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]

-2/3*c*d*(d*csc(b*x+a))^(3/2)*(c*sec(b*x+a))^(1/2)/b+8/3*c*d^3*(c*sec(b*x+a))^(1/2)/b/(d*csc(b*x+a))^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]

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]

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*}

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Mathematica [A]  time = 0.14, 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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fricas [A]  time = 0.96, size = 58, normalized size = 0.84 \[ -\frac {2 \, {\left (4 \, c d^{2} \cos \left (b x + a\right )^{2} - 3 \, c d^{2}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}}{3 \, b \sin \left (b x + a\right )} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]

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)

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maple [A]  time = 1.04, size = 54, normalized size = 0.78 \[ -\frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-3\right ) \cos \left (b x +a \right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {5}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )}{3 b} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \csc \left (b x + a\right )\right )^{\frac {5}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}\,{d x} \]

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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mupad [B]  time = 0.76, size = 61, normalized size = 0.88 \[ \frac {2\,c\,d^2\,\left (2\,\sin \left (a+b\,x\right )-\sin \left (3\,a+3\,b\,x\right )\right )\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}}{3\,b\,{\sin \left (a+b\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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