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]

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

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*}

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Mathematica [A]  time = 0.21, 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)

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

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))

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

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)

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maple [A]  time = 1.04, size = 54, normalized size = 0.78 \[ -\frac {2 \left (4 \left (\cos ^{2}\left (b x +a \right )\right )-1\right ) \cos \left (b x +a \right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {3}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \sin \left (b x +a \right )}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(b*x+a))^(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)

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

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)

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mupad [B]  time = 0.83, size = 64, normalized size = 0.93 \[ -\frac {4\,c^2\,d\,\sqrt {\frac {c}{\cos \left (a+b\,x\right )}}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}\,\left (2\,\cos \left (a+b\,x\right )+\cos \left (3\,a+3\,b\,x\right )\right )}{3\,b\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))**(3/2)*(c*sec(b*x+a))**(5/2),x)

[Out]

Timed out

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