3.238 \(\int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{3/2} \, dx\)

Optimal. Leaf size=104 \[ \frac{64 c d^5 \sqrt{c \sec (a+b x)}}{21 b \sqrt{d \csc (a+b x)}}-\frac{16 c d^3 \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{21 b}-\frac{2 c d \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b} \]

[Out]

(64*c*d^5*Sqrt[c*Sec[a + b*x]])/(21*b*Sqrt[d*Csc[a + b*x]]) - (16*c*d^3*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a +
b*x]])/(21*b) - (2*c*d*(d*Csc[a + b*x])^(7/2)*Sqrt[c*Sec[a + b*x]])/(7*b)

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Rubi [A]  time = 0.162464, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2625, 2619} \[ \frac{64 c d^5 \sqrt{c \sec (a+b x)}}{21 b \sqrt{d \csc (a+b x)}}-\frac{16 c d^3 \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{3/2}}{21 b}-\frac{2 c d \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

(64*c*d^5*Sqrt[c*Sec[a + b*x]])/(21*b*Sqrt[d*Csc[a + b*x]]) - (16*c*d^3*(d*Csc[a + b*x])^(3/2)*Sqrt[c*Sec[a +
b*x]])/(21*b) - (2*c*d*(d*Csc[a + b*x])^(7/2)*Sqrt[c*Sec[a + b*x]])/(7*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))^{9/2} (c \sec (a+b x))^{3/2} \, dx &=-\frac{2 c d (d \csc (a+b x))^{7/2} \sqrt{c \sec (a+b x)}}{7 b}+\frac{1}{7} \left (8 d^2\right ) \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{3/2} \, dx\\ &=-\frac{16 c d^3 (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}{21 b}-\frac{2 c d (d \csc (a+b x))^{7/2} \sqrt{c \sec (a+b x)}}{7 b}+\frac{1}{21} \left (32 d^4\right ) \int \sqrt{d \csc (a+b x)} (c \sec (a+b x))^{3/2} \, dx\\ &=\frac{64 c d^5 \sqrt{c \sec (a+b x)}}{21 b \sqrt{d \csc (a+b x)}}-\frac{16 c d^3 (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)}}{21 b}-\frac{2 c d (d \csc (a+b x))^{7/2} \sqrt{c \sec (a+b x)}}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.298261, size = 57, normalized size = 0.55 \[ -\frac{2 c d^5 \left (3 \csc ^4(a+b x)+8 \csc ^2(a+b x)-32\right ) \sqrt{c \sec (a+b x)}}{21 b \sqrt{d \csc (a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(3/2),x]

[Out]

(-2*c*d^5*(-32 + 8*Csc[a + b*x]^2 + 3*Csc[a + b*x]^4)*Sqrt[c*Sec[a + b*x]])/(21*b*Sqrt[d*Csc[a + b*x]])

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Maple [A]  time = 0.196, size = 64, normalized size = 0.6 \begin{align*}{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}-112\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+42 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{21\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{9}{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))^(9/2)*(c*sec(b*x+a))^(3/2),x)

[Out]

2/21/b*(32*cos(b*x+a)^4-56*cos(b*x+a)^2+21)*cos(b*x+a)*(d/sin(b*x+a))^(9/2)*(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{9}{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))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(3/2), x)

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Fricas [A]  time = 2.44703, size = 203, normalized size = 1.95 \begin{align*} -\frac{2 \,{\left (32 \, c d^{4} \cos \left (b x + a\right )^{4} - 56 \, c d^{4} \cos \left (b x + a\right )^{2} + 21 \, c d^{4}\right )} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}}}{21 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \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))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="fricas")

[Out]

-2/21*(32*c*d^4*cos(b*x + a)^4 - 56*c*d^4*cos(b*x + a)^2 + 21*c*d^4)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a))
/((b*cos(b*x + a)^2 - 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))**(9/2)*(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{9}{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))^(9/2)*(c*sec(b*x+a))^(3/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(3/2), x)