Integrand size = 21, antiderivative size = 205 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}+\frac {5 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {6 a b^2 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a b^2 \cot ^3(c+d x)}{d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {5 b^3 \csc (c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {5 b^3 \csc ^3(c+d x)}{6 d}+\frac {b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{2 d}+\frac {3 a b^2 \tan (c+d x)}{d} \]
3*a^2*b*arctanh(sin(d*x+c))/d+5/2*b^3*arctanh(sin(d*x+c))/d-a^3*cot(d*x+c) /d-6*a*b^2*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^3/d-a*b^2*cot(d*x+c)^3/d-3*a^2* b*csc(d*x+c)/d-5/2*b^3*csc(d*x+c)/d-a^2*b*csc(d*x+c)^3/d-5/6*b^3*csc(d*x+c )^3/d+1/2*b^3*csc(d*x+c)^3*sec(d*x+c)^2/d+3*a*b^2*tan(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(610\) vs. \(2(205)=410\).
Time = 1.44 (sec) , antiderivative size = 610, normalized size of antiderivative = 2.98 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {\csc ^7\left (\frac {1}{2} (c+d x)\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (84 a^2 b+22 b^3+32 a \left (a^2+3 b^2\right ) \cos (c+d x)+8 \left (6 a^2 b+5 b^3\right ) \cos (2 (c+d x))+4 a^3 \cos (3 (c+d x))+48 a b^2 \cos (3 (c+d x))-36 a^2 b \cos (4 (c+d x))-30 b^3 \cos (4 (c+d x))-4 a^3 \cos (5 (c+d x))-48 a b^2 \cos (5 (c+d x))+36 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+30 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-36 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)-30 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+18 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))+15 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-18 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-15 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-18 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))-15 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+18 a^2 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+15 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))\right )}{768 d \left (-1+\cot ^2\left (\frac {1}{2} (c+d x)\right )\right )^2} \]
-1/768*(Csc[(c + d*x)/2]^7*Sec[(c + d*x)/2]^3*(84*a^2*b + 22*b^3 + 32*a*(a ^2 + 3*b^2)*Cos[c + d*x] + 8*(6*a^2*b + 5*b^3)*Cos[2*(c + d*x)] + 4*a^3*Co s[3*(c + d*x)] + 48*a*b^2*Cos[3*(c + d*x)] - 36*a^2*b*Cos[4*(c + d*x)] - 3 0*b^3*Cos[4*(c + d*x)] - 4*a^3*Cos[5*(c + d*x)] - 48*a*b^2*Cos[5*(c + d*x) ] + 36*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 30*b^ 3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 36*a^2*b*Log[Cos [(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 30*b^3*Log[Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 18*a^2*b*Log[Cos[(c + d*x)/2] - Sin[ (c + d*x)/2]]*Sin[3*(c + d*x)] + 15*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d* x)/2]]*Sin[3*(c + d*x)] - 18*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2] ]*Sin[3*(c + d*x)] - 15*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3 *(c + d*x)] - 18*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 18*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 15* b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(-1 + C ot[(c + d*x)/2]^2)^2)
Time = 0.76 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4360, 25, 25, 3042, 25, 3391, 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.
