Integrand size = 21, antiderivative size = 229 \[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {b^3}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac {(a-2 b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac {(a+2 b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac {b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \] Output:
-1/2*b^3/(a^2-b^2)^2/d/(b+a*cos(d*x+c))^2+b^2*(3*a^2+b^2)/(a^2-b^2)^3/d/(b +a*cos(d*x+c))+1/2*(b*(3*a^2+b^2)-a*(a^2+3*b^2)*cos(d*x+c))*csc(d*x+c)^2/( a^2-b^2)^3/d+1/4*(a-2*b)*ln(1-cos(d*x+c))/(a+b)^4/d-1/4*(a+2*b)*ln(1+cos(d *x+c))/(a-b)^4/d+b*(3*a^4+8*a^2*b^2+b^4)*ln(b+a*cos(d*x+c))/(a^2-b^2)^4/d
Result contains complex when optimal does not.
Time = 6.39 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.45 \[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=-\frac {2 i \left (3 a^4 b+8 a^2 b^3+b^5\right ) (c+d x)}{(a-b)^4 (a+b)^4 d}-\frac {i (-a-2 b) \arctan (\tan (c+d x))}{2 (-a+b)^4 d}-\frac {i (a-2 b) \arctan (\tan (c+d x))}{2 (a+b)^4 d}-\frac {b^3}{2 (-a+b)^2 (a+b)^2 d (b+a \cos (c+d x))^2}-\frac {b^2 \left (3 a^2+b^2\right )}{(-a+b)^3 (a+b)^3 d (b+a \cos (c+d x))}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 (a+b)^3 d}+\frac {(-a-2 b) \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{4 (-a+b)^4 d}+\frac {\left (3 a^4 b+8 a^2 b^3+b^5\right ) \log (b+a \cos (c+d x))}{\left (-a^2+b^2\right )^4 d}+\frac {(a-2 b) \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{4 (a+b)^4 d}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 (-a+b)^3 d} \] Input:
Integrate[Csc[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]
Output:
((-2*I)*(3*a^4*b + 8*a^2*b^3 + b^5)*(c + d*x))/((a - b)^4*(a + b)^4*d) - ( (I/2)*(-a - 2*b)*ArcTan[Tan[c + d*x]])/((-a + b)^4*d) - ((I/2)*(a - 2*b)*A rcTan[Tan[c + d*x]])/((a + b)^4*d) - b^3/(2*(-a + b)^2*(a + b)^2*d*(b + a* Cos[c + d*x])^2) - (b^2*(3*a^2 + b^2))/((-a + b)^3*(a + b)^3*d*(b + a*Cos[ c + d*x])) - Csc[(c + d*x)/2]^2/(8*(a + b)^3*d) + ((-a - 2*b)*Log[Cos[(c + d*x)/2]^2])/(4*(-a + b)^4*d) + ((3*a^4*b + 8*a^2*b^3 + b^5)*Log[b + a*Cos [c + d*x]])/((-a^2 + b^2)^4*d) + ((a - 2*b)*Log[Sin[(c + d*x)/2]^2])/(4*(a + b)^4*d) - Sec[(c + d*x)/2]^2/(8*(-a + b)^3*d)
Time = 0.88 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4359, 25, 25, 3042, 25, 3200, 601, 2160, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^3 \left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4359 |
| \(\displaystyle \int -\frac {\cot ^3(c+d x)}{(-a \cos (c+d x)-b)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cot ^3(c+d x)}{(b+a \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cot ^3(c+d x)}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan \left (c+d x-\frac {\pi }{2}\right )^3}{\left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}{\left (b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle -\frac {\int \frac {a^3 \cos ^3(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle -\frac {-\frac {\int \frac {-\frac {\left (a^2+3 b^2\right ) \cos ^3(c+d x) a^7}{\left (a^2-b^2\right )^3}+\frac {b \left (3 a^2-7 b^2\right ) \cos ^2(c+d x) a^6}{\left (a^2-b^2\right )^3}+\frac {b^3 \left (a^2+3 b^2\right ) a^4}{\left (a^2-b^2\right )^3}+\frac {b^2 \left (3 a^4+3 b^2 a^2-2 b^4\right ) \cos (c+d x) a^3}{\left (a^2-b^2\right )^3}}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 a^2}-\frac {a^2 \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle -\frac {-\frac {\int \left (\frac {2 a^2 b^3}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^3}-\frac {2 a^2 \left (3 a^2+b^2\right ) b^2}{\left (a^2-b^2\right )^3 (b+a \cos (c+d x))^2}+\frac {2 a^2 \left (3 a^4+8 b^2 a^2+b^4\right ) b}{\left (a^2-b^2\right )^4 (b+a \cos (c+d x))}-\frac {a^2 (a-2 b)}{2 (a+b)^4 (a-a \cos (c+d x))}-\frac {a^2 (a+2 b)}{2 (a-b)^4 (\cos (c+d x) a+a)}\right )d(a \cos (c+d x))}{2 a^2}-\frac {a^2 \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {a^2 \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}-\frac {\frac {2 a^2 b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {a^2 b^3}{\left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac {a^2 (a-2 b) \log (a-a \cos (c+d x))}{2 (a+b)^4}-\frac {a^2 (a+2 b) \log (a \cos (c+d x)+a)}{2 (a-b)^4}+\frac {2 a^2 b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}}{2 a^2}}{d}\) |
