\(\int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [227]

Optimal result

Integrand size = 21, antiderivative size = 313 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {a^2 b^3}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {3 a (a-3 b) \log (1-\cos (c+d x))}{16 (a+b)^5 d}-\frac {3 a (a+3 b) \log (1+\cos (c+d x))}{16 (a-b)^5 d}+\frac {3 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5 d} \]

[Out]

-1/2*a^2*b^3/(a^2-b^2)^3/d/(b+a*cos(d*x+c))^2+3*a^2*b^2*(a^2+b^2)/(a^2-b^2)^4/d/(b+a*cos(d*x+c))+1/8*(4*b*(3*a
^4+8*a^2*b^2+b^4)-3*a*(a^4+10*a^2*b^2+5*b^4)*cos(d*x+c))*csc(d*x+c)^2/(a^2-b^2)^4/d+1/4*(b*(3*a^2+b^2)-a*(a^2+
3*b^2)*cos(d*x+c))*csc(d*x+c)^4/(a^2-b^2)^3/d+3/16*a*(a-3*b)*ln(1-cos(d*x+c))/(a+b)^5/d-3/16*a*(a+3*b)*ln(1+co
s(d*x+c))/(a-b)^5/d+3*a^2*b*(a^4+5*a^2*b^2+2*b^4)*ln(b+a*cos(d*x+c))/(a^2-b^2)^5/d

Rubi [A] (verified)

Time = 1.46 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2916, 12, 1661, 1643} \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {3 a^2 b^2 \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^4 (a \cos (c+d x)+b)}+\frac {\csc ^4(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 d \left (a^2-b^2\right )^3}-\frac {a^2 b^3}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)^2}+\frac {3 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^5}+\frac {\csc ^2(c+d x) \left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right )}{8 d \left (a^2-b^2\right )^4}+\frac {3 a (a-3 b) \log (1-\cos (c+d x))}{16 d (a+b)^5}-\frac {3 a (a+3 b) \log (\cos (c+d x)+1)}{16 d (a-b)^5} \]

[In]

Int[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]

[Out]

-1/2*(a^2*b^3)/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])^2) + (3*a^2*b^2*(a^2 + b^2))/((a^2 - b^2)^4*d*(b + a*Cos[
c + d*x])) + ((4*b*(3*a^4 + 8*a^2*b^2 + b^4) - 3*a*(a^4 + 10*a^2*b^2 + 5*b^4)*Cos[c + d*x])*Csc[c + d*x]^2)/(8
*(a^2 - b^2)^4*d) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)^3*d) + (3
*a*(a - 3*b)*Log[1 - Cos[c + d*x]])/(16*(a + b)^5*d) - (3*a*(a + 3*b)*Log[1 + Cos[c + d*x]])/(16*(a - b)^5*d)
+ (3*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^5*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1643

