Integrand size = 27, antiderivative size = 162 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \csc (c+d x)}{d}-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {59 a \log (1-\sin (c+d x))}{16 d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {11 a \log (1+\sin (c+d x))}{16 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}+\frac {5 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a+a \sin (c+d x))} \]
-3*a*csc(d*x+c)/d-1/2*a*csc(d*x+c)^2/d-1/3*a*csc(d*x+c)^3/d-59/16*a*ln(1-s in(d*x+c))/d+3*a*ln(sin(d*x+c))/d+11/16*a*ln(1+sin(d*x+c))/d+1/8*a^3/d/(a- a*sin(d*x+c))^2+5/4*a^2/d/(a-a*sin(d*x+c))-1/8*a^2/d/(a+a*sin(d*x+c))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.64 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},3,-\frac {1}{2},\sin ^2(c+d x)\right )}{3 d}-\frac {3 a \log (\cos (c+d x))}{d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {a \sec ^2(c+d x)}{d}+\frac {a \sec ^4(c+d x)}{4 d} \]
-1/2*(a*Csc[c + d*x]^2)/d - (a*Csc[c + d*x]^3*Hypergeometric2F1[-3/2, 3, - 1/2, Sin[c + d*x]^2])/(3*d) - (3*a*Log[Cos[c + d*x]])/d + (3*a*Log[Sin[c + d*x]])/d + (a*Sec[c + d*x]^2)/d + (a*Sec[c + d*x]^4)/(4*d)
Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3315, 27, 99, 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.
| \(\displaystyle \int \csc ^4(c+d x) \sec ^5(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a \sin (c+d x)+a}{\sin (c+d x)^4 \cos (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle \frac {a^5 \int \frac {\csc ^4(c+d x)}{(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^9 \int \frac {\csc ^4(c+d x)}{a^4 (a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {a^9 \int \left (\frac {\csc ^4(c+d x)}{a^9}+\frac {\csc ^3(c+d x)}{a^9}+\frac {3 \csc ^2(c+d x)}{a^9}+\frac {3 \csc (c+d x)}{a^9}+\frac {59}{16 a^8 (a-a \sin (c+d x))}+\frac {11}{16 a^8 (\sin (c+d x) a+a)}+\frac {5}{4 a^7 (a-a \sin (c+d x))^2}+\frac {1}{8 a^7 (\sin (c+d x) a+a)^2}+\frac {1}{4 a^6 (a-a \sin (c+d x))^3}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^9 \left (-\frac {\csc ^3(c+d x)}{3 a^8}-\frac {\csc ^2(c+d x)}{2 a^8}-\frac {3 \csc (c+d x)}{a^8}+\frac {3 \log (a \sin (c+d x))}{a^8}-\frac {59 \log (a-a \sin (c+d x))}{16 a^8}+\frac {11 \log (a \sin (c+d x)+a)}{16 a^8}+\frac {5}{4 a^7 (a-a \sin (c+d x))}-\frac {1}{8 a^7 (a \sin (c+d x)+a)}+\frac {1}{8 a^6 (a-a \sin (c+d x))^2}\right )}{d}\) |
(a^9*((-3*Csc[c + d*x])/a^8 - Csc[c + d*x]^2/(2*a^8) - Csc[c + d*x]^3/(3*a ^8) + (3*Log[a*Sin[c + d*x]])/a^8 - (59*Log[a - a*Sin[c + d*x]])/(16*a^8) + (11*Log[a + a*Sin[c + d*x]])/(16*a^8) + 1/(8*a^6*(a - a*Sin[c + d*x])^2) + 5/(4*a^7*(a - a*Sin[c + d*x])) - 1/(8*a^7*(a + a*Sin[c + d*x]))))/d
3.9.60.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
Time = 0.68 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91
| method | result | size |
| derivativedivides | \(\frac {a \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\frac {1}{4 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}-\frac {7}{12 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {35}{24 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {35}{8 \sin \left (d x +c \right )}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(147\) |
| default | \(\frac {a \left (\frac {1}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}+\frac {3}{4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )^{2}}+3 \ln \left (\tan \left (d x +c \right )\right )\right )+a \left (\frac {1}{4 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}-\frac {7}{12 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {35}{24 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {35}{8 \sin \left (d x +c \right )}+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}\) | \(147\) |
