3.283 \(\int (a \cot (x)+b \csc (x))^5 \, dx\)

Optimal. Leaf size=152 \[ \frac{1}{8} a^2 b \left (7 a^2-3 b^2\right ) \cos (x)+\frac{1}{16} (a+b)^3 \left (8 a^2-9 a b+3 b^2\right ) \log (1-\cos (x))+\frac{1}{16} (a-b)^3 \left (8 a^2+9 a b+3 b^2\right ) \log (\cos (x)+1)+\frac{1}{8} \csc ^2(x) (a \cos (x)+b)^2 \left (b \left (5 a^2-3 b^2\right ) \cos (x)+2 a \left (2 a^2-b^2\right )\right )-\frac{1}{4} \csc ^4(x) (a \cos (x)+b)^4 (a+b \cos (x)) \]

[Out]

(a^2*b*(7*a^2 - 3*b^2)*Cos[x])/8 + ((b + a*Cos[x])^2*(2*a*(2*a^2 - b^2) + b*(5*a^2 - 3*b^2)*Cos[x])*Csc[x]^2)/
8 - ((b + a*Cos[x])^4*(a + b*Cos[x])*Csc[x]^4)/4 + ((a + b)^3*(8*a^2 - 9*a*b + 3*b^2)*Log[1 - Cos[x]])/16 + ((
a - b)^3*(8*a^2 + 9*a*b + 3*b^2)*Log[1 + Cos[x]])/16

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Rubi [A]  time = 0.219501, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {4392, 2668, 739, 819, 774, 633, 31} \[ \frac{1}{8} a^2 b \left (7 a^2-3 b^2\right ) \cos (x)+\frac{1}{16} (a+b)^3 \left (8 a^2-9 a b+3 b^2\right ) \log (1-\cos (x))+\frac{1}{16} (a-b)^3 \left (8 a^2+9 a b+3 b^2\right ) \log (\cos (x)+1)+\frac{1}{8} \csc ^2(x) (a \cos (x)+b)^2 \left (b \left (5 a^2-3 b^2\right ) \cos (x)+2 a \left (2 a^2-b^2\right )\right )-\frac{1}{4} \csc ^4(x) (a \cos (x)+b)^4 (a+b \cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[x] + b*Csc[x])^5,x]

[Out]

(a^2*b*(7*a^2 - 3*b^2)*Cos[x])/8 + ((b + a*Cos[x])^2*(2*a*(2*a^2 - b^2) + b*(5*a^2 - 3*b^2)*Cos[x])*Csc[x]^2)/
8 - ((b + a*Cos[x])^4*(a + b*Cos[x])*Csc[x]^4)/4 + ((a + b)^3*(8*a^2 - 9*a*b + 3*b^2)*Log[1 - Cos[x]])/16 + ((
a - b)^3*(8*a^2 + 9*a*b + 3*b^2)*Log[1 + Cos[x]])/16

