\(\int (a \cot (x)+b \csc (x))^5 \, dx\) [283]

Optimal result

Integrand size = 11, antiderivative size = 152 \[ \int (a \cot (x)+b \csc (x))^5 \, dx=\frac {1}{8} a^2 b \left (7 a^2-3 b^2\right ) \cos (x)+\frac {1}{8} (b+a \cos (x))^2 \left (2 a \left (2 a^2-b^2\right )+b \left (5 a^2-3 b^2\right ) \cos (x)\right ) \csc ^2(x)-\frac {1}{4} (b+a \cos (x))^4 (a+b \cos (x)) \csc ^4(x)+\frac {1}{16} (a+b)^3 \left (8 a^2-9 a b+3 b^2\right ) \log (1-\cos (x))+\frac {1}{16} (a-b)^3 \left (8 a^2+9 a b+3 b^2\right ) \log (1+\cos (x)) \]

[Out]

1/8*a^2*b*(7*a^2-3*b^2)*cos(x)+1/8*(b+a*cos(x))^2*(2*a*(2*a^2-b^2)+b*(5*a^2-3*b^2)*cos(x))*csc(x)^2-1/4*(b+a*c
os(x))^4*(a+b*cos(x))*csc(x)^4+1/16*(a+b)^3*(8*a^2-9*a*b+3*b^2)*ln(1-cos(x))+1/16*(a-b)^3*(8*a^2+9*a*b+3*b^2)*
ln(1+cos(x))

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {4477, 2747, 753, 833, 788, 647, 31} \[ \int (a \cot (x)+b \csc (x))^5 \, dx=\frac {1}{8} a^2 b \left (7 a^2-3 b^2\right ) \cos (x)+\frac {1}{16} (a+b)^3 \left (8 a^2-9 a b+3 b^2\right ) \log (1-\cos (x))+\frac {1}{16} (a-b)^3 \left (8 a^2+9 a b+3 b^2\right ) \log (\cos (x)+1)+\frac {1}{8} \csc ^2(x) (a \cos (x)+b)^2 \left (b \left (5 a^2-3 b^2\right ) \cos (x)+2 a \left (2 a^2-b^2\right )\right )-\frac {1}{4} \csc ^4(x) (a \cos (x)+b)^4 (a+b \cos (x)) \]

[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 31

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

Rule 647

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 753

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 788

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 833

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 2747

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 4477

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]

Rubi steps \begin{align*} \text {integral}& = \int (b+a \cos (x))^5 \csc ^5(x) \, dx \\ & = -\left (a^5 \text {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 \text {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 \text {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 \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.60 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.94 \[ \int (a \cot (x)+b \csc (x))^5 \, dx=\frac {1}{64} \left (2 (7 a-3 b) (a+b)^4 \csc ^2\left (\frac {x}{2}\right )-(a+b)^5 \csc ^4\left (\frac {x}{2}\right )+8 (a-b)^3 \left (8 a^2+9 a b+3 b^2\right ) \log \left (\cos \left (\frac {x}{2}\right )\right )+8 (a+b)^3 \left (8 a^2-9 a b+3 b^2\right ) \log \left (\sin \left (\frac {x}{2}\right )\right )+2 (a-b)^4 (7 a+3 b) \sec ^2\left (\frac {x}{2}\right )-(a-b)^5 \sec ^4\left (\frac {x}{2}\right )\right ) \]

[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

Maple [A] (verified)