| risch | \(-\frac {i a \left (-138 i {\mathrm e}^{10 i \left (d x +c \right )}+105 \,{\mathrm e}^{11 i \left (d x +c \right )}+136 i {\mathrm e}^{8 i \left (d x +c \right )}-101 \,{\mathrm e}^{9 i \left (d x +c \right )}+260 i {\mathrm e}^{6 i \left (d x +c \right )}-158 \,{\mathrm e}^{7 i \left (d x +c \right )}+136 i {\mathrm e}^{4 i \left (d x +c \right )}+158 \,{\mathrm e}^{5 i \left (d x +c \right )}-138 i {\mathrm e}^{2 i \left (d x +c \right )}+101 \,{\mathrm e}^{3 i \left (d x +c \right )}-105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} d}+\frac {11 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}-\frac {59 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\) | \(233\) |
| parallelrisch | \(-\frac {\left (59 \left (2-\sin \left (3 d x +3 c \right )-\sin \left (d x +c \right )+2 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+11 \left (\sin \left (3 d x +3 c \right )+\sin \left (d x +c \right )-2 \cos \left (2 d x +2 c \right )-2\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 \left (\sin \left (3 d x +3 c \right )+\sin \left (d x +c \right )-2 \cos \left (2 d x +2 c \right )-2\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+26 \sec \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\left (\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {\cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )}{12}+\frac {\cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{12}+\frac {\cos \left (\frac {7 d x}{2}+\frac {7 c}{2}\right )}{12}+\frac {\cos \left (\frac {9 d x}{2}+\frac {9 c}{2}\right )}{12}\right ) \left (\sec ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-488 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\csc ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2195 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\cos \left (d x +c \right )-\frac {606 \cos \left (2 d x +2 c \right )}{2195}+\frac {101 \cos \left (3 d x +3 c \right )}{2195}-\frac {610}{439}\right )}{3}\right ) a}{8 d \left (2-\sin \left (3 d x +3 c \right )-\sin \left (d x +c \right )+2 \cos \left (2 d x +2 c \right )\right )}\) | \(339\) |
1/d*(a*(1/4/sin(d*x+c)^2/cos(d*x+c)^4+3/4/sin(d*x+c)^2/cos(d*x+c)^2-3/2/si n(d*x+c)^2+3*ln(tan(d*x+c)))+a*(1/4/sin(d*x+c)^3/cos(d*x+c)^4-7/12/sin(d*x +c)^3/cos(d*x+c)^2+35/24/sin(d*x+c)/cos(d*x+c)^2-35/8/sin(d*x+c)+35/8*ln(s ec(d*x+c)+tan(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (150) = 300\).
Time = 0.27 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.12 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {138 \, a \cos \left (d x + c\right )^{4} - 172 \, a \cos \left (d x + c\right )^{2} - 144 \, {\left (a \cos \left (d x + c\right )^{6} - 2 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 33 \, {\left (a \cos \left (d x + c\right )^{6} - 2 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 177 \, {\left (a \cos \left (d x + c\right )^{6} - 2 \, a \cos \left (d x + c\right )^{4} + a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{4} - a \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (105 \, a \cos \left (d x + c\right )^{4} - 104 \, a \cos \left (d x + c\right )^{2} + 3 \, a\right )} \sin \left (d x + c\right ) + 18 \, a}{48 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2} + {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