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 2916

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot ^3(c+d x) \csc ^2(c+d x)}{(-b-a \cos (c+d x))^3} \, dx \\ & = \frac {a^5 \text {Subst}\left (\int \frac {x^3}{a^3 (-b+x)^3 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {a^2 \text {Subst}\left (\int \frac {x^3}{(-b+x)^3 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d} \\ & = \frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^3 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}-\frac {a^2 b^2 \left (3 a^4-3 a^2 b^2-4 b^4\right ) x}{\left (a^2-b^2\right )^3}+\frac {a^4 b \left (3 a^2-23 b^2\right ) x^2}{\left (a^2-b^2\right )^3}+\frac {3 a^4 \left (a^2+3 b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(-b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{4 d} \\ & = \frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^3 \left (5 a^4+34 a^2 b^2+9 b^4\right )}{\left (a^2-b^2\right )^4}-\frac {3 a^4 b^2 \left (5 a^4+18 a^2 b^2-7 b^4\right ) x}{\left (a^2-b^2\right )^4}+\frac {a^4 b \left (15 a^4-26 a^2 b^2-37 b^4\right ) x^2}{\left (a^2-b^2\right )^4}+\frac {3 a^4 \left (a^4+10 a^2 b^2+5 b^4\right ) x^3}{\left (a^2-b^2\right )^4}}{(-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{8 a^2 d} \\ & = \frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \left (\frac {3 a^3 (a+3 b)}{2 (a-b)^5 (a-x)}-\frac {8 a^4 b^3}{\left (a^2-b^2\right )^3 (b-x)^3}+\frac {24 a^4 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 (b-x)^2}-\frac {24 a^4 b \left (a^4+5 a^2 b^2+2 b^4\right )}{\left (a^2-b^2\right )^5 (b-x)}+\frac {3 a^3 (a-3 b)}{2 (a+b)^5 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{8 a^2 d} \\ & = -\frac {a^2 b^3}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))^2}+\frac {3 a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (b+a \cos (c+d x))}+\frac {\left (4 b \left (3 a^4+8 a^2 b^2+b^4\right )-3 a \left (a^4+10 a^2 b^2+5 b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^4 d}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^3 d}+\frac {3 a (a-3 b) \log (1-\cos (c+d x))}{16 (a+b)^5 d}-\frac {3 a (a+3 b) \log (1+\cos (c+d x))}{16 (a-b)^5 d}+\frac {3 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.51 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.58 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {(b+a \cos (c+d x)) \left (\frac {32 a^2 b^3}{(-a+b)^3 (a+b)^3}+\frac {192 a^2 (a-i b) (a+i b) b^2 (b+a \cos (c+d x))}{(a-b)^4 (a+b)^4}-\frac {384 i a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) (c+d x) (b+a \cos (c+d x))^2}{(a-b)^5 (a+b)^5}-\frac {24 i a (a-3 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^2}{(a+b)^5}+\frac {24 i a (a+3 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^2}{(a-b)^5}+\frac {6 (-a+b) (b+a \cos (c+d x))^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^4}-\frac {(b+a \cos (c+d x))^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}-\frac {12 a (a+3 b) (b+a \cos (c+d x))^2 \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^5}+\frac {192 a^2 b \left (a^4+5 a^2 b^2+2 b^4\right ) (b+a \cos (c+d x))^2 \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^5}+\frac {12 a (a-3 b) (b+a \cos (c+d x))^2 \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^5}+\frac {6 (a+b) (b+a \cos (c+d x))^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{(a-b)^4}+\frac {(b+a \cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{(a-b)^3}\right ) \sec ^3(c+d x)}{64 d (a+b \sec (c+d x))^3} \]

[In]

Integrate[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]

[Out]

((b + a*Cos[c + d*x])*((32*a^2*b^3)/((-a + b)^3*(a + b)^3) + (192*a^2*(a - I*b)*(a + I*b)*b^2*(b + a*Cos[c + d
*x]))/((a - b)^4*(a + b)^4) - ((384*I)*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*(c + d*x)*(b + a*Cos[c + d*x])^2)/((a -
b)^5*(a + b)^5) - ((24*I)*a*(a - 3*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^2)/(a + b)^5 + ((24*I)*a*(a +
3*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^2)/(a - b)^5 + (6*(-a + b)*(b + a*Cos[c + d*x])^2*Csc[(c + d*x
)/2]^2)/(a + b)^4 - ((b + a*Cos[c + d*x])^2*Csc[(c + d*x)/2]^4)/(a + b)^3 - (12*a*(a + 3*b)*(b + a*Cos[c + d*x
])^2*Log[Cos[(c + d*x)/2]^2])/(a - b)^5 + (192*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*(b + a*Cos[c + d*x])^2*Log[b +
a*Cos[c + d*x]])/(a^2 - b^2)^5 + (12*a*(a - 3*b)*(b + a*Cos[c + d*x])^2*Log[Sin[(c + d*x)/2]^2])/(a + b)^5 + (
6*(a + b)*(b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^2)/(a - b)^4 + ((b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^4)/(
a - b)^3)*Sec[c + d*x]^3)/(64*d*(a + b*Sec[c + d*x])^3)

Maple [A] (verified)

