3.338 \(\int \csc ^7(x) \, dx\)

Optimal. Leaf size=36 \[ -\frac {5}{16} \tanh ^{-1}(\cos (x))-\frac {1}{6} \cot (x) \csc ^5(x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {5}{16} \cot (x) \csc (x) \]

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3768, 3770} \[ -\frac {5}{16} \tanh ^{-1}(\cos (x))-\frac {1}{6} \cot (x) \csc ^5(x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {5}{16} \cot (x) \csc (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^7,x]

[Out]

(-5*ArcTanh[Cos[x]])/16 - (5*Cot[x]*Csc[x])/16 - (5*Cot[x]*Csc[x]^3)/24 - (Cot[x]*Csc[x]^5)/6

Rule 3768

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \csc ^7(x) \, dx &=-\frac {1}{6} \cot (x) \csc ^5(x)+\frac {5}{6} \int \csc ^5(x) \, dx\\ &=-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x)+\frac {5}{8} \int \csc ^3(x) \, dx\\ &=-\frac {5}{16} \cot (x) \csc (x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x)+\frac {5}{16} \int \csc (x) \, dx\\ &=-\frac {5}{16} \tanh ^{-1}(\cos (x))-\frac {5}{16} \cot (x) \csc (x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x)\\ \end {align*}

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Mathematica [B]  time = 0.01, size = 95, normalized size = 2.64 \[ -\frac {1}{384} \csc ^6\left (\frac {x}{2}\right )-\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {5}{64} \csc ^2\left (\frac {x}{2}\right )+\frac {1}{384} \sec ^6\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {5}{64} \sec ^2\left (\frac {x}{2}\right )+\frac {5}{16} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {5}{16} \log \left (\cos \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^7,x]

[Out]

(-5*Csc[x/2]^2)/64 - Csc[x/2]^4/64 - Csc[x/2]^6/384 - (5*Log[Cos[x/2]])/16 + (5*Log[Sin[x/2]])/16 + (5*Sec[x/2
]^2)/64 + Sec[x/2]^4/64 + Sec[x/2]^6/384

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^7(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Csc[x]^7,x]

[Out]

Could not integrate

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fricas [B]  time = 1.00, size = 93, normalized size = 2.58 \[ \frac {30 \, \cos \relax (x)^{5} - 80 \, \cos \relax (x)^{3} - 15 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 15 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 66 \, \cos \relax (x)}{96 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^7,x, algorithm="fricas")

[Out]

1/96*(30*cos(x)^5 - 80*cos(x)^3 - 15*(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(1/2*cos(x) + 1/2) + 15*(cos(
x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1)*log(-1/2*cos(x) + 1/2) + 66*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 -
1)

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giac [B]  time = 0.62, size = 112, normalized size = 3.11 \[ -\frac {{\left (\frac {9 \, {\left (\cos \relax (x) - 1\right )}}{\cos \relax (x) + 1} - \frac {45 \, {\left (\cos \relax (x) - 1\right )}^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \relax (x) - 1\right )}^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - 1\right )} {\left (\cos \relax (x) + 1\right )}^{3}}{384 \, {\left (\cos \relax (x) - 1\right )}^{3}} - \frac {15 \, {\left (\cos \relax (x) - 1\right )}}{128 \, {\left (\cos \relax (x) + 1\right )}} + \frac {3 \, {\left (\cos \relax (x) - 1\right )}^{2}}{128 \, {\left (\cos \relax (x) + 1\right )}^{2}} - \frac {{\left (\cos \relax (x) - 1\right )}^{3}}{384 \, {\left (\cos \relax (x) + 1\right )}^{3}} + \frac {5}{32} \, \log \left (-\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^7,x, algorithm="giac")

[Out]

-1/384*(9*(cos(x) - 1)/(cos(x) + 1) - 45*(cos(x) - 1)^2/(cos(x) + 1)^2 + 110*(cos(x) - 1)^3/(cos(x) + 1)^3 - 1
)*(cos(x) + 1)^3/(cos(x) - 1)^3 - 15/128*(cos(x) - 1)/(cos(x) + 1) + 3/128*(cos(x) - 1)^2/(cos(x) + 1)^2 - 1/3
84*(cos(x) - 1)^3/(cos(x) + 1)^3 + 5/32*log(-(cos(x) - 1)/(cos(x) + 1))

