3.233 \(\int (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)} \, dx\)

Optimal. Leaf size=31 \[ -\frac{2 c d \sqrt{d \csc (a+b x)}}{b \sqrt{c \sec (a+b x)}} \]

[Out]

(-2*c*d*Sqrt[d*Csc[a + b*x]])/(b*Sqrt[c*Sec[a + b*x]])

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Rubi [A]  time = 0.0486029, antiderivative size = 31, 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 = {2619} \[ -\frac{2 c d \sqrt{d \csc (a+b x)}}{b \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]],x]

[Out]

(-2*c*d*Sqrt[d*Csc[a + b*x]])/(b*Sqrt[c*Sec[a + b*x]])

Rule 2619

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

Rubi steps

\begin{align*} \int (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)} \, dx &=-\frac{2 c d \sqrt{d \csc (a+b x)}}{b \sqrt{c \sec (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.0633391, size = 31, normalized size = 1. \[ -\frac{2 c d \sqrt{d \csc (a+b x)}}{b \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a + b*x]],x]

[Out]

(-2*c*d*Sqrt[d*Csc[a + b*x]])/(b*Sqrt[c*Sec[a + b*x]])

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Maple [A]  time = 0.164, size = 42, normalized size = 1.4 \begin{align*} -2\,{\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{3/2}\sqrt{{\frac{c}{\cos \left ( bx+a \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*csc(b*x + a))^(3/2)*sqrt(c*sec(b*x + a)), x)

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Fricas [A]  time = 1.70862, size = 85, normalized size = 2.74 \begin{align*} -\frac{2 \, d \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((d*csc(b*x+a))**(3/2)*(c*sec(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 \left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \sqrt{c \sec \left (b x + a\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*csc(b*x + a))^(3/2)*sqrt(c*sec(b*x + a)), x)