Integrand size = 21, antiderivative size = 171 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {21 \text {arctanh}(\sin (c+d x))}{128 a^3 d}+\frac {1}{128 a d (a-a \sin (c+d x))^2}-\frac {a^2}{40 d (a+a \sin (c+d x))^5}-\frac {3 a}{64 d (a+a \sin (c+d x))^4}-\frac {1}{16 d (a+a \sin (c+d x))^3}-\frac {5}{64 a d (a+a \sin (c+d x))^2}+\frac {3}{64 d \left (a^3-a^3 \sin (c+d x)\right )}-\frac {15}{128 d \left (a^3+a^3 \sin (c+d x)\right )} \]
21/128*arctanh(sin(d*x+c))/a^3/d+1/128/a/d/(a-a*sin(d*x+c))^2-1/40*a^2/d/( a+a*sin(d*x+c))^5-3/64*a/d/(a+a*sin(d*x+c))^4-1/16/d/(a+a*sin(d*x+c))^3-5/ 64/a/d/(a+a*sin(d*x+c))^2+3/64/d/(a^3-a^3*sin(d*x+c))-15/128/d/(a^3+a^3*si n(d*x+c))
Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\sec ^4(c+d x) \left (-176+105 \text {arctanh}(\sin (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{10}+7 \sin (c+d x)+469 \sin ^2(c+d x)+420 \sin ^3(c+d x)-140 \sin ^4(c+d x)-315 \sin ^5(c+d x)-105 \sin ^6(c+d x)\right )}{640 a^3 d (1+\sin (c+d x))^3} \]
(Sec[c + d*x]^4*(-176 + 105*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[ (c + d*x)/2])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^10 + 7*Sin[c + d*x] + 469*Sin[c + d*x]^2 + 420*Sin[c + d*x]^3 - 140*Sin[c + d*x]^4 - 315*Sin[c + d*x]^5 - 105*Sin[c + d*x]^6))/(640*a^3*d*(1 + Sin[c + d*x])^3)
Time = 0.34 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3146, 54, 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.
| \(\displaystyle \int \frac {\sec ^5(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^5 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle \frac {a^5 \int \frac {1}{(a-a \sin (c+d x))^3 (\sin (c+d x) a+a)^6}d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {a^5 \int \left (\frac {3}{64 a^7 (a-a \sin (c+d x))^2}+\frac {15}{128 a^7 (\sin (c+d x) a+a)^2}+\frac {1}{64 a^6 (a-a \sin (c+d x))^3}+\frac {5}{32 a^6 (\sin (c+d x) a+a)^3}+\frac {3}{16 a^5 (\sin (c+d x) a+a)^4}+\frac {3}{16 a^4 (\sin (c+d x) a+a)^5}+\frac {1}{8 a^3 (\sin (c+d x) a+a)^6}+\frac {21}{128 a^7 \left (a^2-a^2 \sin ^2(c+d x)\right )}\right )d(a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^5 \left (\frac {21 \text {arctanh}(\sin (c+d x))}{128 a^8}+\frac {3}{64 a^7 (a-a \sin (c+d x))}-\frac {15}{128 a^7 (a \sin (c+d x)+a)}+\frac {1}{128 a^6 (a-a \sin (c+d x))^2}-\frac {5}{64 a^6 (a \sin (c+d x)+a)^2}-\frac {1}{16 a^5 (a \sin (c+d x)+a)^3}-\frac {3}{64 a^4 (a \sin (c+d x)+a)^4}-\frac {1}{40 a^3 (a \sin (c+d x)+a)^5}\right )}{d}\) |
(a^5*((21*ArcTanh[Sin[c + d*x]])/(128*a^8) + 1/(128*a^6*(a - a*Sin[c + d*x ])^2) + 3/(64*a^7*(a - a*Sin[c + d*x])) - 1/(40*a^3*(a + a*Sin[c + d*x])^5 ) - 3/(64*a^4*(a + a*Sin[c + d*x])^4) - 1/(16*a^5*(a + a*Sin[c + d*x])^3) - 5/(64*a^6*(a + a*Sin[c + d*x])^2) - 15/(128*a^7*(a + a*Sin[c + d*x]))))/ d
3.1.87.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Time = 1.96 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.67
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{40 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {3}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {15}{128 \left (1+\sin \left (d x +c \right )\right )}+\frac {21 \ln \left (1+\sin \left (d x +c \right )\right )}{256}+\frac {1}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3}{64 \left (\sin \left (d x +c \right )-1\right )}-\frac {21 \ln \left (\sin \left (d x +c \right )-1\right )}{256}}{d \,a^{3}}\) | \(115\) |
