3.3.78 \(\int \frac {1}{(d \csc (a+b x))^{5/2} (c \sec (a+b x))^{5/2}} \, dx\) [278]

Optimal. Leaf size=135 \[ -\frac {c}{5 b d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{7/2}}+\frac {1}{10 b c d (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2}}+\frac {3 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{20 b c^2 d^2 \sqrt {d \csc (a+b x)} \sqrt {c \sec (a+b x)} \sqrt {\sin (2 a+2 b x)}} \]

[Out]

-1/5*c/b/d/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(7/2)+1/10/b/c/d/(d*csc(b*x+a))^(3/2)/(c*sec(b*x+a))^(3/2)-3/20
*(sin(a+1/4*Pi+b*x)^2)^(1/2)/sin(a+1/4*Pi+b*x)*EllipticE(cos(a+1/4*Pi+b*x),2^(1/2))/b/c^2/d^2/(d*csc(b*x+a))^(
1/2)/(c*sec(b*x+a))^(1/2)/sin(2*b*x+2*a)^(1/2)

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Rubi [A]
time = 0.14, 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 = {2707, 2708, 2710, 2652, 2719} \begin {gather*} \frac {3 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{20 b c^2 d^2 \sqrt {\sin (2 a+2 b x)} \sqrt {c \sec (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {c}{5 b d (c \sec (a+b x))^{7/2} (d \csc (a+b x))^{3/2}}+\frac {1}{10 b c d (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(5/2)),x]

[Out]

-1/5*c/(b*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x])^(7/2)) + 1/(10*b*c*d*(d*Csc[a + b*x])^(3/2)*(c*Sec[a + b*x
])^(3/2)) + (3*EllipticE[a - Pi/4 + b*x, 2])/(20*b*c^2*d^2*Sqrt[d*Csc[a + b*x]]*Sqrt[c*Sec[a + b*x]]*Sqrt[Sin[
2*a + 2*b*x]])

Rule 2652

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

Rule 2707

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[b*(a*Csc[e +
f*x])^(m + 1)*((b*Sec[e + f*x])^(n - 1)/(a*f*(m + n))), x] + Dist[(m + 1)/(a^2*(m + n)), Int[(a*Csc[e + f*x])
^(m + 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && NeQ[m + n, 0] && IntegersQ[2
*m, 2*n]

Rule 2708

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)/(b*f*(m + n))), x] + Dist[(n + 1)/(b^2*(m + n)), Int[(a*Csc[e + f*
x])^m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m + n, 0] && Integers
Q[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 2719

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.68, size = 90, normalized size = 0.67 \begin {gather*} \frac {\left (-2 \cos ^2(a+b x) \cos (2 (a+b x))+3 \sqrt [4]{-\cot ^2(a+b x)} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\csc ^2(a+b x)\right )\right ) \sqrt {c \sec (a+b x)}}{20 b c^3 d (d \csc (a+b x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d*Csc[a + b*x])^(5/2)*(c*Sec[a + b*x])^(5/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(535\) vs. \(2(140)=280\).
time = 36.21, size = 536, normalized size = 3.97

method	result	size
default	\(\frac {\left (4 \sqrt {2}\, \left (\cos ^{6}\left (b x +a \right )\right )-6 \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )+3 \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 {\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 {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-6 \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 {\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 )}}\, \EllipticE \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 )+3 \sqrt {\frac {1-\cos \left (b x +a \right )+\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 {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )-6 \sqrt {\frac {1-\cos \left (b x +a \right )+\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 )}}\, \EllipticE \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 ^{2}\left (b x +a \right )\right ) \sqrt {2}+3 \sqrt {2}\, \cos \left (b x +a \right )\right ) \sqrt {2}}{40 b \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{3} \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {5}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}}}\)	\(536\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/40/b*(4*2^(1/2)*cos(b*x+a)^6-6*cos(b*x+a)^4*2^(1/2)+3*cos(b*x+a)*((1-cos(b*x+a)+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(((1-cos(b*x+a)+sin
(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-6*cos(b*x+a)*((1-cos(b*x+a)+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)*EllipticE(((1-cos(b*x+a)+sin(b*x+a))/sin(b
*x+a))^(1/2),1/2*2^(1/2))+3*((1-cos(b*x+a)+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(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))
-6*((1-cos(b*x+a)+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)*EllipticE(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2*2^(1/2))-cos(b*x+a)^2*2^(1/2)+3*2
^(1/2)*cos(b*x+a))/sin(b*x+a)^3/cos(b*x+a)^3/(d/sin(b*x+a))^(5/2)/(c/cos(b*x+a))^(5/2)*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(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((d*csc(b*x + a))^(5/2)*(c*sec(b*x + a))^(5/2)), x)

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Fricas [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(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*csc(b*x + a))*sqrt(c*sec(b*x + a))/(c^3*d^3*csc(b*x + a)^3*sec(b*x + a)^3), x)

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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(1/(d*csc(b*x+a))**(5/2)/(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(1/(d*csc(b*x+a))^(5/2)/(c*sec(b*x+a))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((d*csc(b*x + a))^(5/2)*(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 \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(5/2)),x)

[Out]

int(1/((c/cos(a + b*x))^(5/2)*(d/sin(a + b*x))^(5/2)), x)

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