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 819

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

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))^5 \, dx &=\int (b+a \cos (x))^5 \csc ^5(x) \, dx\\ &=-\left (a^5 \operatorname{Subst}\left (\int \frac{(b+x)^5}{\left (a^2-x^2\right )^3} \, dx,x,a \cos (x)\right )\right )\\ &=-\frac{1}{4} (b+a \cos (x))^4 (a+b \cos (x)) \csc ^4(x)+\frac{1}{4} a^3 \operatorname{Subst}\left (\int \frac{(b+x)^3 \left (4 a^2-3 b^2+b x\right )}{\left (a^2-x^2\right )^2} \, dx,x,a \cos (x)\right )\\ &=\frac{1}{8} (b+a \cos (x))^2 \left (2 a \left (2 a^2-b^2\right )+b \left (5 a^2-3 b^2\right ) \cos (x)\right ) \csc ^2(x)-\frac{1}{4} (b+a \cos (x))^4 (a+b \cos (x)) \csc ^4(x)-\frac{1}{8} a \operatorname{Subst}\left (\int \frac{(b+x) \left (8 a^4-7 a^2 b^2+3 b^4+b \left (7 a^2-3 b^2\right ) x\right )}{a^2-x^2} \, dx,x,a \cos (x)\right )\\ &=\frac{1}{8} a^2 b \left (7 a^2-3 b^2\right ) \cos (x)+\frac{1}{8} (b+a \cos (x))^2 \left (2 a \left (2 a^2-b^2\right )+b \left (5 a^2-3 b^2\right ) \cos (x)\right ) \csc ^2(x)-\frac{1}{4} (b+a \cos (x))^4 (a+b \cos (x)) \csc ^4(x)+\frac{1}{8} a \operatorname{Subst}\left (\int \frac{-a^2 b \left (7 a^2-3 b^2\right )-b \left (8 a^4-7 a^2 b^2+3 b^4\right )-\left (8 a^4-7 a^2 b^2+3 b^4+b^2 \left (7 a^2-3 b^2\right )\right ) x}{a^2-x^2} \, dx,x,a \cos (x)\right )\\ &=\frac{1}{8} a^2 b \left (7 a^2-3 b^2\right ) \cos (x)+\frac{1}{8} (b+a \cos (x))^2 \left (2 a \left (2 a^2-b^2\right )+b \left (5 a^2-3 b^2\right ) \cos (x)\right ) \csc ^2(x)-\frac{1}{4} (b+a \cos (x))^4 (a+b \cos (x)) \csc ^4(x)-\frac{1}{16} \left ((a+b)^3 \left (8 a^2-9 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-x} \, dx,x,a \cos (x)\right )-\frac{1}{16} \left ((a-b)^3 \left (8 a^2+9 a b+3 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a-x} \, dx,x,a \cos (x)\right )\\ &=\frac{1}{8} a^2 b \left (7 a^2-3 b^2\right ) \cos (x)+\frac{1}{8} (b+a \cos (x))^2 \left (2 a \left (2 a^2-b^2\right )+b \left (5 a^2-3 b^2\right ) \cos (x)\right ) \csc ^2(x)-\frac{1}{4} (b+a \cos (x))^4 (a+b \cos (x)) \csc ^4(x)+\frac{1}{16} (a+b)^3 \left (8 a^2-9 a b+3 b^2\right ) \log (1-\cos (x))+\frac{1}{16} (a-b)^3 \left (8 a^2+9 a b+3 b^2\right ) \log (1+\cos (x))\\ \end{align*}

Mathematica [A]  time = 0.73585, size = 143, normalized size = 0.94 \[ \frac{1}{64} \left (8 (a+b)^3 \left (8 a^2-9 a b+3 b^2\right ) \log \left (\sin \left (\frac{x}{2}\right )\right )+8 \left (8 a^2+9 a b+3 b^2\right ) (a-b)^3 \log \left (\cos \left (\frac{x}{2}\right )\right )-(a+b)^5 \csc ^4\left (\frac{x}{2}\right )+2 (7 a-3 b) (a+b)^4 \csc ^2\left (\frac{x}{2}\right )+(a-b)^5 \left (-\sec ^4\left (\frac{x}{2}\right )\right )+2 (7 a+3 b) (a-b)^4 \sec ^2\left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[x] + b*Csc[x])^5,x]

[Out]

(2*(7*a - 3*b)*(a + b)^4*Csc[x/2]^2 - (a + b)^5*Csc[x/2]^4 + 8*(a - b)^3*(8*a^2 + 9*a*b + 3*b^2)*Log[Cos[x/2]]
+ 8*(a + b)^3*(8*a^2 - 9*a*b + 3*b^2)*Log[Sin[x/2]] + 2*(a - b)^4*(7*a + 3*b)*Sec[x/2]^2 - (a - b)^5*Sec[x/2]
^4)/64

