3.264 \(\int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}} \]

[Out]

(2*(c*Sin[a + b*x])^(3/2))/(3*b*c*d*(d*Cos[a + b*x])^(3/2))

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Rubi [A]  time = 0.0529633, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2563} \[ \frac{2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*Sin[a + b*x]]/(d*Cos[a + b*x])^(5/2),x]

[Out]

(2*(c*Sin[a + b*x])^(3/2))/(3*b*c*d*(d*Cos[a + b*x])^(3/2))

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{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}} \, dx &=\frac{2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0826124, size = 37, normalized size = 1. \[ \frac{2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c*Sin[a + b*x]]/(d*Cos[a + b*x])^(5/2),x]

[Out]

(2*(c*Sin[a + b*x])^(3/2))/(3*b*c*d*(d*Cos[a + b*x])^(3/2))

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Maple [A]  time = 0.108, size = 38, normalized size = 1. \begin{align*}{\frac{2\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{3\,b}\sqrt{c\sin \left ( bx+a \right ) } \left ( d\cos \left ( bx+a \right ) \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*sin(b*x+a))^(1/2)/(d*cos(b*x+a))^(5/2),x)

[Out]

2/3/b*sin(b*x+a)*cos(b*x+a)*(c*sin(b*x+a))^(1/2)/(d*cos(b*x+a))^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(1/2)/(d*cos(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*sin(b*x + a))/(d*cos(b*x + a))^(5/2), x)

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Fricas [A]  time = 3.2955, size = 112, normalized size = 3.03 \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{3 \, b d^{3} \cos \left (b x + a\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(1/2)/(d*cos(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

2/3*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*sin(b*x + a)/(b*d^3*cos(b*x + a)^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((c*sin(b*x+a))**(1/2)/(d*cos(b*x+a))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*sin(b*x+a))^(1/2)/(d*cos(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*sin(b*x + a))/(d*cos(b*x + a))^(5/2), x)