Time = 1.85 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {-\frac {1}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +3 b}{16 \left (a +b \right )^{4} \left (\cos \left (d x +c \right )-1\right )}+\frac {3 a \left (a -3 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}-\frac {b^{3} a^{2}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a -b \right )^{5} \left (a +b \right )^{5}}+\frac {3 a^{2} b^{2} \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}+\frac {1}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -3 b}{16 \left (a -b \right )^{4} \left (\cos \left (d x +c \right )+1\right )}-\frac {3 a \left (a +3 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{5}}}{d}\)	\(255\)
default	\(\frac {-\frac {1}{16 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )^{2}}-\frac {-3 a +3 b}{16 \left (a +b \right )^{4} \left (\cos \left (d x +c \right )-1\right )}+\frac {3 a \left (a -3 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}-\frac {b^{3} a^{2}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {3 a^{2} b \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a -b \right )^{5} \left (a +b \right )^{5}}+\frac {3 a^{2} b^{2} \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (b +a \cos \left (d x +c \right )\right )}+\frac {1}{16 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )^{2}}-\frac {-3 a -3 b}{16 \left (a -b \right )^{4} \left (\cos \left (d x +c \right )+1\right )}-\frac {3 a \left (a +3 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{16 \left (a -b \right )^{5}}}{d}\)	\(255\)
norman	\(\frac {-\frac {1}{64 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{64 d \left (a -b \right )}-\frac {\left (3 a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{32 d \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (3 a +b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{32 d \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (15 a^{7}+459 a^{5} b^{2}+621 b^{4} a^{3}+57 a \,b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32 \left (a^{8}-2 a^{7} b -2 a^{6} b^{2}+6 a^{5} b^{3}-6 a^{3} b^{5}+2 a^{2} b^{6}+2 a \,b^{7}-b^{8}\right ) d}-\frac {\left (15 a^{7}+45 b \,a^{6}+459 a^{5} b^{2}+249 a^{4} b^{3}+621 b^{4} a^{3}+87 b^{5} a^{2}+57 a \,b^{6}+3 b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32 d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}+\frac {3 a \left (a -3 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right )}+\frac {3 b \,a^{2} \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{10}-5 a^{8} b^{2}+10 a^{6} b^{4}-10 b^{6} a^{4}+5 b^{8} a^{2}-b^{10}\right )}\)	\(519\)
parallelrisch	\(\frac {3 b \,a^{2} \left (a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )+\frac {3 a \left (a -3 b \right ) \left (a -b \right )^{5} \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (a +b \right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (\left (5 a^{7} b -\frac {313}{16} a^{6} b^{2}-\frac {251}{2} a^{5} b^{3}-\frac {647}{16} a^{4} b^{4}-\frac {359}{2} a^{3} b^{5}-\frac {459}{16} a^{2} b^{6}-24 a \,b^{7}-\frac {5}{16} a^{8}+5 b^{8}\right ) \cos \left (2 d x +2 c \right )+\left (-\frac {14}{3} a^{7} b -\frac {317}{24} a^{6} b^{2}+\frac {83}{3} a^{5} b^{3}-\frac {367}{24} a^{4} b^{4}+\frac {137}{3} a^{3} b^{5}+\frac {89}{24} a^{2} b^{6}+\frac {10}{3} a \,b^{7}-\frac {5}{8} a^{8}+\frac {17}{12} b^{8}\right ) \cos \left (4 d x +4 c \right )-\frac {5 \left (a +b \right ) \left (a^{6}+\frac {1}{4} a^{5} b +\frac {611}{20} a^{4} b^{2}+\frac {15}{2} a^{3} b^{3}+\frac {467}{10} a^{2} b^{4}-\frac {31}{4} a \,b^{5}+\frac {163}{20} b^{6}\right ) a \cos \left (3 d x +3 c \right )}{3}+\left (a^{6}-\frac {3}{4} a^{5} b +\frac {241}{12} a^{4} b^{2}-\frac {7}{6} a^{3} b^{3}+\frac {51}{2} a^{2} b^{4}+\frac {23}{12} a \,b^{5}+\frac {17}{12} b^{6}\right ) \left (a +b \right ) a \cos \left (5 d x +5 c \right )+\frac {5 \left (a^{6}+\frac {16}{15} a^{5} b +\frac {443}{15} a^{4} b^{2}+\frac {136}{15} a^{3} b^{3}+\frac {97}{3} a^{2} b^{4}+\frac {8}{3} a \,b^{5}+\frac {17}{15} b^{6}\right ) a^{2} \cos \left (6 d x +6 c \right )}{16}+\left (-10 a^{8}+\frac {25}{2} a^{7} b +64 a^{6} b^{2}+\frac {25}{2} a^{5} b^{3}+34 a^{4} b^{4}+\frac {139}{2} a^{3} b^{5}+8 a^{2} b^{6}+\frac {3}{2} a \,b^{7}\right ) \cos \left (d x +c \right )+\frac {17 b^{8}}{4}-\frac {59 a^{2} b^{6}}{8}+10 a^{7} b +10 a \,b^{7}+\frac {5 a^{8}}{8}+\frac {103 a^{6} b^{2}}{8}+63 a^{5} b^{3}+\frac {621 a^{4} b^{4}}{8}+165 a^{3} b^{5}\right )}{1024}}{\left (a -b \right )^{5} \left (a +b \right )^{5} d \left (\cos \left (2 d x +2 c \right ) a^{2}+4 \cos \left (d x +c \right ) a b +a^{2}+2 b^{2}\right )}\)	\(665\)
risch	\(\text {Expression too large to display}\)	\(1848\)