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Maple [A]  time = 0.059, size = 204, normalized size = 1.3 \begin{align*} -{\frac{{a}^{5} \left ( \cot \left ( x \right ) \right ) ^{4}}{4}}+{\frac{{a}^{5} \left ( \cot \left ( x \right ) \right ) ^{2}}{2}}+{a}^{5}\ln \left ( \sin \left ( x \right ) \right ) -{\frac{5\,{a}^{4}b \left ( \cos \left ( x \right ) \right ) ^{5}}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}+{\frac{5\,{a}^{4}b \left ( \cos \left ( x \right ) \right ) ^{5}}{8\, \left ( \sin \left ( x \right ) \right ) ^{2}}}+{\frac{5\, \left ( \cos \left ( x \right ) \right ) ^{3}{a}^{4}b}{8}}+{\frac{15\,{a}^{4}b\cos \left ( x \right ) }{8}}+{\frac{15\,{a}^{4}b\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{8}}-{\frac{5\,{a}^{3}{b}^{2} \left ( \cos \left ( x \right ) \right ) ^{4}}{2\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{5\,{a}^{2}{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{3}}{2\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{5\,{a}^{2}{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{3}}{4\, \left ( \sin \left ( x \right ) \right ) ^{2}}}-{\frac{5\,\cos \left ( x \right ){a}^{2}{b}^{3}}{4}}-{\frac{5\,{a}^{2}{b}^{3}\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{4}}-{\frac{5\,a{b}^{4}}{4\, \left ( \sin \left ( x \right ) \right ) ^{4}}}-{\frac{{b}^{5}\cot \left ( x \right ) \left ( \csc \left ( x \right ) \right ) ^{3}}{4}}-{\frac{3\,{b}^{5}\csc \left ( x \right ) \cot \left ( x \right ) }{8}}+{\frac{3\,{b}^{5}\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*cot(x)+b*csc(x))^5,x)

[Out]

-1/4*a^5*cot(x)^4+1/2*a^5*cot(x)^2+a^5*ln(sin(x))-5/4*a^4*b/sin(x)^4*cos(x)^5+5/8*a^4*b/sin(x)^2*cos(x)^5+5/8*
cos(x)^3*a^4*b+15/8*a^4*b*cos(x)+15/8*a^4*b*ln(csc(x)-cot(x))-5/2*a^3*b^2/sin(x)^4*cos(x)^4-5/2*a^2*b^3/sin(x)
^4*cos(x)^3-5/4*a^2*b^3/sin(x)^2*cos(x)^3-5/4*cos(x)*a^2*b^3-5/4*a^2*b^3*ln(csc(x)-cot(x))-5/4*a*b^4/sin(x)^4-
1/4*b^5*cot(x)*csc(x)^3-3/8*b^5*csc(x)*cot(x)+3/8*b^5*ln(csc(x)-cot(x))

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Maxima [A]  time = 1.00049, size = 254, normalized size = 1.67 \begin{align*} -\frac{5}{2} \, a^{3} b^{2} \cot \left (x\right )^{4} - \frac{5}{16} \, a^{4} b{\left (\frac{2 \,{\left (5 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1} + 3 \, \log \left (\cos \left (x\right ) + 1\right ) - 3 \, \log \left (\cos \left (x\right ) - 1\right )\right )} + \frac{1}{16} \, b^{5}{\left (\frac{2 \,{\left (3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1} - 3 \, \log \left (\cos \left (x\right ) + 1\right ) + 3 \, \log \left (\cos \left (x\right ) - 1\right )\right )} - \frac{5}{8} \, a^{2} b^{3}{\left (\frac{2 \,{\left (\cos \left (x\right )^{3} + \cos \left (x\right )\right )}}{\cos \left (x\right )^{4} - 2 \, \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} \, a^{5}{\left (\frac{4 \, \sin \left (x\right )^{2} - 1}{\sin \left (x\right )^{4}} + 2 \, \log \left (\sin \left (x\right )^{2}\right )\right )} - \frac{5 \, a b^{4}}{4 \, \sin \left (x\right )^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)+b*csc(x))^5,x, algorithm="maxima")

