3.18 \(\int \frac {\csc ^3(x)}{a+b \cot (x)} \, dx\)

Optimal. Leaf size=53 \[ \frac {\sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{b^2}+\frac {a \tanh ^{-1}(\cos (x))}{b^2}-\frac {\csc (x)}{b} \]

[Out]

a*arctanh(cos(x))/b^2-csc(x)/b+arctanh((b-a*cot(x))*sin(x)/(a^2+b^2)^(1/2))*(a^2+b^2)^(1/2)/b^2

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Rubi [A]  time = 0.08, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3510, 3486, 3770, 3509, 206} \[ \frac {\sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {\sin (x) (b-a \cot (x))}{\sqrt {a^2+b^2}}\right )}{b^2}+\frac {a \tanh ^{-1}(\cos (x))}{b^2}-\frac {\csc (x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(a + b*Cot[x]),x]

[Out]

(a*ArcTanh[Cos[x]])/b^2 + (Sqrt[a^2 + b^2]*ArcTanh[((b - a*Cot[x])*Sin[x])/Sqrt[a^2 + b^2]])/b^2 - Csc[x]/b

Rule 206

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

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 3509

Int[sec[(e_.) + (f_.)*(x_)]/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Dist[f^(-1), Subst[Int[1/(a^
2 + b^2 - x^2), x], x, (b - a*Tan[e + f*x])/Sec[e + f*x]], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 + b^2, 0]

Rule 3510

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Dist[d^2/b^2, I
nt[(d*Sec[e + f*x])^(m - 2)*(a - b*Tan[e + f*x]), x], x] + Dist[(d^2*(a^2 + b^2))/b^2, Int[(d*Sec[e + f*x])^(m
- 2)/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 1]

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 \frac {\csc ^3(x)}{a+b \cot (x)} \, dx &=-\frac {\int (a-b \cot (x)) \csc (x) \, dx}{b^2}+\frac {\left (a^2+b^2\right ) \int \frac {\csc (x)}{a+b \cot (x)} \, dx}{b^2}\\ &=-\frac {\csc (x)}{b}-\frac {a \int \csc (x) \, dx}{b^2}-\frac {\left (a^2+b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,(-b+a \cot (x)) \sin (x)\right )}{b^2}\\ &=\frac {a \tanh ^{-1}(\cos (x))}{b^2}+\frac {\sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {(b-a \cot (x)) \sin (x)}{\sqrt {a^2+b^2}}\right )}{b^2}-\frac {\csc (x)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 67, normalized size = 1.26 \[ \frac {2 \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {b \tan \left (\frac {x}{2}\right )-a}{\sqrt {a^2+b^2}}\right )+a \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )-b \csc (x)}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(a + b*Cot[x]),x]

[Out]

(2*Sqrt[a^2 + b^2]*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]] - b*Csc[x] + a*(Log[Cos[x/2]] - Log[Sin[x/2]]))/
b^2

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fricas [B]  time = 0.70, size = 135, normalized size = 2.55 \[ \frac {a \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \sin \relax (x) - a \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \sin \relax (x) + \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} - a^{2} - 2 \, b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (a \cos \relax (x) - b \sin \relax (x)\right )}}{2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}}\right ) \sin \relax (x) - 2 \, b}{2 \, b^{2} \sin \relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*log(1/2*cos(x) + 1/2)*sin(x) - a*log(-1/2*cos(x) + 1/2)*sin(x) + sqrt(a^2 + b^2)*log(-(2*a*b*cos(x)*sin
(x) - (a^2 - b^2)*cos(x)^2 - a^2 - 2*b^2 + 2*sqrt(a^2 + b^2)*(a*cos(x) - b*sin(x)))/(2*a*b*cos(x)*sin(x) - (a^
2 - b^2)*cos(x)^2 + a^2))*sin(x) - 2*b)/(b^2*sin(x))