-1/48*(138*a*cos(d*x + c)^4 - 172*a*cos(d*x + c)^2 - 144*(a*cos(d*x + c)^6 - 2*a*cos(d*x + c)^4 + a*cos(d*x + c)^2 + (a*cos(d*x + c)^4 - a*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*sin(d*x + c)) - 33*(a*cos(d*x + c)^6 - 2*a*co s(d*x + c)^4 + a*cos(d*x + c)^2 + (a*cos(d*x + c)^4 - a*cos(d*x + c)^2)*si n(d*x + c))*log(sin(d*x + c) + 1) + 177*(a*cos(d*x + c)^6 - 2*a*cos(d*x + c)^4 + a*cos(d*x + c)^2 + (a*cos(d*x + c)^4 - a*cos(d*x + c)^2)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 2*(105*a*cos(d*x + c)^4 - 104*a*cos(d*x + c)^ 2 + 3*a)*sin(d*x + c) + 18*a)/(d*cos(d*x + c)^6 - 2*d*cos(d*x + c)^4 + d*c os(d*x + c)^2 + (d*cos(d*x + c)^4 - d*cos(d*x + c)^2)*sin(d*x + c))
Timed out. \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {33 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 177 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac {2 \, {\left (105 \, a \sin \left (d x + c\right )^{5} - 69 \, a \sin \left (d x + c\right )^{4} - 106 \, a \sin \left (d x + c\right )^{3} + 52 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 8 \, a\right )}}{\sin \left (d x + c\right )^{6} - \sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{4} + \sin \left (d x + c\right )^{3}}}{48 \, d} \]
1/48*(33*a*log(sin(d*x + c) + 1) - 177*a*log(sin(d*x + c) - 1) + 144*a*log (sin(d*x + c)) - 2*(105*a*sin(d*x + c)^5 - 69*a*sin(d*x + c)^4 - 106*a*sin (d*x + c)^3 + 52*a*sin(d*x + c)^2 + 4*a*sin(d*x + c) + 8*a)/(sin(d*x + c)^ 6 - sin(d*x + c)^5 - sin(d*x + c)^4 + sin(d*x + c)^3))/d
Time = 0.36 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.92 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {66 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 354 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 288 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {6 \, {\left (11 \, a \sin \left (d x + c\right ) + 13 \, a\right )}}{\sin \left (d x + c\right ) + 1} + \frac {3 \, {\left (177 \, a \sin \left (d x + c\right )^{2} - 394 \, a \sin \left (d x + c\right ) + 221 \, a\right )}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {16 \, {\left (33 \, a \sin \left (d x + c\right )^{3} + 18 \, a \sin \left (d x + c\right )^{2} + 3 \, a \sin \left (d x + c\right ) + 2 \, a\right )}}{\sin \left (d x + c\right )^{3}}}{96 \, d} \]
1/96*(66*a*log(abs(sin(d*x + c) + 1)) - 354*a*log(abs(sin(d*x + c) - 1)) + 288*a*log(abs(sin(d*x + c))) - 6*(11*a*sin(d*x + c) + 13*a)/(sin(d*x + c) + 1) + 3*(177*a*sin(d*x + c)^2 - 394*a*sin(d*x + c) + 221*a)/(sin(d*x + c ) - 1)^2 - 16*(33*a*sin(d*x + c)^3 + 18*a*sin(d*x + c)^2 + 3*a*sin(d*x + c ) + 2*a)/sin(d*x + c)^3)/d
Time = 0.13 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90 \[ \int \csc ^4(c+d x) \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3\,a\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {\frac {35\,a\,{\sin \left (c+d\,x\right )}^5}{8}-\frac {23\,a\,{\sin \left (c+d\,x\right )}^4}{8}-\frac {53\,a\,{\sin \left (c+d\,x\right )}^3}{12}+\frac {13\,a\,{\sin \left (c+d\,x\right )}^2}{6}+\frac {a\,\sin \left (c+d\,x\right )}{6}+\frac {a}{3}}{d\,\left ({\sin \left (c+d\,x\right )}^6-{\sin \left (c+d\,x\right )}^5-{\sin \left (c+d\,x\right )}^4+{\sin \left (c+d\,x\right )}^3\right )}-\frac {59\,a\,\ln \left (\sin \left (c+d\,x\right )-1\right )}{16\,d}+\frac {11\,a\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{16\,d} \]
(3*a*log(sin(c + d*x)))/d - (a/3 + (a*sin(c + d*x))/6 + (13*a*sin(c + d*x) ^2)/6 - (53*a*sin(c + d*x)^3)/12 - (23*a*sin(c + d*x)^4)/8 + (35*a*sin(c + d*x)^5)/8)/(d*(sin(c + d*x)^3 - sin(c + d*x)^4 - sin(c + d*x)^5 + sin(c + d*x)^6)) - (59*a*log(sin(c + d*x) - 1))/(16*d) + (11*a*log(sin(c + d*x) + 1))/(16*d)