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maple [A]  time = 0.36, size = 32, normalized size = 0.89

method	result	size
default	\(\left (-\frac {\left (\csc ^{5}\relax (x )\right )}{6}-\frac {5 \left (\csc ^{3}\relax (x )\right )}{24}-\frac {5 \csc \relax (x )}{16}\right ) \cot \relax (x )+\frac {5 \ln \left (\csc \relax (x )-\cot \relax (x )\right )}{16}\)	\(32\)
norman	\(\frac {-\frac {1}{384}-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{128}-\frac {15 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{128}+\frac {15 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{128}+\frac {3 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{128}+\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{384}}{\tan \left (\frac {x}{2}\right )^{6}}+\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{16}\)	\(58\)
risch	\(\frac {15 \,{\mathrm e}^{11 i x}-85 \,{\mathrm e}^{9 i x}+198 \,{\mathrm e}^{7 i x}+198 \,{\mathrm e}^{5 i x}-85 \,{\mathrm e}^{3 i x}+15 \,{\mathrm e}^{i x}}{24 \left ({\mathrm e}^{2 i x}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i x}+1\right )}{16}+\frac {5 \ln \left ({\mathrm e}^{i x}-1\right )}{16}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^7,x,method=_RETURNVERBOSE)

[Out]

(-1/6*csc(x)^5-5/24*csc(x)^3-5/16*csc(x))*cot(x)+5/16*ln(csc(x)-cot(x))

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maxima [A]  time = 0.51, size = 54, normalized size = 1.50 \[ \frac {15 \, \cos \relax (x)^{5} - 40 \, \cos \relax (x)^{3} + 33 \, \cos \relax (x)}{48 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )}} - \frac {5}{32} \, \log \left (\cos \relax (x) + 1\right ) + \frac {5}{32} \, \log \left (\cos \relax (x) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^7,x, algorithm="maxima")

[Out]

1/48*(15*cos(x)^5 - 40*cos(x)^3 + 33*cos(x))/(cos(x)^6 - 3*cos(x)^4 + 3*cos(x)^2 - 1) - 5/32*log(cos(x) + 1) +
5/32*log(cos(x) - 1)

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mupad [B]  time = 0.25, size = 44, normalized size = 1.22 \[ \frac {\frac {5\,{\cos \relax (x)}^5}{16}-\frac {5\,{\cos \relax (x)}^3}{6}+\frac {11\,\cos \relax (x)}{16}}{{\cos \relax (x)}^6-3\,{\cos \relax (x)}^4+3\,{\cos \relax (x)}^2-1}-\frac {5\,\mathrm {atanh}\left (\cos \relax (x)\right )}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x)^7,x)

[Out]

((11*cos(x))/16 - (5*cos(x)^3)/6 + (5*cos(x)^5)/16)/(3*cos(x)^2 - 3*cos(x)^4 + cos(x)^6 - 1) - (5*atanh(cos(x)
))/16

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sympy [A]  time = 0.17, size = 60, normalized size = 1.67 \[ - \frac {- 15 \cos ^{5}{\relax (x )} + 40 \cos ^{3}{\relax (x )} - 33 \cos {\relax (x )}}{48 \cos ^{6}{\relax (x )} - 144 \cos ^{4}{\relax (x )} + 144 \cos ^{2}{\relax (x )} - 48} + \frac {5 \log {\left (\cos {\relax (x )} - 1 \right )}}{32} - \frac {5 \log {\left (\cos {\relax (x )} + 1 \right )}}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**7,x)

[Out]

-(-15*cos(x)**5 + 40*cos(x)**3 - 33*cos(x))/(48*cos(x)**6 - 144*cos(x)**4 + 144*cos(x)**2 - 48) + 5*log(cos(x)
- 1)/32 - 5*log(cos(x) + 1)/32

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