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

Optimal. Leaf size=313 \[ \frac{3 a^2 b^2 \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^4 (a \cos (c+d x)+b)}-\frac{a^2 b^3}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)^2}+\frac{3 a^2 b \left (5 a^2 b^2+a^4+2 b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^5}+\frac{\csc ^4(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 d \left (a^2-b^2\right )^3}+\frac{\csc ^2(c+d x) \left (4 b \left (8 a^2 b^2+3 a^4+b^4\right )-3 a \left (10 a^2 b^2+a^4+5 b^4\right ) \cos (c+d x)\right )}{8 d \left (a^2-b^2\right )^4}+\frac{3 a (a-3 b) \log (1-\cos (c+d x))}{16 d (a+b)^5}-\frac{3 a (a+3 b) \log (\cos (c+d x)+1)}{16 d (a-b)^5} \]

[Out]

-(a^2*b^3)/(2*(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)

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Rubi [A]  time = 1.00667, antiderivative size = 313, 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} \[ \frac{3 a^2 b^2 \left (a^2+b^2\right )}{d \left (a^2-b^2\right )^4 (a \cos (c+d x)+b)}-\frac{a^2 b^3}{2 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)^2}+\frac{3 a^2 b \left (5 a^2 b^2+a^4+2 b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^5}+\frac{\csc ^4(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{4 d \left (a^2-b^2\right )^3}+\frac{\csc ^2(c+d x) \left (4 b \left (8 a^2 b^2+3 a^4+b^4\right )-3 a \left (10 a^2 b^2+a^4+5 b^4\right ) \cos (c+d x)\right )}{8 d \left (a^2-b^2\right )^4}+\frac{3 a (a-3 b) \log (1-\cos (c+d x))}{16 d (a+b)^5}-\frac{3 a (a+3 b) \log (\cos (c+d x)+1)}{16 d (a-b)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*b^3)/(2*(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 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))^3} \, dx &=-\int \frac{\cot ^3(c+d x) \csc ^2(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac{a^5 \operatorname{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 \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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]  time = 4.50544, size = 496, normalized size = 1.58 \[ \frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left (-\frac{384 i a^2 b \left (5 a^2 b^2+a^4+2 b^4\right ) (c+d x) (a \cos (c+d x)+b)^2}{(a-b)^5 (a+b)^5}+\frac{192 a^2 b^2 (a-i b) (a+i b) (a \cos (c+d x)+b)}{(a-b)^4 (a+b)^4}+\frac{192 a^2 b \left (5 a^2 b^2+a^4+2 b^4\right ) (a \cos (c+d x)+b)^2 \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^5}+\frac{32 a^2 b^3}{(b-a)^3 (a+b)^3}-\frac{12 a (a+3 b) \log \left (\cos ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^2}{(a-b)^5}+\frac{24 i a (a+3 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)^2}{(a-b)^5}-\frac{24 i a (a-3 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)^2}{(a+b)^5}-\frac{\csc ^4\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2}{(a+b)^3}+\frac{6 (b-a) \csc ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2}{(a+b)^4}+\frac{\sec ^4\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2}{(a-b)^3}+\frac{6 (a+b) \sec ^2\left (\frac{1}{2} (c+d x)\right ) (a \cos (c+d x)+b)^2}{(a-b)^4}+\frac{12 a (a-3 b) \log \left (\sin ^2\left (\frac{1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b)^2}{(a+b)^5}\right )}{64 d (a+b \sec (c+d x))^3} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.093, size = 427, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{b}^{3}}{2\,d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3} \left ( b+a\cos \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{{a}^{4}{b}^{2}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+3\,{\frac{{a}^{2}{b}^{4}}{d \left ( a+b \right ) ^{4} \left ( a-b \right ) ^{4} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+3\,{\frac{{a}^{6}b\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{5} \left ( a-b \right ) ^{5}}}+15\,{\frac{{a}^{4}{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{5} \left ( a-b \right ) ^{5}}}+6\,{\frac{{a}^{2}{b}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{5} \left ( a-b \right ) ^{5}}}+{\frac{1}{16\,d \left ( a-b \right ) ^{3} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2}}}+{\frac{3\,a}{16\,d \left ( a-b \right ) ^{4} \left ( \cos \left ( dx+c \right ) +1 \right ) }}+{\frac{3\,b}{16\,d \left ( a-b \right ) ^{4} \left ( \cos \left ( dx+c \right ) +1 \right ) }}-{\frac{3\,{a}^{2}\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{16\,d \left ( a-b \right ) ^{5}}}-{\frac{9\,a\ln \left ( \cos \left ( dx+c \right ) +1 \right ) b}{16\,d \left ( a-b \right ) ^{5}}}-{\frac{1}{16\,d \left ( a+b \right ) ^{3} \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a}{16\,d \left ( a+b \right ) ^{4} \left ( -1+\cos \left ( dx+c \right ) \right ) }}-{\frac{3\,b}{16\,d \left ( a+b \right ) ^{4} \left ( -1+\cos \left ( dx+c \right ) \right ) }}+{\frac{3\,{a}^{2}\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{16\,d \left ( a+b \right ) ^{5}}}-{\frac{9\,a\ln \left ( -1+\cos \left ( dx+c \right ) \right ) b}{16\,d \left ( a+b \right ) ^{5}}} \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))^3,x)

[Out]

-1/2/d*b^3/(a+b)^3*a^2/(a-b)^3/(b+a*cos(d*x+c))^2+3/d*a^4*b^2/(a+b)^4/(a-b)^4/(b+a*cos(d*x+c))+3/d*a^2*b^4/(a+
b)^4/(a-b)^4/(b+a*cos(d*x+c))+3/d*b*a^6/(a+b)^5/(a-b)^5*ln(b+a*cos(d*x+c))+15/d*b^3*a^4/(a+b)^5/(a-b)^5*ln(b+a
*cos(d*x+c))+6/d*b^5*a^2/(a+b)^5/(a-b)^5*ln(b+a*cos(d*x+c))+1/16/d/(a-b)^3/(cos(d*x+c)+1)^2+3/16/d/(a-b)^4/(co
s(d*x+c)+1)*a+3/16/d/(a-b)^4/(cos(d*x+c)+1)*b-3/16/d*a^2/(a-b)^5*ln(cos(d*x+c)+1)-9/16/d*a/(a-b)^5*ln(cos(d*x+
c)+1)*b-1/16/d/(a+b)^3/(-1+cos(d*x+c))^2+3/16/d/(a+b)^4/(-1+cos(d*x+c))*a-3/16/d/(a+b)^4/(-1+cos(d*x+c))*b+3/1
6/d*a^2/(a+b)^5*ln(-1+cos(d*x+c))-9/16/d*a/(a+b)^5*ln(-1+cos(d*x+c))*b

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Maxima [B]  time = 1.04808, size = 954, normalized size = 3.05 \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))^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

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Fricas [B]  time = 5.87997, size = 3970, normalized size = 12.68 \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))^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)

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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))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.60742, size = 2094, normalized size = 6.69 \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))^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