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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.102423, 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 = {2626, 2619} \[ \frac{2 c d (c \sec (a+b x))^{3/2} \sqrt{d \csc (a+b x)}}{3 b}-\frac{8 c^3 d \sqrt{d \csc (a+b x)}}{3 b \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2626

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

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

Mathematica [A]  time = 0.207171, size = 45, normalized size = 0.65 \[ -\frac{2 c d (2 \cos (2 (a+b x))+1) (c \sec (a+b x))^{3/2} \sqrt{d \csc (a+b x)}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*d*(1 + 2*Cos[2*(a + b*x)])*Sqrt[d*Csc[a + b*x]]*(c*Sec[a + b*x])^(3/2))/(3*b)

________________________________________________________________________________________

Maple [A]  time = 0.161, size = 54, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-2 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{3\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{3}{2}}} \left ({\frac{c}{\cos \left ( bx+a \right ) }} \right ) ^{{\frac{5}{2}}}} \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))^(5/2),x)

[Out]

-2/3/b*(4*cos(b*x+a)^2-1)*cos(b*x+a)*(d/sin(b*x+a))^(3/2)*(c/cos(b*x+a))^(5/2)*sin(b*x+a)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{5}{2}}\,{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))^(5/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.54961, size = 132, normalized size = 1.91 \begin{align*} -\frac{2 \,{\left (4 \, c^{2} d \cos \left (b x + a\right )^{2} - c^{2} d\right )} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}}}{3 \, b \cos \left (b x + a\right )} \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))^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \csc \left (b x + a\right )\right )^{\frac{3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{5}{2}}\,{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))^(5/2),x, algorithm="giac")

[Out]

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