3.1.8 \(\int \frac {\csc ^3(x)}{a+a \cos (x)} \, dx\) [8]

Optimal. Leaf size=49 \[ -\frac {3 \tanh ^{-1}(\cos (x))}{8 a}-\frac {1}{8 (a-a \cos (x))}+\frac {a}{8 (a+a \cos (x))^2}+\frac {1}{4 (a+a \cos (x))} \]

[Out]

-3/8*arctanh(cos(x))/a-1/8/(a-a*cos(x))+1/8*a/(a+a*cos(x))^2+1/4/(a+a*cos(x))

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Rubi [A]
time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2746, 46, 212} \begin {gather*} \frac {a}{8 (a \cos (x)+a)^2}-\frac {1}{8 (a-a \cos (x))}+\frac {1}{4 (a \cos (x)+a)}-\frac {3 \tanh ^{-1}(\cos (x))}{8 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + a*Cos[x]),x]

[Out]

(-3*ArcTanh[Cos[x]])/(8*a) - 1/(8*(a - a*Cos[x])) + a/(8*(a + a*Cos[x])^2) + 1/(4*(a + a*Cos[x]))

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \frac {\csc ^3(x)}{a+a \cos (x)} \, dx &=-\left (a^3 \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,a \cos (x)\right )\right )\\ &=-\left (a^3 \text {Subst}\left (\int \left (\frac {1}{8 a^3 (a-x)^2}+\frac {1}{4 a^2 (a+x)^3}+\frac {1}{4 a^3 (a+x)^2}+\frac {3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \cos (x)\right )\right )\\ &=-\frac {1}{8 (a-a \cos (x))}+\frac {a}{8 (a+a \cos (x))^2}+\frac {1}{4 (a+a \cos (x))}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \cos (x)\right )\\ &=-\frac {3 \tanh ^{-1}(\cos (x))}{8 a}-\frac {1}{8 (a-a \cos (x))}+\frac {a}{8 (a+a \cos (x))^2}+\frac {1}{4 (a+a \cos (x))}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 60, normalized size = 1.22 \begin {gather*} \frac {4-2 \cot ^2\left (\frac {x}{2}\right )-12 \cos ^2\left (\frac {x}{2}\right ) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+\sec ^2\left (\frac {x}{2}\right )}{16 a (1+\cos (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a + a*Cos[x]),x]

[Out]

(4 - 2*Cot[x/2]^2 - 12*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]) + Sec[x/2]^2)/(16*a*(1 + Cos[x]))

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Maple [A]
time = 0.10, size = 44, normalized size = 0.90

method	result	size
default	\(\frac {\frac {1}{-8+8 \cos \left (x \right )}+\frac {3 \ln \left (-1+\cos \left (x \right )\right )}{16}+\frac {1}{8 \left (\cos \left (x \right )+1\right )^{2}}+\frac {1}{4 \cos \left (x \right )+4}-\frac {3 \ln \left (\cos \left (x \right )+1\right )}{16}}{a}\)	\(44\)
norman	\(\frac {-\frac {1}{16 a}+\frac {3 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16 a}+\frac {\tan ^{6}\left (\frac {x}{2}\right )}{32 a}}{\tan \left (\frac {x}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{8 a}\)	\(47\)
risch	\(\frac {3 \,{\mathrm e}^{5 i x}+6 \,{\mathrm e}^{4 i x}-2 \,{\mathrm e}^{3 i x}+6 \,{\mathrm e}^{2 i x}+3 \,{\mathrm e}^{i x}}{4 \left ({\mathrm e}^{i x}+1\right )^{4} a \left ({\mathrm e}^{i x}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right )}{8 a}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^3/(a+a*cos(x)),x,method=_RETURNVERBOSE)

[Out]

1/a*(1/8/(-1+cos(x))+3/16*ln(-1+cos(x))+1/8/(cos(x)+1)^2+1/4/(cos(x)+1)-3/16*ln(cos(x)+1))

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Maxima [A]
time = 0.27, size = 58, normalized size = 1.18 \begin {gather*} \frac {3 \, \cos \left (x\right )^{2} + 3 \, \cos \left (x\right ) - 2}{8 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} - \frac {3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} + \frac {3 \, \log \left (\cos \left (x\right ) - 1\right )}{16 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*cos(x)),x, algorithm="maxima")

[Out]

1/8*(3*cos(x)^2 + 3*cos(x) - 2)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a) - 3/16*log(cos(x) + 1)/a + 3/16*log(c
os(x) - 1)/a

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Fricas [A]
time = 0.37, size = 83, normalized size = 1.69 \begin {gather*} \frac {6 \, \cos \left (x\right )^{2} - 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 6 \, \cos \left (x\right ) - 4}{16 \, {\left (a \cos \left (x\right )^{3} + a \cos \left (x\right )^{2} - a \cos \left (x\right ) - a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*cos(x)),x, algorithm="fricas")

[Out]

1/16*(6*cos(x)^2 - 3*(cos(x)^3 + cos(x)^2 - cos(x) - 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^3 + cos(x)^2 - cos(x
) - 1)*log(-1/2*cos(x) + 1/2) + 6*cos(x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\csc ^{3}{\left (x \right )}}{\cos {\left (x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**3/(a+a*cos(x)),x)

[Out]

Integral(csc(x)**3/(cos(x) + 1), x)/a

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Giac [A]
time = 0.40, size = 52, normalized size = 1.06 \begin {gather*} -\frac {3 \, \log \left (\cos \left (x\right ) + 1\right )}{16 \, a} + \frac {3 \, \log \left (-\cos \left (x\right ) + 1\right )}{16 \, a} + \frac {3 \, \cos \left (x\right )^{2} + 3 \, \cos \left (x\right ) - 2}{8 \, a {\left (\cos \left (x\right ) + 1\right )}^{2} {\left (\cos \left (x\right ) - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(a+a*cos(x)),x, algorithm="giac")

[Out]

-3/16*log(cos(x) + 1)/a + 3/16*log(-cos(x) + 1)/a + 1/8*(3*cos(x)^2 + 3*cos(x) - 2)/(a*(cos(x) + 1)^2*(cos(x)
- 1))

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Mupad [B]
time = 0.29, size = 45, normalized size = 0.92 \begin {gather*} -\frac {\frac {3\,{\cos \left (x\right )}^2}{8}+\frac {3\,\cos \left (x\right )}{8}-\frac {1}{4}}{-a\,{\cos \left (x\right )}^3-a\,{\cos \left (x\right )}^2+a\,\cos \left (x\right )+a}-\frac {3\,\mathrm {atanh}\left (\cos \left (x\right )\right )}{8\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^3*(a + a*cos(x))),x)

[Out]

- ((3*cos(x))/8 + (3*cos(x)^2)/8 - 1/4)/(a + a*cos(x) - a*cos(x)^2 - a*cos(x)^3) - (3*atanh(cos(x)))/(8*a)

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