[In]

int(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/16/(a+b)^3/(cos(d*x+c)-1)^2-1/16*(-3*a+3*b)/(a+b)^4/(cos(d*x+c)-1)+3/16*a*(a-3*b)/(a+b)^5*ln(cos(d*x+c
)-1)-1/2*b^3/(a+b)^3*a^2/(a-b)^3/(b+a*cos(d*x+c))^2+3*a^2*b*(a^4+5*a^2*b^2+2*b^4)/(a-b)^5/(a+b)^5*ln(b+a*cos(d
*x+c))+3*a^2*b^2*(a^2+b^2)/(a+b)^4/(a-b)^4/(b+a*cos(d*x+c))+1/16/(a-b)^3/(cos(d*x+c)+1)^2-1/16*(-3*a-3*b)/(a-b
)^4/(cos(d*x+c)+1)-3/16*a*(a+3*b)/(a-b)^5*ln(cos(d*x+c)+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1803 vs. \(2 (303) = 606\).

Time = 0.56 (sec) , antiderivative size = 1803, normalized size of antiderivative = 5.76 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/16*(76*a^6*b^3 + 36*a^4*b^5 - 108*a^2*b^7 - 4*b^9 + 6*(a^9 + 17*a^7*b^2 - 5*a^5*b^4 - 13*a^3*b^6)*cos(d*x +
c)^5 - 12*(a^8*b - 9*a^6*b^3 - a^4*b^5 + 9*a^2*b^7)*cos(d*x + c)^4 - 2*(5*a^9 + 98*a^7*b^2 - 12*a^5*b^4 - 98*a
^3*b^6 + 7*a*b^8)*cos(d*x + c)^3 + 8*(2*a^8*b - 25*a^6*b^3 - 3*a^4*b^5 + 25*a^2*b^7 + b^9)*cos(d*x + c)^2 + 2*
(55*a^7*b^2 - 9*a^5*b^4 - 51*a^3*b^6 + 5*a*b^8)*cos(d*x + c) + 48*(a^6*b^3 + 5*a^4*b^5 + 2*a^2*b^7 + (a^8*b +
5*a^6*b^3 + 2*a^4*b^5)*cos(d*x + c)^6 + 2*(a^7*b^2 + 5*a^5*b^4 + 2*a^3*b^6)*cos(d*x + c)^5 - (2*a^8*b + 9*a^6*
b^3 - a^4*b^5 - 2*a^2*b^7)*cos(d*x + c)^4 - 4*(a^7*b^2 + 5*a^5*b^4 + 2*a^3*b^6)*cos(d*x + c)^3 + (a^8*b + 3*a^
6*b^3 - 8*a^4*b^5 - 4*a^2*b^7)*cos(d*x + c)^2 + 2*(a^7*b^2 + 5*a^5*b^4 + 2*a^3*b^6)*cos(d*x + c))*log(a*cos(d*
x + c) + b) - 3*(a^7*b^2 + 8*a^6*b^3 + 25*a^5*b^4 + 40*a^4*b^5 + 35*a^3*b^6 + 16*a^2*b^7 + 3*a*b^8 + (a^9 + 8*
a^8*b + 25*a^7*b^2 + 40*a^6*b^3 + 35*a^5*b^4 + 16*a^4*b^5 + 3*a^3*b^6)*cos(d*x + c)^6 + 2*(a^8*b + 8*a^7*b^2 +
25*a^6*b^3 + 40*a^5*b^4 + 35*a^4*b^5 + 16*a^3*b^6 + 3*a^2*b^7)*cos(d*x + c)^5 - (2*a^9 + 16*a^8*b + 49*a^7*b^
2 + 72*a^6*b^3 + 45*a^5*b^4 - 8*a^4*b^5 - 29*a^3*b^6 - 16*a^2*b^7 - 3*a*b^8)*cos(d*x + c)^4 - 4*(a^8*b + 8*a^7