Time = 30.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.10

method	result	size
default	\(b^{5} \left (\left (-\frac {\csc \left (x \right )^{3}}{4}-\frac {3 \csc \left (x \right )}{8}\right ) \cot \left (x \right )+\frac {3 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\right )-\frac {5 a \,b^{4}}{4 \sin \left (x \right )^{4}}+10 a^{2} b^{3} \left (-\frac {\cos \left (x \right )^{3}}{4 \sin \left (x \right )^{4}}-\frac {\cos \left (x \right )^{3}}{8 \sin \left (x \right )^{2}}-\frac {\cos \left (x \right )}{8}-\frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\right )-\frac {5 a^{3} b^{2} \cos \left (x \right )^{4}}{2 \sin \left (x \right )^{4}}+5 a^{4} b \left (-\frac {\cos \left (x \right )^{5}}{4 \sin \left (x \right )^{4}}+\frac {\cos \left (x \right )^{5}}{8 \sin \left (x \right )^{2}}+\frac {\cos \left (x \right )^{3}}{8}+\frac {3 \cos \left (x \right )}{8}+\frac {3 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\right )+a^{5} \left (-\frac {\cot \left (x \right )^{4}}{4}+\frac {\cot \left (x \right )^{2}}{2}+\ln \left (\sin \left (x \right )\right )\right )\)	\(167\)
parts	\(a^{5} \left (-\frac {\cot \left (x \right )^{4}}{4}+\frac {\cot \left (x \right )^{2}}{2}-\frac {\ln \left (1+\cot \left (x \right )^{2}\right )}{2}\right )+b^{5} \left (\left (-\frac {\csc \left (x \right )^{3}}{4}-\frac {3 \csc \left (x \right )}{8}\right ) \cot \left (x \right )+\frac {3 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\right )+5 a^{4} b \left (-\frac {\cos \left (x \right )^{5}}{4 \sin \left (x \right )^{4}}+\frac {\cos \left (x \right )^{5}}{8 \sin \left (x \right )^{2}}+\frac {\cos \left (x \right )^{3}}{8}+\frac {3 \cos \left (x \right )}{8}+\frac {3 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\right )-\frac {5 a^{3} b^{2} \cot \left (x \right )^{4}}{2}+10 a^{2} b^{3} \left (-\frac {\cos \left (x \right )^{3}}{4 \sin \left (x \right )^{4}}-\frac {\cos \left (x \right )^{3}}{8 \sin \left (x \right )^{2}}-\frac {\cos \left (x \right )}{8}-\frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{8}\right )-\frac {5 b^{4} \csc \left (x \right )^{4} a}{4}\)	\(169\)
risch	\(-i a^{5} x -\frac {{\mathrm e}^{i x} \left (25 a^{4} b \,{\mathrm e}^{6 i x}+10 a^{2} b^{3} {\mathrm e}^{6 i x}-3 b^{5} {\mathrm e}^{6 i x}+16 a^{5} {\mathrm e}^{5 i x}+80 b^{2} a^{3} {\mathrm e}^{5 i x}+15 a^{4} b \,{\mathrm e}^{4 i x}+70 a^{2} b^{3} {\mathrm e}^{4 i x}+11 b^{5} {\mathrm e}^{4 i x}-16 a^{5} {\mathrm e}^{3 i x}+80 a \,b^{4} {\mathrm e}^{3 i x}+15 a^{4} b \,{\mathrm e}^{2 i x}+70 a^{2} b^{3} {\mathrm e}^{2 i x}+11 b^{5} {\mathrm e}^{2 i x}+16 a^{5} {\mathrm e}^{i x}+80 a^{3} b^{2} {\mathrm e}^{i x}+25 a^{4} b +10 a^{2} b^{3}-3 b^{5}\right )}{4 \left ({\mathrm e}^{2 i x}-1\right )^{4}}+\ln \left ({\mathrm e}^{i x}-1\right ) a^{5}+\frac {15 \ln \left ({\mathrm e}^{i x}-1\right ) a^{4} b}{8}-\frac {5 \ln \left ({\mathrm e}^{i x}-1\right ) a^{2} b^{3}}{4}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) b^{5}}{8}+\ln \left ({\mathrm e}^{i x}+1\right ) a^{5}-\frac {15 \ln \left ({\mathrm e}^{i x}+1\right ) a^{4} b}{8}+\frac {5 \ln \left ({\mathrm e}^{i x}+1\right ) a^{2} b^{3}}{4}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) b^{5}}{8}\)	\(352\)

[In]

