3.215 \(\int \frac{\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=259 \[ \frac{a^3 b^2}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 d (a+b)^4}-\frac{\left (3 a^2+4 a b-b^2\right ) \log (\cos (c+d x)+1)}{16 d (a-b)^4}+\frac{2 a^3 b \left (a^2+2 b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^4(c+d x) \left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right )}{4 d \left (a^2-b^2\right )^2}+\frac{\csc ^2(c+d x) \left (8 a b \left (a^2+b^2\right )-\left (12 a^2 b^2+3 a^4+b^4\right ) \cos (c+d x)\right )}{8 d \left (a^2-b^2\right )^3} \]

[Out]

(a^3*b^2)/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((8*a*b*(a^2 + b^2) - (3*a^4 + 12*a^2*b^2 + b^4)*Cos[c + d*
x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^3*d) + ((2*a*b - (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)^2
*d) + ((3*a^2 - 4*a*b - b^2)*Log[1 - Cos[c + d*x]])/(16*(a + b)^4*d) - ((3*a^2 + 4*a*b - b^2)*Log[1 + Cos[c +
d*x]])/(16*(a - b)^4*d) + (2*a^3*b*(a^2 + 2*b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)

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Rubi [A]  time = 0.741104, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3872, 2837, 12, 1647, 1629} \[ \frac{a^3 b^2}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 d (a+b)^4}-\frac{\left (3 a^2+4 a b-b^2\right ) \log (\cos (c+d x)+1)}{16 d (a-b)^4}+\frac{2 a^3 b \left (a^2+2 b^2\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac{\csc ^4(c+d x) \left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right )}{4 d \left (a^2-b^2\right )^2}+\frac{\csc ^2(c+d x) \left (8 a b \left (a^2+b^2\right )-\left (12 a^2 b^2+3 a^4+b^4\right ) \cos (c+d x)\right )}{8 d \left (a^2-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]

[Out]

(a^3*b^2)/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((8*a*b*(a^2 + b^2) - (3*a^4 + 12*a^2*b^2 + b^4)*Cos[c + d*
x])*Csc[c + d*x]^2)/(8*(a^2 - b^2)^3*d) + ((2*a*b - (a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^4)/(4*(a^2 - b^2)^2
*d) + ((3*a^2 - 4*a*b - b^2)*Log[1 - Cos[c + d*x]])/(16*(a + b)^4*d) - ((3*a^2 + 4*a*b - b^2)*Log[1 + Cos[c +
d*x]])/(16*(a - b)^4*d) + (2*a^3*b*(a^2 + 2*b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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]

Rule 2837

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*x)/b)^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 12

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

Rule 1647

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 1629

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]

Rubi steps

\begin{align*} \int \frac{\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cot ^2(c+d x) \csc ^3(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x^2}{a^2 (-b+x)^2 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{x^2}{(-b+x)^2 \left (a^2-x^2\right )^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac{\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac{a \operatorname{Subst}\left (\int \frac{-\frac{a^2 b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^2}+\frac{2 a^2 b \left (a^2-3 b^2\right ) x}{\left (a^2-b^2\right )^2}+\frac{3 a^2 \left (a^2+b^2\right ) x^2}{\left (a^2-b^2\right )^2}}{(-b+x)^2 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{4 d}\\ &=\frac{\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac{\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^2 b^2 \left (5 a^4+12 a^2 b^2-b^4\right )}{\left (a^2-b^2\right )^3}+\frac{2 a^2 b \left (5 a^2+b^2\right ) x}{\left (a^2-b^2\right )^2}+\frac{a^2 \left (3 a^4+12 a^2 b^2+b^4\right ) x^2}{\left (a^2-b^2\right )^3}}{(-b+x)^2 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{8 a d}\\ &=\frac{\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac{\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac{\operatorname{Subst}\left (\int \left (\frac{a \left (3 a^2+4 a b-b^2\right )}{2 (a-b)^4 (a-x)}+\frac{8 a^4 b^2}{\left (a^2-b^2\right )^3 (b-x)^2}-\frac{16 a^4 b \left (a^2+2 b^2\right )}{\left (a^2-b^2\right )^4 (b-x)}+\frac{a \left (3 a^2-4 a b-b^2\right )}{2 (a+b)^4 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{8 a d}\\ &=\frac{a^3 b^2}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac{\left (8 a b \left (a^2+b^2\right )-\left (3 a^4+12 a^2 b^2+b^4\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{8 \left (a^2-b^2\right )^3 d}+\frac{\left (2 a b-\left (a^2+b^2\right ) \cos (c+d x)\right ) \csc ^4(c+d x)}{4 \left (a^2-b^2\right )^2 d}+\frac{\left (3 a^2-4 a b-b^2\right ) \log (1-\cos (c+d x))}{16 (a+b)^4 d}-\frac{\left (3 a^2+4 a b-b^2\right ) \log (1+\cos (c+d x))}{16 (a-b)^4 d}+\frac{2 a^3 b \left (a^2+2 b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end{align*}