*b^2 + 25*a^6*b^3 + 40*a^5*b^4 + 35*a^4*b^5 + 16*a^3*b^6 + 3*a^2*b^7)*cos(d*x + c)^3 + (a^9 + 8*a^8*b + 23*a^7
*b^2 + 24*a^6*b^3 - 15*a^5*b^4 - 64*a^4*b^5 - 67*a^3*b^6 - 32*a^2*b^7 - 6*a*b^8)*cos(d*x + c)^2 + 2*(a^8*b + 8
*a^7*b^2 + 25*a^6*b^3 + 40*a^5*b^4 + 35*a^4*b^5 + 16*a^3*b^6 + 3*a^2*b^7)*cos(d*x + c))*log(1/2*cos(d*x + c) +
1/2) + 3*(a^7*b^2 - 8*a^6*b^3 + 25*a^5*b^4 - 40*a^4*b^5 + 35*a^3*b^6 - 16*a^2*b^7 + 3*a*b^8 + (a^9 - 8*a^8*b
+ 25*a^7*b^2 - 40*a^6*b^3 + 35*a^5*b^4 - 16*a^4*b^5 + 3*a^3*b^6)*cos(d*x + c)^6 + 2*(a^8*b - 8*a^7*b^2 + 25*a^
6*b^3 - 40*a^5*b^4 + 35*a^4*b^5 - 16*a^3*b^6 + 3*a^2*b^7)*cos(d*x + c)^5 - (2*a^9 - 16*a^8*b + 49*a^7*b^2 - 72
*a^6*b^3 + 45*a^5*b^4 + 8*a^4*b^5 - 29*a^3*b^6 + 16*a^2*b^7 - 3*a*b^8)*cos(d*x + c)^4 - 4*(a^8*b - 8*a^7*b^2 +
25*a^6*b^3 - 40*a^5*b^4 + 35*a^4*b^5 - 16*a^3*b^6 + 3*a^2*b^7)*cos(d*x + c)^3 + (a^9 - 8*a^8*b + 23*a^7*b^2 -
24*a^6*b^3 - 15*a^5*b^4 + 64*a^4*b^5 - 67*a^3*b^6 + 32*a^2*b^7 - 6*a*b^8)*cos(d*x + c)^2 + 2*(a^8*b - 8*a^7*b
^2 + 25*a^6*b^3 - 40*a^5*b^4 + 35*a^4*b^5 - 16*a^3*b^6 + 3*a^2*b^7)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2)
)/((a^12 - 5*a^10*b^2 + 10*a^8*b^4 - 10*a^6*b^6 + 5*a^4*b^8 - a^2*b^10)*d*cos(d*x + c)^6 + 2*(a^11*b - 5*a^9*b
^3 + 10*a^7*b^5 - 10*a^5*b^7 + 5*a^3*b^9 - a*b^11)*d*cos(d*x + c)^5 - (2*a^12 - 11*a^10*b^2 + 25*a^8*b^4 - 30*
a^6*b^6 + 20*a^4*b^8 - 7*a^2*b^10 + b^12)*d*cos(d*x + c)^4 - 4*(a^11*b - 5*a^9*b^3 + 10*a^7*b^5 - 10*a^5*b^7 +
5*a^3*b^9 - a*b^11)*d*cos(d*x + c)^3 + (a^12 - 7*a^10*b^2 + 20*a^8*b^4 - 30*a^6*b^6 + 25*a^4*b^8 - 11*a^2*b^1
0 + 2*b^12)*d*cos(d*x + c)^2 + 2*(a^11*b - 5*a^9*b^3 + 10*a^7*b^5 - 10*a^5*b^7 + 5*a^3*b^9 - a*b^11)*d*cos(d*x
+ c) + (a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12)*d)

Sympy [F]

\[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate(csc(d*x+c)**5/(a+b*sec(d*x+c))**3,x)

[Out]

Integral(csc(c + d*x)**5/(a + b*sec(c + d*x))**3, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (303) = 606\).