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

[Out]

b^5*((-1/4*csc(x)^3-3/8*csc(x))*cot(x)+3/8*ln(csc(x)-cot(x)))-5/4*a*b^4/sin(x)^4+10*a^2*b^3*(-1/4/sin(x)^4*cos
(x)^3-1/8/sin(x)^2*cos(x)^3-1/8*cos(x)-1/8*ln(csc(x)-cot(x)))-5/2*a^3*b^2/sin(x)^4*cos(x)^4+5*a^4*b*(-1/4/sin(
x)^4*cos(x)^5+1/8/sin(x)^2*cos(x)^5+1/8*cos(x)^3+3/8*cos(x)+3/8*ln(csc(x)-cot(x)))+a^5*(-1/4*cot(x)^4+1/2*cot(
x)^2+ln(sin(x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (142) = 284\).

Time = 0.27 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.92 \[ \int (a \cot (x)+b \csc (x))^5 \, dx=\frac {12 \, a^{5} + 40 \, a^{3} b^{2} - 20 \, a b^{4} - 2 \, {\left (25 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{3} - 16 \, {\left (a^{5} + 5 \, a^{3} b^{2}\right )} \cos \left (x\right )^{2} + 10 \, {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (x\right ) + {\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5} + {\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{4} - 2 \, {\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5} + {\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (x\right )^{4} - 2 \, {\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (x\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{16 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (141) = 282\).

Time = 51.39 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.03 \[ \int (a \cot (x)+b \csc (x))^5 \, dx=- \frac {a^{5} \log {\left (\csc ^{2}{\left (x \right )} \right )}}{2} - \frac {a^{5} \csc ^{4}{\left (x \right )}}{4} + a^{5} \csc ^{2}{\left (x \right )} + \frac {15 a^{4} b \log {\left (\cos {\left (x \right )} - 1 \right )}}{16} - \frac {15 a^{4} b \log {\left (\cos {\left (x \right )} + 1 \right )}}{16} - \frac {25 a^{4} b \cos ^{3}{\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} + \frac {15 a^{4} b \cos {\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} - \frac {5 a^{3} b^{2} \cot ^{4}{\left (x \right )}}{2} - \frac {5 a^{2} b^{3} \log {\left (\cos {\left (x \right )} - 1 \right )}}{8} + \frac {5 a^{2} b^{3} \log {\left (\cos {\left (x \right )} + 1 \right )}}{8} - \frac {10 a^{2} b^{3} \cos ^{3}{\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} - \frac {10 a^{2} b^{3} \cos {\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} - \frac {5 a b^{4} \csc ^{4}{\left (x \right )}}{4} + \frac {3 b^{5} \log {\left (\cos {\left (x \right )} - 1 \right )}}{16} - \frac {3 b^{5} \log {\left (\cos {\left (x \right )} + 1 \right )}}{16} + \frac {3 b^{5} \cos ^{3}{\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} - \frac {5 b^{5} \cos {\left (x \right )}}{8 \cos ^{4}{\left (x \right )} - 16 \cos ^{2}{\left (x \right )} + 8} \]

[In]

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

[Out]

