))/(a*b*f*(m + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2,
0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sin ^{\frac{3}{2}}(x)}{\cos ^{\frac{7}{2}}(x)} \, dx &=\frac{2 \sin ^{\frac{5}{2}}(x)}{5 \cos ^{\frac{5}{2}}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0184556, size = 16, normalized size = 1. \[ \frac{2 \sin ^{\frac{5}{2}}(x)}{5 \cos ^{\frac{5}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^(3/2)/Cos[x]^(7/2),x]

[Out]

(2*Sin[x]^(5/2))/(5*Cos[x]^(5/2))

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Maple [B]  time = 0.048, size = 33, normalized size = 2.1 \begin{align*}{\frac{- \left ( \sin \left ( x \right ) \right ) ^{2}+ \left ( \cos \left ( x \right ) \right ) ^{2}-2\,\cos \left ( x \right ) +1}{-5+5\,\cos \left ( x \right ) } \left ( \sin \left ( x \right ) \right ) ^{{\frac{5}{2}}} \left ( \cos \left ( x \right ) \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^(3/2)/cos(x)^(7/2),x)

[Out]

1/5*(-sin(x)^2+cos(x)^2-2*cos(x)+1)*sin(x)^(5/2)/(-1+cos(x))/cos(x)^(7/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )^{\frac{3}{2}}}{\cos \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^(3/2)/cos(x)^(7/2),x, algorithm="maxima")

[Out]

integrate(sin(x)^(3/2)/cos(x)^(7/2), x)

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Fricas [A]  time = 2.69549, size = 63, normalized size = 3.94 \begin{align*} -\frac{2 \,{\left (\cos \left (x\right )^{2} - 1\right )} \sqrt{\sin \left (x\right )}}{5 \, \cos \left (x\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^(3/2)/cos(x)^(7/2),x, algorithm="fricas")

[Out]

-2/5*(cos(x)^2 - 1)*sqrt(sin(x))/cos(x)^(5/2)

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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(sin(x)**(3/2)/cos(x)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (x\right )^{\frac{3}{2}}}{\cos \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^(3/2)/cos(x)^(7/2),x, algorithm="giac")

[Out]

integrate(sin(x)^(3/2)/cos(x)^(7/2), x)