Time = 0.23 (sec) , antiderivative size = 707, normalized size of antiderivative = 2.26 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {48 \, {\left (a^{6} b + 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}} - \frac {3 \, {\left (a^{2} + 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} + \frac {3 \, {\left (a^{2} - 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} + \frac {2 \, {\left (38 \, a^{4} b^{3} + 56 \, a^{2} b^{5} + 2 \, b^{7} + 3 \, {\left (a^{7} + 18 \, a^{5} b^{2} + 13 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{5} - 6 \, {\left (a^{6} b - 8 \, a^{4} b^{3} - 9 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} - {\left (5 \, a^{7} + 103 \, a^{5} b^{2} + 91 \, a^{3} b^{4} - 7 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (2 \, a^{6} b - 23 \, a^{4} b^{3} - 26 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (55 \, a^{5} b^{2} + 46 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \cos \left (d x + c\right )\right )}}{a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10} + {\left (a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \cos \left (d x + c\right )^{5} - {\left (2 \, a^{10} - 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} - 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} - b^{10}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{10} - 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} - 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} - 2 \, b^{10}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \cos \left (d x + c\right )}}{16 \, d} \]

[In]

integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/16*(48*(a^6*b + 5*a^4*b^3 + 2*a^2*b^5)*log(a*cos(d*x + c) + b)/(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 +
5*a^2*b^8 - b^10) - 3*(a^2 + 3*a*b)*log(cos(d*x + c) + 1)/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4
- b^5) + 3*(a^2 - 3*a*b)*log(cos(d*x + c) - 1)/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5) + 2*(
38*a^4*b^3 + 56*a^2*b^5 + 2*b^7 + 3*(a^7 + 18*a^5*b^2 + 13*a^3*b^4)*cos(d*x + c)^5 - 6*(a^6*b - 8*a^4*b^3 - 9*
a^2*b^5)*cos(d*x + c)^4 - (5*a^7 + 103*a^5*b^2 + 91*a^3*b^4 - 7*a*b^6)*cos(d*x + c)^3 + 4*(2*a^6*b - 23*a^4*b^
3 - 26*a^2*b^5 - b^7)*cos(d*x + c)^2 + (55*a^5*b^2 + 46*a^3*b^4 - 5*a*b^6)*cos(d*x + c))/(a^8*b^2 - 4*a^6*b^4
+ 6*a^4*b^6 - 4*a^2*b^8 + b^10 + (a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*cos(d*x + c)^6 + 2*(a^9*
b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)^5 - (2*a^10 - 9*a^8*b^2 + 16*a^6*b^4 - 14*a^4*b^6
+ 6*a^2*b^8 - b^10)*cos(d*x + c)^4 - 4*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)^3 + (a
^10 - 6*a^8*b^2 + 14*a^6*b^4 - 16*a^4*b^6 + 9*a^2*b^8 - 2*b^10)*cos(d*x + c)^2 + 2*(a^9*b - 4*a^7*b^3 + 6*a^5*
b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1551 vs. \(2 (303) = 606\).

Time = 0.48 (sec) , antiderivative size = 1551, normalized size of antiderivative = 4.96 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/64*(12*(a^2 - 3*a*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*
b^3 + 5*a*b^4 + b^5) + 192*(a^6*b + 5*a^4*b^3 + 2*a^2*b^5)*log(abs(-a - b - a*(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) + b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^
10) - (8*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 12*a^2*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 4*b^3*(c
os(d*x + c) - 1)/(cos(d*x + c) + 1) - a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 3*a^2*b*(cos(d*x + c) -
1)^2/(cos(d*x + c) + 1)^2 - 3*a*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + b^3*(cos(d*x + c) - 1)^2/(cos(
d*x + c) + 1)^2)/(a^6 - 6*a^5*b + 15*a^4*b^2 - 20*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6) - (a^8 - 2*a^7*b - 2*a
^6*b^2 + 6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8 - 6*a^8*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 20*
a^7*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 12*a^6*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 28*a^5*b^3*(c
os(d*x + c) - 1)/(cos(d*x + c) + 1) + 40*a^4*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 4*a^3*b^5*(cos(d*x +
c) - 1)/(cos(d*x + c) + 1) - 20*a^2*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 12*a*b^7*(cos(d*x + c) - 1)/(c
os(d*x + c) + 1) - 2*b^8*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 6*a^8*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)
^2 + 163*a^7*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 257*a^6*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)
^2 + 339*a^5*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 203*a^4*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) +
1)^2 - 223*a^3*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 309*a^2*b^6*(cos(d*x + c) - 1)^2/(cos(d*x + c)
+ 1)^2 - 23*a*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 7*b^8*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
+ 10*a^8*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 186*a^7*b*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 274
*a^6*b^2*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 890*a^5*b^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 8
94*a^4*b^4*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 478*a^3*b^5*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 -
374*a^2*b^6*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 18*a*b^7*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 -
4*b^8*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 9*a^8*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 45*a^7*b*(
cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 45*a^6*b^2*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 63*a^5*b^3*(
cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 117*a^4*b^4*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 9*a^3*b^5*(
cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 63*a^2*b^6*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 27*a*b^7*(co
s(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4)/((a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*
b^6 + 4*a^2*b^7 + a*b^8 - b^9)*(a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x + c) - 1)/(cos(d*x + c) +
1) + a*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)^2))/d

