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

Optimal. Leaf size=135 \[ -\frac {2 d^4 \sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{21 b c^2}+\frac {2 d^3 (d \csc (a+b x))^{3/2}}{21 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{7/2}}{7 b c \sqrt {c \sec (a+b x)}} \]

[Out]

2/21*d^3*(d*csc(b*x+a))^(3/2)/b/c/(c*sec(b*x+a))^(1/2)-2/7*d*(d*csc(b*x+a))^(7/2)/b/c/(c*sec(b*x+a))^(1/2)+2/2
1*d^4*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticF(cos(a+1/4*Pi+b*x),2^(1/2))*(d*csc(b*x+a))^(1/2)*
(c*sec(b*x+a))^(1/2)*sin(2*b*x+2*a)^(1/2)/b/c^2

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Rubi [A]  time = 0.20, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2623, 2625, 2630, 2573, 2641} \[ -\frac {2 d^4 \sqrt {\sin (2 a+2 b x)} F\left (\left .a+b x-\frac {\pi }{4}\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}{21 b c^2}+\frac {2 d^3 (d \csc (a+b x))^{3/2}}{21 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{7/2}}{7 b c \sqrt {c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^3*(d*Csc[a + b*x])^(3/2))/(21*b*c*Sqrt[c*Sec[a + b*x]]) - (2*d*(d*Csc[a + b*x])^(7/2))/(7*b*c*Sqrt[c*Sec[
a + b*x]]) - (2*d^4*Sqrt[d*Csc[a + b*x]]*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[2*a + 2*b*
x]])/(21*b*c^2)

Rule 2573

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]

Rule 2623

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(a*(a*Csc[e
+ f*x])^(m - 1)*(b*Sec[e + f*x])^(n + 1))/(f*b*(m - 1)), x] + Dist[(a^2*(n + 1))/(b^2*(m - 1)), Int[(a*Csc[e +
f*x])^(m - 2)*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && Intege
rsQ[2*m, 2*n]

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 2630

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Csc[e + f*
x])^m*(b*Sec[e + f*x])^n*(a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n, Int[1/((a*Sin[e + f*x])^m*(b*Cos[e + f*x])^n),
x], x] /; FreeQ[{a, b, e, f, m, n}, x] && IntegerQ[m - 1/2] && IntegerQ[n - 1/2]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {(d \csc (a+b x))^{9/2}}{(c \sec (a+b x))^{3/2}} \, dx &=-\frac {2 d (d \csc (a+b x))^{7/2}}{7 b c \sqrt {c \sec (a+b x)}}-\frac {d^2 \int (d \csc (a+b x))^{5/2} \sqrt {c \sec (a+b x)} \, dx}{7 c^2}\\ &=\frac {2 d^3 (d \csc (a+b x))^{3/2}}{21 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{7/2}}{7 b c \sqrt {c \sec (a+b x)}}-\frac {\left (2 d^4\right ) \int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \, dx}{21 c^2}\\ &=\frac {2 d^3 (d \csc (a+b x))^{3/2}}{21 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{7/2}}{7 b c \sqrt {c \sec (a+b x)}}-\frac {\left (2 d^4 \sqrt {c \cos (a+b x)} \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {d \sin (a+b x)}\right ) \int \frac {1}{\sqrt {c \cos (a+b x)} \sqrt {d \sin (a+b x)}} \, dx}{21 c^2}\\ &=\frac {2 d^3 (d \csc (a+b x))^{3/2}}{21 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{7/2}}{7 b c \sqrt {c \sec (a+b x)}}-\frac {\left (2 d^4 \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{21 c^2}\\ &=\frac {2 d^3 (d \csc (a+b x))^{3/2}}{21 b c \sqrt {c \sec (a+b x)}}-\frac {2 d (d \csc (a+b x))^{7/2}}{7 b c \sqrt {c \sec (a+b x)}}-\frac {2 d^4 \sqrt {d \csc (a+b x)} F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{21 b c^2}\\ \end {align*}

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Mathematica [C]  time = 1.46, size = 119, normalized size = 0.88 \[ -\frac {d^3 \cos (2 (a+b x)) (d \csc (a+b x))^{3/2} \left ((\cos (2 (a+b x))+5) \csc ^4(a+b x)-2 \left (-\cot ^2(a+b x)\right )^{3/4} \sec ^2(a+b x) \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\csc ^2(a+b x)\right )\right )}{21 b c \left (\csc ^2(a+b x)-2\right ) \sqrt {c \sec (a+b x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(d^3*Cos[2*(a + b*x)]*(d*Csc[a + b*x])^(3/2)*((5 + Cos[2*(a + b*x)])*Csc[a + b*x]^4 - 2*(-Cot[a + b*x]^2
)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, Csc[a + b*x]^2]*Sec[a + b*x]^2))/(b*c*(-2 + Csc[a + b*x]^2)*Sqrt[c*Se
c[a + b*x]])

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fricas [F]  time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d \csc \left (b x + a\right )} \sqrt {c \sec \left (b x + a\right )} d^{4} \csc \left (b x + a\right )^{4}}{c^{2} \sec \left (b x + a\right )^{2}}, x\right ) \]

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]

integral(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a))*d^4*csc(b*x + a)^4/(c^2*sec(b*x + a)^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}}}{\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]

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)

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maple [B]  time = 1.23, size = 542, normalized size = 4.01 \[ \frac {\left (2 \sin \left (b x +a \right ) \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+2 \sin \left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-\left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}-2 \cos \left (b x +a \right ) \sqrt {2}\right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {9}{2}} \sin \left (b x +a \right ) \sqrt {2}}{21 b \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \cos \left (b x +a \right )^{2}} \]

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]

1/21/b*(2*sin(b*x+a)*cos(b*x+a)^3*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin
(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2
^(1/2))+2*sin(b*x+a)*cos(b*x+a)^2*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin
(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2
^(1/2))-2*sin(b*x+a)*cos(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b
*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(
1/2))-2*sin(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*
((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-cos(b*x
+a)^3*2^(1/2)-2*cos(b*x+a)*2^(1/2))*(d/sin(b*x+a))^(9/2)*sin(b*x+a)/(c/cos(b*x+a))^(3/2)/cos(b*x+a)^2*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \csc \left (b x + a\right )\right )^{\frac {9}{2}}}{\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{9/2}}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d/sin(a + b*x))^(9/2)/(c/cos(a + b*x))^(3/2),x)

[Out]

int((d/sin(a + b*x))^(9/2)/(c/cos(a + b*x))^(3/2), x)

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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))**(9/2)/(c*sec(b*x+a))**(3/2),x)

[Out]

Timed out

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