Input:
Int[Csc[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]
Output:
-((-1/2*(a^2*(b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x]))/((a^2 - b^2 )^3*(a^2 - a^2*Cos[c + d*x]^2)) - (-((a^2*b^3)/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2)) + (2*a^2*b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*(b + a*Cos[c + d*x ])) + (a^2*(a - 2*b)*Log[a - a*Cos[c + d*x]])/(2*(a + b)^4) - (a^2*(a + 2* b)*Log[a + a*Cos[c + d*x]])/(2*(a - b)^4) + (2*a^2*b*(3*a^4 + 8*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^4)/(2*a^2))/d)
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(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] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m _.), x_Symbol] :> Int[Cot[e + f*x]^p*(b + a*Sin[e + f*x])^m, x] /; FreeQ[{a , b, e, f, p}, x] && IntegerQ[m] && EqQ[m, p]
Time = 1.23 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{4 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -2 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{4 \left (a -b \right )^{4}}-\frac {b^{3}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{4}+8 a^{2} b^{2}+b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {1}{4 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (a -2 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{4 \left (a +b \right )^{4}}}{d}\) | \(196\) |
| default | \(\frac {\frac {1}{4 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -2 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{4 \left (a -b \right )^{4}}-\frac {b^{3}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{4}+8 a^{2} b^{2}+b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}+\frac {1}{4 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (a -2 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{4 \left (a +b \right )^{4}}}{d}\) | \(196\) |
| norman | \(\frac {-\frac {1}{8 d \left (a +b \right )}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 d \left (a -b \right )}+\frac {\left (a^{5}+22 a^{3} b^{2}+13 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 d \left (a^{6}-2 a^{5} b -a^{4} b^{2}+4 a^{3} b^{3}-a^{2} b^{4}-2 a \,b^{5}+b^{6}\right )}-\frac {\left (2 a^{5}+5 a^{4} b +44 a^{3} b^{2}+18 a^{2} b^{3}+26 a \,b^{4}+b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}+\frac {b \left (3 a^{4}+8 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}+\frac {\left (a -2 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}\) | \(382\) |
| parallelrisch | \(\frac {192 b \left (a^{4}+\frac {8}{3} a^{2} b^{2}+\frac {1}{3} b^{4}\right ) \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )+32 \left (a -2 b \right ) \left (a -b \right )^{4} \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\left (\left (8 a^{6} b -4 a^{5} b^{2}-96 a^{4} b^{3}-88 a^{3} b^{4}-48 a^{2} b^{5}-52 a \,b^{6}\right ) \cos \left (2 d x +2 c \right )-4 a \left (a^{4}-a^{3} b +11 a^{2} b^{2}+a \,b^{3}+6 b^{4}\right ) \left (a +b \right )^{2} \cos \left (3 d x +3 c \right )-a^{2} \left (a +b \right ) \left (a^{4}+a^{3} b +21 a^{2} b^{2}+5 a \,b^{3}+8 b^{4}\right ) \cos \left (4 d x +4 c \right )-12 \left (a^{4}-\frac {7}{3} a^{3} b -\frac {11}{3} a^{2} b^{2}+\frac {7}{3} a \,b^{3}-\frac {10}{3} b^{4}\right ) a \left (a +b \right )^{2} \cos \left (d x +c \right )+\left (a^{2}+a b +2 b^{2}\right ) \left (a^{5}+9 a^{4} b +15 a^{3} b^{2}+41 a^{2} b^{3}+30 a \,b^{4}-8 b^{5}\right )\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+64 b^{7}\right ) \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-64 b^{7} \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{64 \left (a -b \right )^{4} \left (a +b \right )^{4} d \left (a^{2} \cos \left (2 d x +2 c \right )+4 a b \cos \left (d x +c \right )+a^{2}+2 b^{2}\right )}\) | \(468\) |
| risch | \(\text {Expression too large to display}\) | \(1332\) |
Input:
int(csc(d*x+c)^3/(a+b*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(1/4/(a-b)^3/(1+cos(d*x+c))+1/4/(a-b)^4*(-a-2*b)*ln(1+cos(d*x+c))-1/2* b^3/(a+b)^2/(a-b)^2/(b+a*cos(d*x+c))^2+b*(3*a^4+8*a^2*b^2+b^4)/(a+b)^4/(a- b)^4*ln(b+a*cos(d*x+c))+b^2*(3*a^2+b^2)/(a+b)^3/(a-b)^3/(b+a*cos(d*x+c))+1 /4/(a+b)^3/(-1+cos(d*x+c))+1/4*(a-2*b)/(a+b)^4*ln(-1+cos(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (221) = 442\).
