3.3.46 \(\int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx\) [246]

Optimal. Leaf size=166 \[ \frac {40 c d^5 (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}}-\frac {20 c d^3 (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {40 c^2 d^4 \sqrt {d \csc (a+b x)} F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}}{21 b} \]

[Out]

-20/21*c*d^3*(d*csc(b*x+a))^(3/2)*(c*sec(b*x+a))^(3/2)/b-2/7*c*d*(d*csc(b*x+a))^(7/2)*(c*sec(b*x+a))^(3/2)/b+4
0/21*c*d^5*(c*sec(b*x+a))^(3/2)/b/(d*csc(b*x+a))^(1/2)-40/21*c^2*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

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Rubi [A]
time = 0.18, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2705, 2706, 2710, 2653, 2720} \begin {gather*} \frac {40 c^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}+\frac {40 c d^5 (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}}-\frac {20 c d^3 (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}}{21 b}-\frac {2 c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{7/2}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(5/2),x]

[Out]

(40*c*d^5*(c*Sec[a + b*x])^(3/2))/(21*b*Sqrt[d*Csc[a + b*x]]) - (20*c*d^3*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*
x])^(3/2))/(21*b) - (2*c*d*(d*Csc[a + b*x])^(7/2)*(c*Sec[a + b*x])^(3/2))/(7*b) + (40*c^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)

Rule 2653

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 2705

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[(-a)*b*(a*Cs
c[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 2706

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]

Rule 2710

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 2720

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

Rubi steps

\begin {align*} \int (d \csc (a+b x))^{9/2} (c \sec (a+b x))^{5/2} \, dx &=-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {1}{7} \left (10 d^2\right ) \int (d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2} \, dx\\ &=-\frac {20 c d^3 (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {1}{7} \left (20 d^4\right ) \int \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{5/2} \, dx\\ &=\frac {40 c d^5 (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}}-\frac {20 c d^3 (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {1}{21} \left (40 c^2 d^4\right ) \int \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \, dx\\ &=\frac {40 c d^5 (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}}-\frac {20 c d^3 (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {1}{21} \left (40 c^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\\ &=\frac {40 c d^5 (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}}-\frac {20 c d^3 (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {1}{21} \left (40 c^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\\ &=\frac {40 c d^5 (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}}-\frac {20 c d^3 (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}{21 b}-\frac {2 c d (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2}}{7 b}+\frac {40 c^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}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 1.51, size = 92, normalized size = 0.55 \begin {gather*} -\frac {2 c d^5 \left (-7+\cot ^2(a+b x) \left (13+3 \csc ^2(a+b x)\right )+20 \left (-\cot ^2(a+b x)\right )^{3/4} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {3}{2};\csc ^2(a+b x)\right )\right ) (c \sec (a+b x))^{3/2}}{21 b \sqrt {d \csc (a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*Csc[a + b*x])^(9/2)*(c*Sec[a + b*x])^(5/2),x]

[Out]

(-2*c*d^5*(-7 + Cot[a + b*x]^2*(13 + 3*Csc[a + b*x]^2) + 20*(-Cot[a + b*x]^2)^(3/4)*Hypergeometric2F1[1/2, 3/4
, 3/2, Csc[a + b*x]^2])*(c*Sec[a + b*x])^(3/2))/(21*b*Sqrt[d*Csc[a + b*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(562\) vs. \(2(165)=330\).
time = 55.39, size = 563, normalized size = 3.39

method	result	size
default	\(\frac {\left (-40 \left (\cos ^{4}\left (b x +a \right )\right ) \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\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 {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-40 \left (\cos ^{3}\left (b x +a \right )\right ) \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\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 {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+40 \left (\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\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 {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+40 \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\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 {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+20 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )-30 \left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}+7 \sqrt {2}\right ) \cos \left (b x +a \right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {9}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \sin \left (b x +a \right ) \sqrt {2}}{21 b}\)	\(563\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/21/b*(-40*cos(b*x+a)^4*sin(b*x+a)*(-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1+sin(b*x+a))/s
in(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),1/
2*2^(1/2))-40*cos(b*x+a)^3*sin(b*x+a)*(-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1+sin(b*x+a))
/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2),
1/2*2^(1/2))+40*cos(b*x+a)^2*sin(b*x+a)*(-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1+sin(b*x+a
))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2
),1/2*2^(1/2))+40*cos(b*x+a)*sin(b*x+a)*(-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2)*((cos(b*x+a)-1+sin(b*x+a
))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticF((-(cos(b*x+a)-1-sin(b*x+a))/sin(b*x+a))^(1/2
),1/2*2^(1/2))+20*2^(1/2)*cos(b*x+a)^4-30*cos(b*x+a)^2*2^(1/2)+7*2^(1/2))*cos(b*x+a)*(d/sin(b*x+a))^(9/2)*(c/c
os(b*x+a))^(5/2)*sin(b*x+a)*2^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(5/2), x)

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Fricas [C] Result contains complex when optimal does not.
time = 0.88, size = 223, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (10 \, {\left (i \, c^{2} d^{4} \cos \left (b x + a\right )^{3} - i \, c^{2} d^{4} \cos \left (b x + a\right )\right )} \sqrt {-4 i \, c d} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) \sin \left (b x + a\right ) + 10 \, {\left (-i \, c^{2} d^{4} \cos \left (b x + a\right )^{3} + i \, c^{2} d^{4} \cos \left (b x + a\right )\right )} \sqrt {4 i \, c d} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) \sin \left (b x + a\right ) + {\left (20 \, c^{2} d^{4} \cos \left (b x + a\right )^{4} - 30 \, c^{2} d^{4} \cos \left (b x + a\right )^{2} + 7 \, c^{2} d^{4}\right )} \sqrt {\frac {c}{\cos \left (b x + a\right )}} \sqrt {\frac {d}{\sin \left (b x + a\right )}}\right )}}{21 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

-2/21*(10*(I*c^2*d^4*cos(b*x + a)^3 - I*c^2*d^4*cos(b*x + a))*sqrt(-4*I*c*d)*ellipticF(cos(b*x + a) + I*sin(b*
x + a), -1)*sin(b*x + a) + 10*(-I*c^2*d^4*cos(b*x + a)^3 + I*c^2*d^4*cos(b*x + a))*sqrt(4*I*c*d)*ellipticF(cos
(b*x + a) - I*sin(b*x + a), -1)*sin(b*x + a) + (20*c^2*d^4*cos(b*x + a)^4 - 30*c^2*d^4*cos(b*x + a)^2 + 7*c^2*
d^4)*sqrt(c/cos(b*x + a))*sqrt(d/sin(b*x + a)))/((b*cos(b*x + a)^3 - b*cos(b*x + a))*sin(b*x + a))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))**(9/2)*(c*sec(b*x+a))**(5/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*csc(b*x+a))^(9/2)*(c*sec(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate((d*csc(b*x + a))^(9/2)*(c*sec(b*x + a))^(5/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{9/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(9/2),x)

[Out]

int((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(9/2), x)

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