3.231 \(\int (d \csc (a+b x))^{7/2} \sqrt{c \sec (a+b x)} \, dx\)

Optimal. Leaf size=69 \[ -\frac{8 c d^3 \sqrt{d \csc (a+b x)}}{5 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b \sqrt{c \sec (a+b x)}} \]

[Out]

(-8*c*d^3*Sqrt[d*Csc[a + b*x]])/(5*b*Sqrt[c*Sec[a + b*x]]) - (2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*Sqrt[c*Sec[a
+ b*x]])

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Rubi [A]  time = 0.0970203, 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} \[ -\frac{8 c d^3 \sqrt{d \csc (a+b x)}}{5 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b \sqrt{c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(7/2)*Sqrt[c*Sec[a + b*x]],x]

[Out]

(-8*c*d^3*Sqrt[d*Csc[a + b*x]])/(5*b*Sqrt[c*Sec[a + b*x]]) - (2*c*d*(d*Csc[a + b*x])^(5/2))/(5*b*Sqrt[c*Sec[a
+ b*x]])

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))^{7/2} \sqrt{c \sec (a+b x)} \, dx &=-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b \sqrt{c \sec (a+b x)}}+\frac{1}{5} \left (4 d^2\right ) \int (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)} \, dx\\ &=-\frac{8 c d^3 \sqrt{d \csc (a+b x)}}{5 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b \sqrt{c \sec (a+b x)}}\\ \end{align*}

Mathematica [A]  time = 0.116126, size = 56, normalized size = 0.81 \[ -\frac{2 d^3 \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)} (4 \cos (a+b x)+\cot (a+b x) \csc (a+b x))}{5 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(7/2)*Sqrt[c*Sec[a + b*x]],x]

[Out]

(-2*d^3*Sqrt[d*Csc[a + b*x]]*(4*Cos[a + b*x] + Cot[a + b*x]*Csc[a + b*x])*Sqrt[c*Sec[a + b*x]])/(5*b)

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Maple [A]  time = 0.191, size = 54, normalized size = 0.8 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-10 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{5\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{7}{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))^(7/2)*(c*sec(b*x+a))^(1/2),x)

[Out]

2/5/b*(4*cos(b*x+a)^2-5)*cos(b*x+a)*(d/sin(b*x+a))^(7/2)*(c/cos(b*x+a))^(1/2)*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{7}{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))^(7/2)*(c*sec(b*x+a))^(1/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(7/2)*sqrt(c*sec(b*x + a)), x)

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Fricas [A]  time = 1.82622, size = 155, normalized size = 2.25 \begin{align*} -\frac{2 \,{\left (4 \, d^{3} \cos \left (b x + a\right )^{3} - 5 \, d^{3} \cos \left (b x + a\right )\right )} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}}}{5 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(7/2)*(c*sec(b*x+a))^(1/2),x, algorithm="fricas")

[Out]

-2/5*(4*d^3*cos(b*x + a)^3 - 5*d^3*cos(b*x + a))*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))/(b*cos(b*x + a)^2 -
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))**(7/2)*(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{7}{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))^(7/2)*(c*sec(b*x+a))^(1/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(7/2)*sqrt(c*sec(b*x + a)), x)