3.285 \(\int (a \cot (x)+b \csc (x))^3 \, dx\)

Optimal. Leaf size=77 \[ -\frac{1}{2} a^2 b \cos (x)-\frac{1}{4} (2 a-b) (a+b)^2 \log (1-\cos (x))-\frac{1}{4} (a-b)^2 (2 a+b) \log (\cos (x)+1)-\frac{1}{2} \csc ^2(x) (a \cos (x)+b)^2 (a+b \cos (x)) \]

[Out]

-(a^2*b*Cos[x])/2 - ((b + a*Cos[x])^2*(a + b*Cos[x])*Csc[x]^2)/2 - ((2*a - b)*(a + b)^2*Log[1 - Cos[x]])/4 - (
(a - b)^2*(2*a + b)*Log[1 + Cos[x]])/4

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Rubi [A]  time = 0.135766, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4392, 2668, 739, 774, 633, 31} \[ -\frac{1}{2} a^2 b \cos (x)-\frac{1}{4} (2 a-b) (a+b)^2 \log (1-\cos (x))-\frac{1}{4} (a-b)^2 (2 a+b) \log (\cos (x)+1)-\frac{1}{2} \csc ^2(x) (a \cos (x)+b)^2 (a+b \cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x] + b*Csc[x])^3,x]

[Out]

-(a^2*b*Cos[x])/2 - ((b + a*Cos[x])^2*(a + b*Cos[x])*Csc[x]^2)/2 - ((2*a - b)*(a + b)^2*Log[1 - Cos[x]])/4 - (
(a - b)^2*(2*a + b)*Log[1 + Cos[x]])/4

Rule 4392

Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b_.))^(p_)*(u_.), x_Symbol] :> Int[A
ctivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a*Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2668

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*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 739

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 774

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/c, x] + Dist[1
/c, Int[(c*d*f - a*e*g + c*(e*f + d*g)*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x]

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.273392, size = 79, normalized size = 1.03 \[ \frac{1}{8} \left (-(a+b)^3 \csc ^2\left (\frac{x}{2}\right )+(a-b)^3 \left (-\sec ^2\left (\frac{x}{2}\right )\right )-4 (2 a-b) (a+b)^2 \log \left (\sin \left (\frac{x}{2}\right )\right )-4 (2 a+b) (a-b)^2 \log \left (\cos \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x] + b*Csc[x])^3,x]

[Out]

(-((a + b)^3*Csc[x/2]^2) - 4*(a - b)^2*(2*a + b)*Log[Cos[x/2]] - 4*(2*a - b)*(a + b)^2*Log[Sin[x/2]] - (a - b)
^3*Sec[x/2]^2)/8

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Maple [A]  time = 0.051, size = 87, normalized size = 1.1 \begin{align*} -{\frac{{a}^{3} \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}-{a}^{3}\ln \left ( \sin \left ( x \right ) \right ) -{\frac{3\,{a}^{2}b \left ( \cos \left ( x \right ) \right ) ^{3}}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{3\,{a}^{2}b\cos \left ( x \right ) }{2}}-{\frac{3\,{a}^{2}b\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}}-{\frac{3\,a{b}^{2}}{2\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{{b}^{3}\csc \left ( x \right ) \cot \left ( x \right ) }{2}}+{\frac{{b}^{3}\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988473, size = 117, normalized size = 1.52 \begin{align*} -\frac{3}{2} \, a b^{2} \cot \left (x\right )^{2} + \frac{3}{4} \, a^{2} b{\left (\frac{2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} + \log \left (\cos \left (x\right ) + 1\right ) - \log \left (\cos \left (x\right ) - 1\right )\right )} + \frac{1}{4} \, b^{3}{\left (\frac{2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \log \left (\cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right ) - 1\right )\right )} - \frac{1}{2} \, a^{3}{\left (\frac{1}{\sin \left (x\right )^{2}} + \log \left (\sin \left (x\right )^{2}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.10249, size = 313, normalized size = 4.06 \begin{align*} \frac{2 \, a^{3} + 6 \, a b^{2} + 2 \,{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (x\right ) +{\left (2 \, a^{3} - 3 \, a^{2} b + b^{3} -{\left (2 \, a^{3} - 3 \, a^{2} b + b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3} -{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{4 \,{\left (\cos \left (x\right )^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 46.8825, size = 122, normalized size = 1.58 \begin{align*} \frac{a^{3} \log{\left (\csc ^{2}{\left (x \right )} \right )}}{2} - \frac{a^{3} \csc ^{2}{\left (x \right )}}{2} - \frac{3 a^{2} b \log{\left (\cos{\left (x \right )} - 1 \right )}}{4} + \frac{3 a^{2} b \log{\left (\cos{\left (x \right )} + 1 \right )}}{4} + \frac{3 a^{2} b \cos{\left (x \right )}}{2 \cos ^{2}{\left (x \right )} - 2} - \frac{3 a b^{2} \csc ^{2}{\left (x \right )}}{2} + \frac{b^{3} \log{\left (\cos{\left (x \right )} - 1 \right )}}{4} - \frac{b^{3} \log{\left (\cos{\left (x \right )} + 1 \right )}}{4} + \frac{b^{3} \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((a*cot(x)+b*csc(x))**3,x)

[Out]

a**3*log(csc(x)**2)/2 - a**3*csc(x)**2/2 - 3*a**2*b*log(cos(x) - 1)/4 + 3*a**2*b*log(cos(x) + 1)/4 + 3*a**2*b*
cos(x)/(2*cos(x)**2 - 2) - 3*a*b**2*csc(x)**2/2 + b**3*log(cos(x) - 1)/4 - b**3*log(cos(x) + 1)/4 + b**3*cos(x
)/(2*cos(x)**2 - 2)

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Giac [A]  time = 1.18298, size = 116, normalized size = 1.51 \begin{align*} -\frac{1}{4} \,{\left (2 \, a^{3} - 3 \, a^{2} b + b^{3}\right )} \log \left (\cos \left (x\right ) + 1\right ) - \frac{1}{4} \,{\left (2 \, a^{3} + 3 \, a^{2} b - b^{3}\right )} \log \left (-\cos \left (x\right ) + 1\right ) + \frac{a^{3} + 3 \, a b^{2} +{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}{\left (\cos \left (x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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