3.256 \(\int \frac{(d \csc (a+b x))^{5/2}}{\sqrt{c \sec (a+b x)}} \, dx\)

Optimal. Leaf size=33 \[ -\frac{2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0480831, antiderivative size = 33, 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} \[ -\frac{2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.111938, size = 33, normalized size = 1. \[ -\frac{2 c d (d \csc (a+b x))^{3/2}}{3 b (c \sec (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.153, size = 42, normalized size = 1.3 \begin{align*} -{\frac{2\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{3\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}}{\frac{1}{\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))^(5/2)/(c*sec(b*x+a))^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \csc \left (b x + a\right )\right )^{\frac{5}{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))^(5/2)/(c*sec(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.06628, size = 116, normalized size = 3.52 \begin{align*} -\frac{2 \, d^{2} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )^{2}}{3 \, b c \sin \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*d^2*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))*cos(b*x + a)^2/(b*c*sin(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((d*csc(b*x+a))**(5/2)/(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 \frac{\left (d \csc \left (b x + a\right )\right )^{\frac{5}{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))^(5/2)/(c*sec(b*x+a))^(1/2),x, algorithm="giac")

[Out]

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