3.297 \(\int \frac{1}{(d \cos (a+b x))^{3/2} \sqrt{c \sin (a+b x)}} \, dx\)

Optimal. Leaf size=35 \[ \frac{2 \sqrt{c \sin (a+b x)}}{b c d \sqrt{d \cos (a+b x)}} \]

[Out]

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

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Rubi [A]  time = 0.0539481, antiderivative size = 35, 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} \[ \frac{2 \sqrt{c \sin (a+b x)}}{b c d \sqrt{d \cos (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]]),x]

[Out]

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

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

Mathematica [A]  time = 0.0621568, size = 36, normalized size = 1.03 \[ \frac{\sin (2 (a+b x))}{b \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]]),x]

[Out]

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

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Maple [A]  time = 0.069, size = 38, normalized size = 1.1 \begin{align*} 2\,{\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{b \left ( d\cos \left ( bx+a \right ) \right ) ^{3/2}\sqrt{c\sin \left ( bx+a \right ) }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} \sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*cos(b*x+a))^(3/2)/(c*sin(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.61902, size = 92, normalized size = 2.63 \begin{align*} \frac{2 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{c \sin \left (b x + a\right )}}{b c d^{2} \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*cos(b*x+a))^(3/2)/(c*sin(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \cos \left (b x + a\right )\right )^{\frac{3}{2}} \sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*cos(b*x+a))^(3/2)/(c*sin(b*x+a))^(1/2),x, algorithm="giac")

[Out]

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