Time = 0.26 (sec) , antiderivative size = 1071, normalized size of antiderivative = 4.68 \[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(csc(d*x+c)^3/(a+b*sec(d*x+c))^3,x, algorithm="fricas")
Output:
-1/4*(16*a^4*b^3 - 8*a^2*b^5 - 8*b^7 - 2*(a^7 + 8*a^5*b^2 - 7*a^3*b^4 - 2* a*b^6)*cos(d*x + c)^3 + 2*(a^6*b - 11*a^4*b^3 + 7*a^2*b^5 + 3*b^7)*cos(d*x + c)^2 + 2*(11*a^5*b^2 - 10*a^3*b^4 - a*b^6)*cos(d*x + c) + 4*(3*a^4*b^3 + 8*a^2*b^5 + b^7 - (3*a^6*b + 8*a^4*b^3 + a^2*b^5)*cos(d*x + c)^4 - 2*(3* a^5*b^2 + 8*a^3*b^4 + a*b^6)*cos(d*x + c)^3 + (3*a^6*b + 5*a^4*b^3 - 7*a^2 *b^5 - b^7)*cos(d*x + c)^2 + 2*(3*a^5*b^2 + 8*a^3*b^4 + a*b^6)*cos(d*x + c ))*log(a*cos(d*x + c) + b) - (a^5*b^2 + 6*a^4*b^3 + 14*a^3*b^4 + 16*a^2*b^ 5 + 9*a*b^6 + 2*b^7 - (a^7 + 6*a^6*b + 14*a^5*b^2 + 16*a^4*b^3 + 9*a^3*b^4 + 2*a^2*b^5)*cos(d*x + c)^4 - 2*(a^6*b + 6*a^5*b^2 + 14*a^4*b^3 + 16*a^3* b^4 + 9*a^2*b^5 + 2*a*b^6)*cos(d*x + c)^3 + (a^7 + 6*a^6*b + 13*a^5*b^2 + 10*a^4*b^3 - 5*a^3*b^4 - 14*a^2*b^5 - 9*a*b^6 - 2*b^7)*cos(d*x + c)^2 + 2* (a^6*b + 6*a^5*b^2 + 14*a^4*b^3 + 16*a^3*b^4 + 9*a^2*b^5 + 2*a*b^6)*cos(d* x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^5*b^2 - 6*a^4*b^3 + 14*a^3*b^4 - 16*a^2*b^5 + 9*a*b^6 - 2*b^7 - (a^7 - 6*a^6*b + 14*a^5*b^2 - 16*a^4*b^3 + 9*a^3*b^4 - 2*a^2*b^5)*cos(d*x + c)^4 - 2*(a^6*b - 6*a^5*b^2 + 14*a^4*b^3 - 16*a^3*b^4 + 9*a^2*b^5 - 2*a*b^6)*cos(d*x + c)^3 + (a^7 - 6*a^6*b + 13*a ^5*b^2 - 10*a^4*b^3 - 5*a^3*b^4 + 14*a^2*b^5 - 9*a*b^6 + 2*b^7)*cos(d*x + c)^2 + 2*(a^6*b - 6*a^5*b^2 + 14*a^4*b^3 - 16*a^3*b^4 + 9*a^2*b^5 - 2*a*b^ 6)*cos(d*x + c))*log(-1/2*cos(d*x + c) + 1/2))/((a^10 - 4*a^8*b^2 + 6*a^6* b^4 - 4*a^4*b^6 + a^2*b^8)*d*cos(d*x + c)^4 + 2*(a^9*b - 4*a^7*b^3 + 6*...