| \(\displaystyle \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \csc ^4(c+d x) \sec ^3(c+d x) \left (-(-a \cos (c+d x)-b)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(b+a \cos (c+d x))^3 \csc ^4(c+d x) \sec ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \csc ^4(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}{\sin \left (c+d x-\frac {\pi }{2}\right )^3 \cos \left (c+d x-\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^4 \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3391 |
| \(\displaystyle -\int \left (-a^3 \sec ^4\left (\frac {1}{2} (2 c-\pi )+d x\right )-b^3 \sec ^3(c+d x) \sec ^4\left (\frac {1}{2} (2 c-\pi )+d x\right )-3 a b^2 \sec ^2(c+d x) \sec ^4\left (\frac {1}{2} (2 c-\pi )+d x\right )-3 a^2 b \sec (c+d x) \sec ^4\left (\frac {1}{2} (2 c-\pi )+d x\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \text {arctanh}(\sin (c+d x))}{d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {3 a^2 b \csc (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}-\frac {a b^2 \cot ^3(c+d x)}{d}-\frac {6 a b^2 \cot (c+d x)}{d}+\frac {5 b^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {5 b^3 \csc ^3(c+d x)}{6 d}-\frac {5 b^3 \csc (c+d x)}{2 d}+\frac {b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{2 d}\) |
(3*a^2*b*ArcTanh[Sin[c + d*x]])/d + (5*b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a^3*Cot[c + d*x])/d - (6*a*b^2*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3* d) - (a*b^2*Cot[c + d*x]^3)/d - (3*a^2*b*Csc[c + d*x])/d - (5*b^3*Csc[c + d*x])/(2*d) - (a^2*b*Csc[c + d*x]^3)/d - (5*b^3*Csc[c + d*x]^3)/(6*d) + (b ^3*Csc[c + d*x]^3*Sec[c + d*x]^2)/(2*d) + (3*a*b^2*Tan[c + d*x])/d
3.2.94.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (G tQ[m, 0] || IntegerQ[n])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 1.60 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+b^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {5}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )}+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(187\) |
| default | \(\frac {a^{3} \left (-\frac {2}{3}-\frac {\csc \left (d x +c \right )^{2}}{3}\right ) \cot \left (d x +c \right )+3 a^{2} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+3 a \,b^{2} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+b^{3} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {5}{6 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {5}{2 \sin \left (d x +c \right )}+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(187\) |
| parallelrisch | \(\frac {-576 \left (1+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {5 b^{2}}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+576 \left (1+\cos \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {5 b^{2}}{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 \csc \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\left (-12 a^{2} b -10 b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (-a^{3}-12 a \,b^{2}\right ) \cos \left (3 d x +3 c \right )+\left (9 a^{2} b +\frac {15}{2} b^{3}\right ) \cos \left (4 d x +4 c \right )+\left (a^{3}+12 a \,b^{2}\right ) \cos \left (5 d x +5 c \right )+\left (-8 a^{3}-24 a \,b^{2}\right ) \cos \left (d x +c \right )-21 a^{2} b -\frac {11 b^{3}}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(231\) |
| norman | \(\frac {-\frac {a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}{24 d}+\frac {\left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{24 d}+\frac {\left (7 a^{3}-39 a^{2} b +57 a \,b^{2}-25 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{24 d}-\frac {\left (7 a^{3}+39 a^{2} b +57 a \,b^{2}+25 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{24 d}-\frac {\left (13 a^{3}-21 a^{2} b +165 a \,b^{2}-25 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{12 d}+\frac {\left (13 a^{3}+21 a^{2} b +165 a \,b^{2}+25 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{12 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\frac {b \left (6 a^{2}+5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \left (6 a^{2}+5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(305\) |
| risch | \(-\frac {i \left (18 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}+15 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-24 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-20 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-12 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}-84 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-22 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-20 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-96 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-24 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}-20 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-4 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}-48 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+18 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}+15 b^{3} {\mathrm e}^{i \left (d x +c \right )}+4 a^{3}+48 a \,b^{2}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a^{2}}{d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a^{2}}{d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(349\) |