Mupad [B] (verification not implemented)

Time = 14.98 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.15 \[ \int \frac {\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3}{16\,{\left (a+b\right )}^3}-\frac {15\,b}{16\,{\left (a+b\right )}^4}+\frac {3\,b^2}{4\,{\left (a+b\right )}^5}\right )}{d}+\frac {\frac {19\,a^4\,b^3+28\,a^2\,b^5+b^7}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (-2\,a^6\,b+23\,a^4\,b^3+26\,a^2\,b^5+b^7\right )}{2\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\cos \left (c+d\,x\right )}^4\,\left (-a^6\,b+8\,a^4\,b^3+9\,a^2\,b^5\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\cos \left (c+d\,x\right )}^5\,\left (a^7+18\,a^5\,b^2+13\,a^3\,b^4\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {a\,\cos \left (c+d\,x\right )\,\left (55\,a^4\,b^2+46\,a^2\,b^4-5\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {a\,{\cos \left (c+d\,x\right )}^3\,\left (5\,a^6+103\,a^4\,b^2+91\,a^2\,b^4-7\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2\,\left (a^2-2\,b^2\right )-{\cos \left (c+d\,x\right )}^4\,\left (2\,a^2-b^2\right )+b^2+a^2\,{\cos \left (c+d\,x\right )}^6+2\,a\,b\,\cos \left (c+d\,x\right )-4\,a\,b\,{\cos \left (c+d\,x\right )}^3+2\,a\,b\,{\cos \left (c+d\,x\right )}^5\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{4\,{\left (a-b\right )}^5}+\frac {15\,b}{16\,{\left (a-b\right )}^4}+\frac {3}{16\,{\left (a-b\right )}^3}\right )}{d}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (3\,a^6\,b+15\,a^4\,b^3+6\,a^2\,b^5\right )}{d\,\left (a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}\right )} \]

[In]

int(1/(sin(c + d*x)^5*(a + b/cos(c + d*x))^3),x)

[Out]

(log(cos(c + d*x) - 1)*(3/(16*(a + b)^3) - (15*b)/(16*(a + b)^4) + (3*b^2)/(4*(a + b)^5)))/d + ((b^7 + 28*a^2*
b^5 + 19*a^4*b^3)/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (cos(c + d*x)^2*(b^7 - 2*a^6*b + 26*a^
2*b^5 + 23*a^4*b^3))/(2*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (3*cos(c + d*x)^4*(9*a^2*b^5 - a^6*
b + 8*a^4*b^3))/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (3*cos(c + d*x)^5*(a^7 + 13*a^3*b^4 + 18
*a^5*b^2))/(8*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (a*cos(c + d*x)*(46*a^2*b^4 - 5*b^6 + 55*a^4*
b^2))/(8*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (a*cos(c + d*x)^3*(5*a^6 - 7*b^6 + 91*a^2*b^4 + 10
3*a^4*b^2))/(8*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)))/(d*(cos(c + d*x)^2*(a^2 - 2*b^2) - cos(c + d*
x)^4*(2*a^2 - b^2) + b^2 + a^2*cos(c + d*x)^6 + 2*a*b*cos(c + d*x) - 4*a*b*cos(c + d*x)^3 + 2*a*b*cos(c + d*x)
^5)) - (log(cos(c + d*x) + 1)*((3*b^2)/(4*(a - b)^5) + (15*b)/(16*(a - b)^4) + 3/(16*(a - b)^3)))/d + (log(b +
a*cos(c + d*x))*(3*a^6*b + 6*a^2*b^5 + 15*a^4*b^3))/(d*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5
*a^8*b^2))