Mathematica [A]  time = 1.38311, size = 320, normalized size = 1.24 \[ \frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left (\frac{8 \left (-3 a^2-4 a b+b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a-b)^4}+\frac{128 a^3 b \left (a^2+2 b^2\right ) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}+\frac{8 \left (3 a^2-4 a b-b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)}{(a+b)^4}+\frac{64 a^3 b^2}{(a-b)^3 (a+b)^3}-\frac{\csc ^4\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^2}+\frac{2 (b-3 a) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^3}+\frac{\sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^2}+\frac{2 (3 a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^3}\right )}{64 d (a+b \sec (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]

[Out]

((b + a*Cos[c + d*x])*((64*a^3*b^2)/((a - b)^3*(a + b)^3) + (2*(-3*a + b)*(b + a*Cos[c + d*x])*Csc[(c + d*x)/2
]^2)/(a + b)^3 - ((b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^4)/(a + b)^2 + (8*(-3*a^2 - 4*a*b + b^2)*(b + a*Cos[c
+ d*x])*Log[Cos[(c + d*x)/2]])/(a - b)^4 + (128*a^3*b*(a^2 + 2*b^2)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x
]])/(a^2 - b^2)^4 + (8*(3*a^2 - 4*a*b - b^2)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]])/(a + b)^4 + (2*(3*a +
b)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a - b)^3 + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/(a - b)^2)*
Sec[c + d*x]^2)/(64*d*(a + b*Sec[c + d*x])^2)

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Maple [A]  time = 0.086, size = 368, normalized size = 1.4 \begin{align*}{\frac{{a}^{3}{b}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+2\,{\frac{b{a}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+4\,{\frac{{a}^{3}{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4}}}+{\frac{1}{16\,d \left ( a-b \right ) ^{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{3\,a}{16\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{b}{16\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{3\,\ln \left ( \cos \left ( dx+c \right ) +1 \right ){a}^{2}}{16\,d \left ( a-b \right ) ^{4}}}-{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) ab}{4\,d \left ( a-b \right ) ^{4}}}+{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ){b}^{2}}{16\,d \left ( a-b \right ) ^{4}}}-{\frac{1}{16\,d \left ( a+b \right ) ^{2} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a}{16\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{b}{16\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{3\,\ln \left ( -1+\cos \left ( dx+c \right ) \right ){a}^{2}}{16\,d \left ( a+b \right ) ^{4}}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) ab}{4\,d \left ( a+b \right ) ^{4}}}-{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ){b}^{2}}{16\,d \left ( a+b \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x)

[Out]

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

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Maxima [B]  time = 1.04761, size = 690, normalized size = 2.66 \begin{align*} \frac{\frac{32 \,{\left (a^{5} b + 2 \, a^{3} b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac{{\left (3 \, a^{2} + 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac{{\left (3 \, a^{2} - 4 \, a b - b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac{2 \,{\left (20 \, a^{3} b^{2} + 4 \, a b^{4} +{\left (3 \, a^{5} + 20 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{4} -{\left (5 \, a^{4} b - 4 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} -{\left (5 \, a^{5} + 36 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} +{\left (7 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7} +{\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{5} +{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \,{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} +{\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )}}{16 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

1/16*(32*(a^5*b + 2*a^3*b^3)*log(a*cos(d*x + c) + b)/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) - (3*a^2
+ 4*a*b - b^2)*log(cos(d*x + c) + 1)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + (3*a^2 - 4*a*b - b^2)*log(c
os(d*x + c) - 1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 2*(20*a^3*b^2 + 4*a*b^4 + (3*a^5 + 20*a^3*b^2 +
a*b^4)*cos(d*x + c)^4 - (5*a^4*b - 4*a^2*b^3 - b^5)*cos(d*x + c)^3 - (5*a^5 + 36*a^3*b^2 + 7*a*b^4)*cos(d*x +
c)^2 + (7*a^4*b - 8*a^2*b^3 + b^5)*cos(d*x + c))/(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7 + (a^7 - 3*a^5*b^2 + 3*
a^3*b^4 - a*b^6)*cos(d*x + c)^5 + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^4 - 2*(a^7 - 3*a^5*b^2 +
3*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2
+ 3*a^3*b^4 - a*b^6)*cos(d*x + c)))/d