1/d*(a^3*(-2/3-1/3*csc(d*x+c)^2)*cot(d*x+c)+3*a^2*b*(-1/3/sin(d*x+c)^3-1/s in(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))+3*a*b^2*(-1/3/sin(d*x+c)^3/cos(d*x+c) +4/3/sin(d*x+c)/cos(d*x+c)-8/3*cot(d*x+c))+b^3*(-1/3/sin(d*x+c)^3/cos(d*x+ c)^2+5/6/sin(d*x+c)/cos(d*x+c)^2-5/2/sin(d*x+c)+5/2*ln(sec(d*x+c)+tan(d*x+ c))))
Time = 0.27 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.27 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {8 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{4} + 36 \, a b^{2} \cos \left (d x + c\right ) - 12 \, {\left (a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, b^{3} - 8 \, {\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left ({\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - {\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 3 \, {\left ({\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - {\left (6 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
-1/12*(8*(a^3 + 12*a*b^2)*cos(d*x + c)^5 + 6*(6*a^2*b + 5*b^3)*cos(d*x + c )^4 + 36*a*b^2*cos(d*x + c) - 12*(a^3 + 12*a*b^2)*cos(d*x + c)^3 + 6*b^3 - 8*(6*a^2*b + 5*b^3)*cos(d*x + c)^2 - 3*((6*a^2*b + 5*b^3)*cos(d*x + c)^4 - (6*a^2*b + 5*b^3)*cos(d*x + c)^2)*log(sin(d*x + c) + 1)*sin(d*x + c) + 3 *((6*a^2*b + 5*b^3)*cos(d*x + c)^4 - (6*a^2*b + 5*b^3)*cos(d*x + c)^2)*log (-sin(d*x + c) + 1)*sin(d*x + c))/((d*cos(d*x + c)^4 - d*cos(d*x + c)^2)*s in(d*x + c))
\[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{3} \csc ^{4}{\left (c + d x \right )}\, dx \]
Time = 0.20 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.93 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx=-\frac {b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{3}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a b^{2} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a^{3}}{\tan \left (d x + c\right )^{3}}}{12 \, d} \]
-1/12*(b^3*(2*(15*sin(d*x + c)^4 - 10*sin(d*x + c)^2 - 2)/(sin(d*x + c)^5 - sin(d*x + c)^3) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) + 6*a^2*b*(2*(3*sin(d*x + c)^2 + 1)/sin(d*x + c)^3 - 3*log(sin(d*x + c) + 1 ) + 3*log(sin(d*x + c) - 1)) + 12*a*b^2*((6*tan(d*x + c)^2 + 1)/tan(d*x + c)^3 - 3*tan(d*x + c)) + 4*(3*tan(d*x + c)^2 + 1)*a^3/tan(d*x + c)^3)/d
Time = 0.38 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.76 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 63 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, {\left (6 \, a^{2} b + 5 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, {\left (6 \, a^{2} b + 5 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {24 \, {\left (6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 63 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
1/24*(a^3*tan(1/2*d*x + 1/2*c)^3 - 3*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 3*a*b^ 2*tan(1/2*d*x + 1/2*c)^3 - b^3*tan(1/2*d*x + 1/2*c)^3 + 9*a^3*tan(1/2*d*x + 1/2*c) - 45*a^2*b*tan(1/2*d*x + 1/2*c) + 63*a*b^2*tan(1/2*d*x + 1/2*c) - 27*b^3*tan(1/2*d*x + 1/2*c) + 12*(6*a^2*b + 5*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 12*(6*a^2*b + 5*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2 4*(6*a*b^2*tan(1/2*d*x + 1/2*c)^3 - b^3*tan(1/2*d*x + 1/2*c)^3 - 6*a*b^2*t an(1/2*d*x + 1/2*c) - b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 - (9*a^3*tan(1/2*d*x + 1/2*c)^2 + 45*a^2*b*tan(1/2*d*x + 1/2*c)^2 + 6 3*a*b^2*tan(1/2*d*x + 1/2*c)^2 + 27*b^3*tan(1/2*d*x + 1/2*c)^2 + a^3 + 3*a ^2*b + 3*a*b^2 + b^3)/tan(1/2*d*x + 1/2*c)^3)/d
Time = 13.85 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.27 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\left (a-b\right )}^3}{24\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,b\,{\left (a-b\right )}^2}{4}-\frac {3\,{\left (a-b\right )}^3}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {7\,a^3}{3}+13\,a^2\,b+19\,a\,b^2+\frac {25\,b^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {17\,a^3}{3}+29\,a^2\,b+89\,a\,b^2+\frac {77\,b^3}{3}\right )+a\,b^2+a^2\,b+\frac {a^3}{3}+\frac {b^3}{3}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^3+15\,a^2\,b+69\,a\,b^2+b^3\right )}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2\,b\,6{}\mathrm {i}+b^3\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{d} \]
(tan(c/2 + (d*x)/2)^3*(a - b)^3)/(24*d) - (tan(c/2 + (d*x)/2)*((3*b*(a - b )^2)/4 - (3*(a - b)^3)/8))/d - (atanh(tan(c/2 + (d*x)/2))*(a^2*b*6i + b^3* 5i)*1i)/d - (tan(c/2 + (d*x)/2)^2*(19*a*b^2 + 13*a^2*b + (7*a^3)/3 + (25*b ^3)/3) - tan(c/2 + (d*x)/2)^4*(89*a*b^2 + 29*a^2*b + (17*a^3)/3 + (77*b^3) /3) + a*b^2 + a^2*b + a^3/3 + b^3/3 + tan(c/2 + (d*x)/2)^6*(69*a*b^2 + 15* a^2*b + 3*a^3 + b^3))/(d*(8*tan(c/2 + (d*x)/2)^3 - 16*tan(c/2 + (d*x)/2)^5 + 8*tan(c/2 + (d*x)/2)^7))