\[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(csc(d*x+c)**3/(a+b*sec(d*x+c))**3,x)
Output:
Integral(csc(c + d*x)**3/(a + b*sec(c + d*x))**3, x)
Time = 0.05 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.90 \[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {4 \, {\left (3 \, a^{4} b + 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (8 \, a^{2} b^{3} + 4 \, b^{5} - {\left (a^{5} + 9 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{4} b - 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (11 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8} - {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )}}{4 \, d} \] Input:
integrate(csc(d*x+c)^3/(a+b*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/4*(4*(3*a^4*b + 8*a^2*b^3 + b^5)*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) - (a + 2*b)*log(cos(d*x + c) + 1)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + (a - 2*b)*log(cos(d*x + c) - 1)/(a^ 4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 2*(8*a^2*b^3 + 4*b^5 - (a^5 + 9 *a^3*b^2 + 2*a*b^4)*cos(d*x + c)^3 + (a^4*b - 10*a^2*b^3 - 3*b^5)*cos(d*x + c)^2 + (11*a^3*b^2 + a*b^4)*cos(d*x + c))/(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b ^6 - b^8 - (a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*cos(d*x + c)^4 - 2*(a^7 *b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*cos(d*x + c)^3 + (a^8 - 4*a^6*b^2 + 6* a^4*b^4 - 4*a^2*b^6 + b^8)*cos(d*x + c)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b ^5 - a*b^7)*cos(d*x + c)))/d
Time = 0.16 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.56 \[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {{\left (3 \, a^{5} b + 8 \, a^{3} b^{3} + a b^{5}\right )} \log \left ({\left | a \cos \left (d x + c\right ) + b \right |}\right )}{a^{9} d - 4 \, a^{7} b^{2} d + 6 \, a^{5} b^{4} d - 4 \, a^{3} b^{6} d + a b^{8} d} + \frac {{\left (a - 2 \, b\right )} \log \left ({\left | -\cos \left (d x + c\right ) + 1 \right |}\right )}{4 \, {\left (a^{4} d + 4 \, a^{3} b d + 6 \, a^{2} b^{2} d + 4 \, a b^{3} d + b^{4} d\right )}} - \frac {{\left (a + 2 \, b\right )} \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{4 \, {\left (a^{4} d - 4 \, a^{3} b d + 6 \, a^{2} b^{2} d - 4 \, a b^{3} d + b^{4} d\right )}} - \frac {8 \, a^{4} b^{3} - 4 \, a^{2} b^{5} - 4 \, b^{7} - {\left (a^{7} + 8 \, a^{5} b^{2} - 7 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{6} b - 11 \, a^{4} b^{3} + 7 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (11 \, a^{5} b^{2} - 10 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right ) + b\right )}^{2} {\left (a + b\right )}^{4} {\left (a - b\right )}^{4} d {\left (\cos \left (d x + c\right ) + 1\right )} {\left (\cos \left (d x + c\right ) - 1\right )}} \] Input:
integrate(csc(d*x+c)^3/(a+b*sec(d*x+c))^3,x, algorithm="giac")
Output:
(3*a^5*b + 8*a^3*b^3 + a*b^5)*log(abs(a*cos(d*x + c) + b))/(a^9*d - 4*a^7* b^2*d + 6*a^5*b^4*d - 4*a^3*b^6*d + a*b^8*d) + 1/4*(a - 2*b)*log(abs(-cos( d*x + c) + 1))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4*d) - 1/4 *(a + 2*b)*log(abs(-cos(d*x + c) - 1))/(a^4*d - 4*a^3*b*d + 6*a^2*b^2*d - 4*a*b^3*d + b^4*d) - 1/2*(8*a^4*b^3 - 4*a^2*b^5 - 4*b^7 - (a^7 + 8*a^5*b^2 - 7*a^3*b^4 - 2*a*b^6)*cos(d*x + c)^3 + (a^6*b - 11*a^4*b^3 + 7*a^2*b^5 + 3*b^7)*cos(d*x + c)^2 + (11*a^5*b^2 - 10*a^3*b^4 - a*b^6)*cos(d*x + c))/( (a*cos(d*x + c) + b)^2*(a + b)^4*(a - b)^4*d*(cos(d*x + c) + 1)*(cos(d*x + c) - 1))
