3.240 \(\int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2} \, dx\)

Optimal. Leaf size=69 \[ \frac{8 c d^3 \sqrt{c \sec (a+b x)}}{3 b \sqrt{d \csc (a+b x)}}-\frac{2 c d \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.102838, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2625, 2619} \[ \frac{8 c d^3 \sqrt{c \sec (a+b x)}}{3 b \sqrt{d \csc (a+b x)}}-\frac{2 c d \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

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

Rule 2625

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*(m - 1)), x] + Dist[(a^2*(m + n - 2))/(m - 1), Int[(a*Csc[e +
f*x])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && IntegersQ[2*m, 2*n] &&
!GtQ[n, m]

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))^{5/2} (c \sec (a+b x))^{3/2} \, dx &=-\frac{2 c d (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}{3 b}+\frac{1}{3} \left (4 d^2\right ) \int \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2} \, dx\\ &=\frac{8 c d^3 \sqrt{c \sec (a+b x)}}{3 b \sqrt{d \csc (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.135484, size = 45, normalized size = 0.65 \[ -\frac{2 c d^3 \left (\csc ^2(a+b x)-4\right ) \sqrt{c \sec (a+b x)}}{3 b \sqrt{d \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

(-2*c*d^3*(-4 + Csc[a + b*x]^2)*Sqrt[c*Sec[a + b*x]])/(3*b*Sqrt[d*Csc[a + b*x]])

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Maple [A]  time = 0.162, size = 54, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-6 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{3\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}} \left ({\frac{c}{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{3}{2}}}} \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))^(3/2),x)

[Out]

-2/3/b*(4*cos(b*x+a)^2-3)*cos(b*x+a)*(d/sin(b*x+a))^(5/2)*(c/cos(b*x+a))^(3/2)*sin(b*x+a)

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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{5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}\,{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(5/2)*(c*sec(b*x + a))^(3/2), x)

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Fricas [A]  time = 2.06235, size = 135, normalized size = 1.96 \begin{align*} -\frac{2 \,{\left (4 \, c d^{2} \cos \left (b x + a\right )^{2} - 3 \, c d^{2}\right )} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}}}{3 \, b \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))^(3/2),x, algorithm="fricas")

[Out]

-2/3*(4*c*d^2*cos(b*x + a)^2 - 3*c*d^2)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))/(b*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))**(3/2),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{5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}\,{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))^(3/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(5/2)*(c*sec(b*x + a))^(3/2), x)