3.4.84 \(\int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx\) [384]

Optimal. Leaf size=123 \[ -\frac {3 b \csc (e+f x)}{4 f \sqrt {b \sec (e+f x)}}-\frac {3 b \csc ^3(e+f x)}{10 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}+\frac {3 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{4 f} \]

[Out]

-3/4*b*csc(f*x+e)/f/(b*sec(f*x+e))^(1/2)-3/10*b*csc(f*x+e)^3/f/(b*sec(f*x+e))^(1/2)-1/5*b*csc(f*x+e)^5/f/(b*se
c(f*x+e))^(1/2)+3/4*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*EllipticF(sin(1/2*f*x+1/2*e),2^(1/2))*cos(
f*x+e)^(1/2)*(b*sec(f*x+e))^(1/2)/f

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Rubi [A]
time = 0.10, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2705, 3856, 2720} \begin {gather*} -\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}-\frac {3 b \csc ^3(e+f x)}{10 f \sqrt {b \sec (e+f x)}}-\frac {3 b \csc (e+f x)}{4 f \sqrt {b \sec (e+f x)}}+\frac {3 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^6*Sqrt[b*Sec[e + f*x]],x]

[Out]

(-3*b*Csc[e + f*x])/(4*f*Sqrt[b*Sec[e + f*x]]) - (3*b*Csc[e + f*x]^3)/(10*f*Sqrt[b*Sec[e + f*x]]) - (b*Csc[e +
f*x]^5)/(5*f*Sqrt[b*Sec[e + f*x]]) + (3*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/(4
*f)

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 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]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps

\begin {align*} \int \csc ^6(e+f x) \sqrt {b \sec (e+f x)} \, dx &=-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}+\frac {9}{10} \int \csc ^4(e+f x) \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {3 b \csc ^3(e+f x)}{10 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}+\frac {3}{4} \int \csc ^2(e+f x) \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {3 b \csc (e+f x)}{4 f \sqrt {b \sec (e+f x)}}-\frac {3 b \csc ^3(e+f x)}{10 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}+\frac {3}{8} \int \sqrt {b \sec (e+f x)} \, dx\\ &=-\frac {3 b \csc (e+f x)}{4 f \sqrt {b \sec (e+f x)}}-\frac {3 b \csc ^3(e+f x)}{10 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}+\frac {1}{8} \left (3 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx\\ &=-\frac {3 b \csc (e+f x)}{4 f \sqrt {b \sec (e+f x)}}-\frac {3 b \csc ^3(e+f x)}{10 f \sqrt {b \sec (e+f x)}}-\frac {b \csc ^5(e+f x)}{5 f \sqrt {b \sec (e+f x)}}+\frac {3 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{4 f}\\ \end {align*}

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Mathematica [A]
time = 0.32, size = 73, normalized size = 0.59 \begin {gather*} \frac {\left (-\cot (e+f x) \left (15+6 \csc ^2(e+f x)+4 \csc ^4(e+f x)\right )+15 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )\right ) \sqrt {b \sec (e+f x)}}{20 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[e + f*x]^6*Sqrt[b*Sec[e + f*x]],x]

[Out]

((-(Cot[e + f*x]*(15 + 6*Csc[e + f*x]^2 + 4*Csc[e + f*x]^4)) + 15*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]
)*Sqrt[b*Sec[e + f*x]])/(20*f)

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Maple [C] Result contains complex when optimal does not.
time = 0.29, size = 485, normalized size = 3.94

method	result	size
default	\(\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (15 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{5}\left (f x +e \right )\right ) \sin \left (f x +e \right )+15 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-30 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-30 i \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+15 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right ) \cos \left (f x +e \right )+15 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )-15 \left (\cos ^{5}\left (f x +e \right )\right )+36 \left (\cos ^{3}\left (f x +e \right )\right )-25 \cos \left (f x +e \right )\right ) \left (\cos \left (f x +e \right )+1\right )^{2} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}{20 f \sin \left (f x +e \right )^{9}}\)	\(485\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/20/f*(-1+cos(f*x+e))^2*(15*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/
(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^5*sin(f*x+e)+15*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1)
)^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^4*sin(f*x+e)-30*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),
I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^3*sin(f*x+e)-30*I*EllipticF(I*(-1+cos
(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2*sin(f*x+e)+15*I
*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+
e)*cos(f*x+e)+15*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+
1))^(1/2)*sin(f*x+e)-15*cos(f*x+e)^5+36*cos(f*x+e)^3-25*cos(f*x+e))*(cos(f*x+e)+1)^2*(b/cos(f*x+e))^(1/2)/sin(
f*x+e)^9

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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(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^6, x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.12, size = 204, normalized size = 1.66 \begin {gather*} -\frac {15 \, \sqrt {2} {\left (i \, \cos \left (f x + e\right )^{4} - 2 i \, \cos \left (f x + e\right )^{2} + i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 15 \, \sqrt {2} {\left (-i \, \cos \left (f x + e\right )^{4} + 2 i \, \cos \left (f x + e\right )^{2} - i\right )} \sqrt {b} \sin \left (f x + e\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (15 \, \cos \left (f x + e\right )^{5} - 36 \, \cos \left (f x + e\right )^{3} + 25 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{40 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )} \sin \left (f x + e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-1/40*(15*sqrt(2)*(I*cos(f*x + e)^4 - 2*I*cos(f*x + e)^2 + I)*sqrt(b)*sin(f*x + e)*weierstrassPInverse(-4, 0,
cos(f*x + e) + I*sin(f*x + e)) + 15*sqrt(2)*(-I*cos(f*x + e)^4 + 2*I*cos(f*x + e)^2 - I)*sqrt(b)*sin(f*x + e)*
weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e)) + 2*(15*cos(f*x + e)^5 - 36*cos(f*x + e)^3 + 25*cos(
f*x + e))*sqrt(b/cos(f*x + e)))/((f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)*sin(f*x + e))

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)**6*(b*sec(f*x+e))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3004 deep

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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(csc(f*x+e)^6*(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(f*x + e))*csc(f*x + e)^6, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}}{{\sin \left (e+f\,x\right )}^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^6,x)

[Out]

int((b/cos(e + f*x))^(1/2)/sin(e + f*x)^6, x)

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