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giac [B]  time = 0.41, size = 108, normalized size = 2.04 \[ -\frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{b^{2}} - \frac {\tan \left (\frac {1}{2} \, x\right )}{2 \, b} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b \tan \left (\frac {1}{2} \, x\right ) - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{b^{2}} + \frac {2 \, a \tan \left (\frac {1}{2} \, x\right ) - b}{2 \, b^{2} \tan \left (\frac {1}{2} \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*log(abs(tan(1/2*x)))/b^2 - 1/2*tan(1/2*x)/b - sqrt(a^2 + b^2)*log(abs(2*b*tan(1/2*x) - 2*a - 2*sqrt(a^2 + b
^2))/abs(2*b*tan(1/2*x) - 2*a + 2*sqrt(a^2 + b^2)))/b^2 + 1/2*(2*a*tan(1/2*x) - b)/(b^2*tan(1/2*x))

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maple [B]  time = 0.27, size = 107, normalized size = 2.02 \[ -\frac {\tan \left (\frac {x}{2}\right )}{2 b}+\frac {2 \arctanh \left (\frac {2 \tan \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right ) a^{2}}{b^{2} \sqrt {a^{2}+b^{2}}}+\frac {2 \arctanh \left (\frac {2 \tan \left (\frac {x}{2}\right ) b -2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}-\frac {1}{2 b \tan \left (\frac {x}{2}\right )}-\frac {a \ln \left (\tan \left (\frac {x}{2}\right )\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^3/(a+b*cot(x)),x)

[Out]

-1/2/b*tan(1/2*x)+2/b^2/(a^2+b^2)^(1/2)*arctanh(1/2*(2*tan(1/2*x)*b-2*a)/(a^2+b^2)^(1/2))*a^2+2/(a^2+b^2)^(1/2
)*arctanh(1/2*(2*tan(1/2*x)*b-2*a)/(a^2+b^2)^(1/2))-1/2/b/tan(1/2*x)-1/b^2*a*ln(tan(1/2*x))

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maxima [B]  time = 1.12, size = 107, normalized size = 2.02 \[ -\frac {a \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{b^{2}} - \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} + \sqrt {a^{2} + b^{2}}}{a - \frac {b \sin \relax (x)}{\cos \relax (x) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{b^{2}} - \frac {\cos \relax (x) + 1}{2 \, b \sin \relax (x)} - \frac {\sin \relax (x)}{2 \, b {\left (\cos \relax (x) + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*log(sin(x)/(cos(x) + 1))/b^2 - sqrt(a^2 + b^2)*log((a - b*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(a - b*sin
(x)/(cos(x) + 1) - sqrt(a^2 + b^2)))/b^2 - 1/2*(cos(x) + 1)/(b*sin(x)) - 1/2*sin(x)/(b*(cos(x) + 1))

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mupad [B]  time = 0.49, size = 170, normalized size = 3.21 \[ \frac {2\,\mathrm {atanh}\left (\frac {b^3\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}+4\,a^3\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}+3\,a\,b^2\,\sin \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}+2\,a^2\,b\,\cos \left (\frac {x}{2}\right )\,\sqrt {a^2+b^2}}{4\,\sin \left (\frac {x}{2}\right )\,a^4+2\,\cos \left (\frac {x}{2}\right )\,a^3\,b+5\,\sin \left (\frac {x}{2}\right )\,a^2\,b^2+2\,\cos \left (\frac {x}{2}\right )\,a\,b^3+\sin \left (\frac {x}{2}\right )\,b^4}\right )\,\sqrt {a^2+b^2}}{b^2}-\frac {1}{b\,\sin \relax (x)}-\frac {a\,\ln \left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^3*(a + b*cot(x))),x)

[Out]

(2*atanh((b^3*cos(x/2)*(a^2 + b^2)^(1/2) + 4*a^3*sin(x/2)*(a^2 + b^2)^(1/2) + 3*a*b^2*sin(x/2)*(a^2 + b^2)^(1/
2) + 2*a^2*b*cos(x/2)*(a^2 + b^2)^(1/2))/(4*a^4*sin(x/2) + b^4*sin(x/2) + 5*a^2*b^2*sin(x/2) + 2*a*b^3*cos(x/2
) + 2*a^3*b*cos(x/2)))*(a^2 + b^2)^(1/2))/b^2 - 1/(b*sin(x)) - (a*log(sin(x/2)/cos(x/2)))/b^2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{3}{\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**3/(a+b*cot(x)),x)

[Out]

Integral(csc(x)**3/(a + b*cot(x)), x)

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