[Out]

-5/2*a^3*b^2*cot(x)^4 - 5/16*a^4*b*(2*(5*cos(x)^3 - 3*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) + 3*log(cos(x) + 1)
- 3*log(cos(x) - 1)) + 1/16*b^5*(2*(3*cos(x)^3 - 5*cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) - 3*log(cos(x) + 1) + 3
*log(cos(x) - 1)) - 5/8*a^2*b^3*(2*(cos(x)^3 + cos(x))/(cos(x)^4 - 2*cos(x)^2 + 1) - log(cos(x) + 1) + log(cos
(x) - 1)) + 1/4*a^5*((4*sin(x)^2 - 1)/sin(x)^4 + 2*log(sin(x)^2)) - 5/4*a*b^4/sin(x)^4

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Fricas [B]  time = 2.17755, size = 703, normalized size = 4.62 \begin{align*} \frac{12 \, a^{5} + 40 \, a^{3} b^{2} - 20 \, a b^{4} - 2 \,{\left (25 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{3} - 16 \,{\left (a^{5} + 5 \, a^{3} b^{2}\right )} \cos \left (x\right )^{2} + 10 \,{\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (x\right ) +{\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5} +{\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{4} - 2 \,{\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) +{\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5} +{\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (x\right )^{4} - 2 \,{\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{16 \,{\left (\cos \left (x\right )^{4} - 2 \, \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))^5,x, algorithm="fricas")

[Out]

1/16*(12*a^5 + 40*a^3*b^2 - 20*a*b^4 - 2*(25*a^4*b + 10*a^2*b^3 - 3*b^5)*cos(x)^3 - 16*(a^5 + 5*a^3*b^2)*cos(x
)^2 + 10*(3*a^4*b - 2*a^2*b^3 - b^5)*cos(x) + (8*a^5 - 15*a^4*b + 10*a^2*b^3 - 3*b^5 + (8*a^5 - 15*a^4*b + 10*
a^2*b^3 - 3*b^5)*cos(x)^4 - 2*(8*a^5 - 15*a^4*b + 10*a^2*b^3 - 3*b^5)*cos(x)^2)*log(1/2*cos(x) + 1/2) + (8*a^5
+ 15*a^4*b - 10*a^2*b^3 + 3*b^5 + (8*a^5 + 15*a^4*b - 10*a^2*b^3 + 3*b^5)*cos(x)^4 - 2*(8*a^5 + 15*a^4*b - 10
*a^2*b^3 + 3*b^5)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(cos(x)^4 - 2*cos(x)^2 + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)+b*csc(x))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.11573, size = 228, normalized size = 1.5 \begin{align*} \frac{1}{16} \,{\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{16} \,{\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (-\cos \left (x\right ) + 1\right ) + \frac{6 \, a^{5} + 20 \, a^{3} b^{2} - 10 \, a b^{4} -{\left (25 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{3} - 8 \,{\left (a^{5} + 5 \, a^{3} b^{2}\right )} \cos \left (x\right )^{2} + 5 \,{\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (x\right )}{8 \,{\left (\cos \left (x\right ) + 1\right )}^{2}{\left (\cos \left (x\right ) - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cot(x)+b*csc(x))^5,x, algorithm="giac")

[Out]

1/16*(8*a^5 - 15*a^4*b + 10*a^2*b^3 - 3*b^5)*log(cos(x) + 1) + 1/16*(8*a^5 + 15*a^4*b - 10*a^2*b^3 + 3*b^5)*lo
g(-cos(x) + 1) + 1/8*(6*a^5 + 20*a^3*b^2 - 10*a*b^4 - (25*a^4*b + 10*a^2*b^3 - 3*b^5)*cos(x)^3 - 8*(a^5 + 5*a^
3*b^2)*cos(x)^2 + 5*(3*a^4*b - 2*a^2*b^3 - b^5)*cos(x))/((cos(x) + 1)^2*(cos(x) - 1)^2)