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

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

[Out]

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

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Rubi [A]  time = 0.0494305, 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} \[ \frac{2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.111973, size = 33, normalized size = 1. \[ \frac{2 c d (c \sec (a+b x))^{3/2}}{3 b (d \csc (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.149, size = 42, normalized size = 1.3 \begin{align*}{\frac{2\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{3\,b} \left ({\frac{c}{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{d}{\sin \left ( bx+a \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.36281, size = 127, normalized size = 3.85 \begin{align*} -\frac{2 \,{\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}}}{3 \, b d \cos \left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(c^2*cos(b*x + a)^2 - c^2)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))/(b*d*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((c*sec(b*x+a))**(5/2)/(d*csc(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 (c \sec \left (b x + a\right )\right )^{\frac{5}{2}}}{\sqrt{d \csc \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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