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Fricas [B]  time = 4.19684, size = 2653, normalized size = 10.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/16*(40*a^5*b^2 - 32*a^3*b^4 - 8*a*b^6 + 2*(3*a^7 + 17*a^5*b^2 - 19*a^3*b^4 - a*b^6)*cos(d*x + c)^4 - 2*(5*a^
6*b - 9*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^3 - 2*(5*a^7 + 31*a^5*b^2 - 29*a^3*b^4 - 7*a*b^6)*cos(d*x + c)
^2 + 2*(7*a^6*b - 15*a^4*b^3 + 9*a^2*b^5 - b^7)*cos(d*x + c) + 32*(a^5*b^2 + 2*a^3*b^4 + (a^6*b + 2*a^4*b^3)*c
os(d*x + c)^5 + (a^5*b^2 + 2*a^3*b^4)*cos(d*x + c)^4 - 2*(a^6*b + 2*a^4*b^3)*cos(d*x + c)^3 - 2*(a^5*b^2 + 2*a
^3*b^4)*cos(d*x + c)^2 + (a^6*b + 2*a^4*b^3)*cos(d*x + c))*log(a*cos(d*x + c) + b) - (3*a^6*b + 16*a^5*b^2 + 3
3*a^4*b^3 + 32*a^3*b^4 + 13*a^2*b^5 - b^7 + (3*a^7 + 16*a^6*b + 33*a^5*b^2 + 32*a^4*b^3 + 13*a^3*b^4 - a*b^6)*
cos(d*x + c)^5 + (3*a^6*b + 16*a^5*b^2 + 33*a^4*b^3 + 32*a^3*b^4 + 13*a^2*b^5 - b^7)*cos(d*x + c)^4 - 2*(3*a^7
+ 16*a^6*b + 33*a^5*b^2 + 32*a^4*b^3 + 13*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - 2*(3*a^6*b + 16*a^5*b^2 + 33*a^4*
b^3 + 32*a^3*b^4 + 13*a^2*b^5 - b^7)*cos(d*x + c)^2 + (3*a^7 + 16*a^6*b + 33*a^5*b^2 + 32*a^4*b^3 + 13*a^3*b^4
- a*b^6)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (3*a^6*b - 16*a^5*b^2 + 33*a^4*b^3 - 32*a^3*b^4 + 13*a^2
*b^5 - b^7 + (3*a^7 - 16*a^6*b + 33*a^5*b^2 - 32*a^4*b^3 + 13*a^3*b^4 - a*b^6)*cos(d*x + c)^5 + (3*a^6*b - 16*
a^5*b^2 + 33*a^4*b^3 - 32*a^3*b^4 + 13*a^2*b^5 - b^7)*cos(d*x + c)^4 - 2*(3*a^7 - 16*a^6*b + 33*a^5*b^2 - 32*a
^4*b^3 + 13*a^3*b^4 - a*b^6)*cos(d*x + c)^3 - 2*(3*a^6*b - 16*a^5*b^2 + 33*a^4*b^3 - 32*a^3*b^4 + 13*a^2*b^5 -
b^7)*cos(d*x + c)^2 + (3*a^7 - 16*a^6*b + 33*a^5*b^2 - 32*a^4*b^3 + 13*a^3*b^4 - a*b^6)*cos(d*x + c))*log(-1/
2*cos(d*x + c) + 1/2))/((a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c)^5 + (a^8*b - 4*a^6*b^
3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*cos(d*x + c)^4 - 2*(a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(
d*x + c)^3 - 2*(a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*cos(d*x + c)^2 + (a^9 - 4*a^7*b^2 + 6*a^5*b
^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c) + (a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d)

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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(csc(d*x+c)**5/(a+b*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.44167, size = 959, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/64*(4*(3*a^2 - 4*a*b - b^2)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4
*a*b^3 + b^4) + 128*(a^5*b + 2*a^3*b^3)*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^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) - (8*a^2*(cos(d*x + c) - 1)/(co
s(d*x + c) + 1) - 8*a*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2
+ 2*a*b*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)/(a^4 - 4*a^
3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (a^2 + 2*a*b + b^2 - 8*a^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 8*a*b*(c
os(d*x + c) - 1)/(cos(d*x + c) + 1) + 18*a^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 24*a*b*(cos(d*x + c)
- 1)^2/(cos(d*x + c) + 1)^2 - 6*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2)*(cos(d*x + c) + 1)^2/((a^4 + 4*
a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*(cos(d*x + c) - 1)^2) - 128*(a^6*b + a^4*b^3 + 2*a^3*b^4 + a^6*b*(cos(d*x +
c) - 1)/(cos(d*x + c) + 1) - a^5*b^2*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2*a^4*b^3*(cos(d*x + c) - 1)/(co
s(d*x + c) + 1) - 2*a^3*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 +
b^8)*(a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))))/d