Time = 11.82 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.65 \[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}-\frac {1}{4\,{\left (a+b\right )}^3}+\frac {3\,b}{4\,{\left (a-b\right )}^4}+\frac {1}{4\,{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}-\frac {1}{4\,{\left (a+b\right )}^3}\right )}{d}-\frac {\frac {{\cos \left (c+d\,x\right )}^3\,\left (a^5+9\,a^3\,b^2+2\,a\,b^4\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\cos \left (c+d\,x\right )}^2\,\left (-a^4\,b+10\,a^2\,b^3+3\,b^5\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {2\,b\,\left (2\,a^2\,b^2+b^4\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {a\,\cos \left (c+d\,x\right )\,\left (11\,a^2\,b^2+b^4\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2\,\left (a^2-b^2\right )+b^2-a^2\,{\cos \left (c+d\,x\right )}^4+2\,a\,b\,\cos \left (c+d\,x\right )-2\,a\,b\,{\cos \left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (a+2\,b\right )}{4\,d\,{\left (a-b\right )}^4} \] Input:
int(1/(sin(c + d*x)^3*(a + b/cos(c + d*x))^3),x)
Output:
(log(b + a*cos(c + d*x))*((3*b)/(4*(a + b)^4) - 1/(4*(a + b)^3) + (3*b)/(4 *(a - b)^4) + 1/(4*(a - b)^3)))/d - (log(cos(c + d*x) - 1)*((3*b)/(4*(a + b)^4) - 1/(4*(a + b)^3)))/d - ((cos(c + d*x)^3*(2*a*b^4 + a^5 + 9*a^3*b^2) )/(2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (cos(c + d*x)^2*(3*b^5 - a^4*b + 10*a^2*b^3))/(2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (2*b*(b^4 + 2*a^ 2*b^2))/((a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (a*cos(c + d*x)*(b^4 + 11* a^2*b^2))/(2*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)))/(d*(cos(c + d*x)^2*(a^2 - b^2) + b^2 - a^2*cos(c + d*x)^4 + 2*a*b*cos(c + d*x) - 2*a*b*cos(c + d* x)^3)) - (log(cos(c + d*x) + 1)*(a + 2*b))/(4*d*(a - b)^4)
Time = 0.18 (sec) , antiderivative size = 1673, normalized size of antiderivative = 7.31 \[ \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(csc(d*x+c)^3/(a+b*sec(d*x+c))^3,x)
Output:
(24*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b )*sin(c + d*x)**2*a**5*b**2 + 64*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**2*a**3*b**4 + 8*cos(c + d*x)* log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**2 *a*b**6 + 4*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**6*b - 24 *cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**5*b**2 + 56*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**4*b**3 - 64*cos(c + d*x)*lo g(tan((c + d*x)/2))*sin(c + d*x)**2*a**3*b**4 + 36*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**2*a**2*b**5 - 8*cos(c + d*x)*log(tan((c + d*x)/2 ))*sin(c + d*x)**2*a*b**6 + 2*cos(c + d*x)*sin(c + d*x)**2*a**7 - 2*cos(c + d*x)*sin(c + d*x)**2*a**6*b + 20*cos(c + d*x)*sin(c + d*x)**2*a**5*b**2 - 44*cos(c + d*x)*sin(c + d*x)**2*a**4*b**3 + 38*cos(c + d*x)*sin(c + d*x) **2*a**3*b**4 - 26*cos(c + d*x)*sin(c + d*x)**2*a**2*b**5 + 12*cos(c + d*x )*sin(c + d*x)**2*a*b**6 - 2*cos(c + d*x)*a**7 + 6*cos(c + d*x)*a**5*b**2 - 6*cos(c + d*x)*a**3*b**4 + 2*cos(c + d*x)*a*b**6 - 12*log(tan((c + d*x)/ 2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**4*a**6*b - 32*log(t an((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**4*a**4 *b**3 - 4*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**4*a**2*b**5 + 12*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2* b - a - b)*sin(c + d*x)**2*a**6*b + 44*log(tan((c + d*x)/2)**2*a - tan(...