-a**5*log(csc(x)**2)/2 - a**5*csc(x)**4/4 + a**5*csc(x)**2 + 15*a**4*b*log(cos(x) - 1)/16 - 15*a**4*b*log(cos(
x) + 1)/16 - 25*a**4*b*cos(x)**3/(8*cos(x)**4 - 16*cos(x)**2 + 8) + 15*a**4*b*cos(x)/(8*cos(x)**4 - 16*cos(x)*
*2 + 8) - 5*a**3*b**2*cot(x)**4/2 - 5*a**2*b**3*log(cos(x) - 1)/8 + 5*a**2*b**3*log(cos(x) + 1)/8 - 10*a**2*b*
*3*cos(x)**3/(8*cos(x)**4 - 16*cos(x)**2 + 8) - 10*a**2*b**3*cos(x)/(8*cos(x)**4 - 16*cos(x)**2 + 8) - 5*a*b**
4*csc(x)**4/4 + 3*b**5*log(cos(x) - 1)/16 - 3*b**5*log(cos(x) + 1)/16 + 3*b**5*cos(x)**3/(8*cos(x)**4 - 16*cos
(x)**2 + 8) - 5*b**5*cos(x)/(8*cos(x)**4 - 16*cos(x)**2 + 8)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.24 \[ \int (a \cot (x)+b \csc (x))^5 \, dx=-\frac {5}{2} \, a^{3} b^{2} \cot \left (x\right )^{4} - \frac {5}{16} \, a^{4} b {\left (\frac {2 \, {\left (5 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1} + 3 \, \log \left (\cos \left (x\right ) + 1\right ) - 3 \, \log \left (\cos \left (x\right ) - 1\right )\right )} + \frac {1}{16} \, b^{5} {\left (\frac {2 \, {\left (3 \, \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1} - 3 \, \log \left (\cos \left (x\right ) + 1\right ) + 3 \, \log \left (\cos \left (x\right ) - 1\right )\right )} - \frac {5}{8} \, a^{2} b^{3} {\left (\frac {2 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )\right )}}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1} - \log \left (\cos \left (x\right ) + 1\right ) + \log \left (\cos \left (x\right ) - 1\right )\right )} + \frac {1}{4} \, a^{5} {\left (\frac {4 \, \sin \left (x\right )^{2} - 1}{\sin \left (x\right )^{4}} + 2 \, \log \left (\sin \left (x\right )^{2}\right )\right )} - \frac {5 \, a b^{4}}{4 \, \sin \left (x\right )^{4}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.11 \[ \int (a \cot (x)+b \csc (x))^5 \, dx=\frac {1}{16} \, {\left (8 \, a^{5} - 15 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{16} \, {\left (8 \, a^{5} + 15 \, a^{4} b - 10 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (-\cos \left (x\right ) + 1\right ) + \frac {6 \, a^{5} + 20 \, a^{3} b^{2} - 10 \, a b^{4} - {\left (25 \, a^{4} b + 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (x\right )^{3} - 8 \, {\left (a^{5} + 5 \, a^{3} b^{2}\right )} \cos \left (x\right )^{2} + 5 \, {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (x\right )}{8 \, {\left (\cos \left (x\right ) + 1\right )}^{2} {\left (\cos \left (x\right ) - 1\right )}^{2}} \]

[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)

Mupad [B] (verification not implemented)

Time = 28.98 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14 \[ \int (a \cot (x)+b \csc (x))^5 \, dx={\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (\frac {5\,\left (a+b\right )\,{\left (a-b\right )}^4}{32}+\frac {{\left (a-b\right )}^5}{32}\right )-\frac {\frac {5\,a\,b^4}{4}+\frac {5\,a^4\,b}{4}-{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (3\,a^5+10\,a^4\,b+10\,a^3\,b^2-5\,a\,b^4-2\,b^5\right )+\frac {a^5}{4}+\frac {b^5}{4}+\frac {5\,a^2\,b^3}{2}+\frac {5\,a^3\,b^2}{2}}{16\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}-a^5\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )+\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,\left (a^5+\frac {15\,a^4\,b}{8}-\frac {5\,a^2\,b^3}{4}+\frac {3\,b^5}{8}\right )-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,{\left (a-b\right )}^5}{64} \]

[In]

int((b/sin(x) + a*cot(x))^5,x)

[Out]

tan(x/2)^2*((5*(a + b)*(a - b)^4)/32 + (a - b)^5/32) - ((5*a*b^4)/4 + (5*a^4*b)/4 - tan(x/2)^2*(10*a^4*b - 5*a
*b^4 + 3*a^5 - 2*b^5 + 10*a^3*b^2) + a^5/4 + b^5/4 + (5*a^2*b^3)/2 + (5*a^3*b^2)/2)/(16*tan(x/2)^4) - a^5*log(
tan(x/2)^2 + 1) + log(tan(x/2))*((15*a^4*b)/8 + a^5 + (3*b^5)/8 - (5*a^2*b^3)/4) - (tan(x/2)^4*(a - b)^5)/64