3.304 \(\int (\csc (x)-\sin (x))^3 \, dx\)

Optimal. Leaf size=34 \[ -\frac{5 \cos ^3(x)}{6}-\frac{5 \cos (x)}{2}-\frac{1}{2} \cos ^3(x) \cot ^2(x)+\frac{5}{2} \tanh ^{-1}(\cos (x)) \]

[Out]

(5*ArcTanh[Cos[x]])/2 - (5*Cos[x])/2 - (5*Cos[x]^3)/6 - (Cos[x]^3*Cot[x]^2)/2

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Rubi [A]  time = 0.0458312, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4397, 2592, 288, 302, 206} \[ -\frac{5 \cos ^3(x)}{6}-\frac{5 \cos (x)}{2}-\frac{1}{2} \cos ^3(x) \cot ^2(x)+\frac{5}{2} \tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Csc[x] - Sin[x])^3,x]

[Out]

(5*ArcTanh[Cos[x]])/2 - (5*Cos[x])/2 - (5*Cos[x]^3)/6 - (Cos[x]^3*Cot[x]^2)/2

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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])

Rubi steps

\begin{align*} \int (\csc (x)-\sin (x))^3 \, dx &=\int \cos ^3(x) \cot ^3(x) \, dx\\ &=-\operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{2} \cos ^3(x) \cot ^2(x)+\frac{5}{2} \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{2} \cos ^3(x) \cot ^2(x)+\frac{5}{2} \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\cos (x)\right )\\ &=-\frac{5 \cos (x)}{2}-\frac{5 \cos ^3(x)}{6}-\frac{1}{2} \cos ^3(x) \cot ^2(x)+\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (x)\right )\\ &=\frac{5}{2} \tanh ^{-1}(\cos (x))-\frac{5 \cos (x)}{2}-\frac{5 \cos ^3(x)}{6}-\frac{1}{2} \cos ^3(x) \cot ^2(x)\\ \end{align*}

Mathematica [A]  time = 0.0204385, size = 61, normalized size = 1.79 \[ -\frac{9 \cos (x)}{4}-\frac{1}{12} \cos (3 x)-\frac{1}{8} \csc ^2\left (\frac{x}{2}\right )+\frac{1}{8} \sec ^2\left (\frac{x}{2}\right )-\frac{5}{2} \log \left (\sin \left (\frac{x}{2}\right )\right )+\frac{5}{2} \log \left (\cos \left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Csc[x] - Sin[x])^3,x]

[Out]

(-9*Cos[x])/4 - Cos[3*x]/12 - Csc[x/2]^2/8 + (5*Log[Cos[x/2]])/2 - (5*Log[Sin[x/2]])/2 + Sec[x/2]^2/8

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Maple [A]  time = 0.019, size = 32, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+ \left ( \sin \left ( x \right ) \right ) ^{2} \right ) \cos \left ( x \right ) }{3}}-3\,\cos \left ( x \right ) -{\frac{5\,\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}}-{\frac{\cot \left ( x \right ) \csc \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((csc(x)-sin(x))^3,x)

[Out]

1/3*(2+sin(x)^2)*cos(x)-3*cos(x)-5/2*ln(csc(x)-cot(x))-1/2*cot(x)*csc(x)

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Maxima [A]  time = 0.986781, size = 50, normalized size = 1.47 \begin{align*} -\frac{1}{3} \, \cos \left (x\right )^{3} + \frac{\cos \left (x\right )}{2 \,{\left (\cos \left (x\right )^{2} - 1\right )}} - 2 \, \cos \left (x\right ) + \frac{5}{4} \, \log \left (\cos \left (x\right ) + 1\right ) - \frac{5}{4} \, \log \left (\cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)-sin(x))^3,x, algorithm="maxima")

[Out]

-1/3*cos(x)^3 + 1/2*cos(x)/(cos(x)^2 - 1) - 2*cos(x) + 5/4*log(cos(x) + 1) - 5/4*log(cos(x) - 1)

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Fricas [B]  time = 2.12217, size = 197, normalized size = 5.79 \begin{align*} -\frac{4 \, \cos \left (x\right )^{5} + 20 \, \cos \left (x\right )^{3} - 15 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + 15 \,{\left (\cos \left (x\right )^{2} - 1\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - 30 \, \cos \left (x\right )}{12 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)-sin(x))^3,x, algorithm="fricas")

[Out]

-1/12*(4*cos(x)^5 + 20*cos(x)^3 - 15*(cos(x)^2 - 1)*log(1/2*cos(x) + 1/2) + 15*(cos(x)^2 - 1)*log(-1/2*cos(x)
+ 1/2) - 30*cos(x))/(cos(x)^2 - 1)

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Sympy [A]  time = 5.87116, size = 42, normalized size = 1.24 \begin{align*} - \frac{5 \log{\left (\cos{\left (x \right )} - 1 \right )}}{4} + \frac{5 \log{\left (\cos{\left (x \right )} + 1 \right )}}{4} - \frac{\cos ^{3}{\left (x \right )}}{3} - 2 \cos{\left (x \right )} + \frac{\cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)-sin(x))**3,x)

[Out]

-5*log(cos(x) - 1)/4 + 5*log(cos(x) + 1)/4 - cos(x)**3/3 - 2*cos(x) + cos(x)/(2*cos(x)**2 - 2)

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Giac [B]  time = 1.16047, size = 134, normalized size = 3.94 \begin{align*} \frac{{\left (\frac{10 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + 1\right )}{\left (\cos \left (x\right ) + 1\right )}}{8 \,{\left (\cos \left (x\right ) - 1\right )}} - \frac{\cos \left (x\right ) - 1}{8 \,{\left (\cos \left (x\right ) + 1\right )}} - \frac{2 \,{\left (\frac{12 \,{\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac{9 \,{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 7\right )}}{3 \,{\left (\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{3}} - \frac{5}{4} \, \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((csc(x)-sin(x))^3,x, algorithm="giac")

[Out]

1/8*(10*(cos(x) - 1)/(cos(x) + 1) + 1)*(cos(x) + 1)/(cos(x) - 1) - 1/8*(cos(x) - 1)/(cos(x) + 1) - 2/3*(12*(co
s(x) - 1)/(cos(x) + 1) - 9*(cos(x) - 1)^2/(cos(x) + 1)^2 - 7)/((cos(x) - 1)/(cos(x) + 1) - 1)^3 - 5/4*log(-(co
s(x) - 1)/(cos(x) + 1))