| default | \(\frac {-\frac {1}{40 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {3}{64 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{16 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{64 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {15}{128 \left (1+\sin \left (d x +c \right )\right )}+\frac {21 \ln \left (1+\sin \left (d x +c \right )\right )}{256}+\frac {1}{128 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3}{64 \left (\sin \left (d x +c \right )-1\right )}-\frac {21 \ln \left (\sin \left (d x +c \right )-1\right )}{256}}{d \,a^{3}}\) | \(115\) |
| risch | \(-\frac {i \left (630 i {\mathrm e}^{12 i \left (d x +c \right )}+105 \,{\mathrm e}^{13 i \left (d x +c \right )}+210 i {\mathrm e}^{10 i \left (d x +c \right )}-1190 \,{\mathrm e}^{11 i \left (d x +c \right )}-4004 i {\mathrm e}^{8 i \left (d x +c \right )}-3689 \,{\mathrm e}^{9 i \left (d x +c \right )}+4004 i {\mathrm e}^{6 i \left (d x +c \right )}-1716 \,{\mathrm e}^{7 i \left (d x +c \right )}-210 i {\mathrm e}^{4 i \left (d x +c \right )}-3689 \,{\mathrm e}^{5 i \left (d x +c \right )}-630 i {\mathrm e}^{2 i \left (d x +c \right )}-1190 \,{\mathrm e}^{3 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{320 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )^{4} a^{3} d}-\frac {21 \ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{128 a^{3} d}+\frac {21 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 a^{3} d}\) | \(231\) |
| norman | \(\frac {\frac {137 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {137 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {129 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {129 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {107 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}+\frac {107 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {171 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}-\frac {171 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}+\frac {489 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d a}+\frac {489 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d a}+\frac {1619 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{80 d a}-\frac {1559 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}-\frac {1559 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{128 a^{3} d}+\frac {21 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{128 a^{3} d}\) | \(318\) |
| parallelrisch | \(\frac {\left (-1155 \sin \left (5 d x +5 c \right )+105 \sin \left (7 d x +7 c \right )-3990 \cos \left (2 d x +2 c \right )+420 \cos \left (4 d x +4 c \right )+630 \cos \left (6 d x +6 c \right )-2835 \sin \left (d x +c \right )-4095 \sin \left (3 d x +3 c \right )-3780\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (1155 \sin \left (5 d x +5 c \right )-105 \sin \left (7 d x +7 c \right )+3990 \cos \left (2 d x +2 c \right )-420 \cos \left (4 d x +4 c \right )-630 \cos \left (6 d x +6 c \right )+2835 \sin \left (d x +c \right )+4095 \sin \left (3 d x +3 c \right )+3780\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+676 \sin \left (5 d x +5 c \right )-176 \sin \left (7 d x +7 c \right )-690 \cos \left (2 d x +2 c \right )-3084 \cos \left (4 d x +4 c \right )-846 \cos \left (6 d x +6 c \right )+12760 \sin \left (d x +c \right )+6444 \sin \left (3 d x +3 c \right )+4620}{640 a^{3} d \left (38 \cos \left (2 d x +2 c \right )-6 \cos \left (6 d x +6 c \right )-\sin \left (7 d x +7 c \right )+27 \sin \left (d x +c \right )+11 \sin \left (5 d x +5 c \right )+39 \sin \left (3 d x +3 c \right )-4 \cos \left (4 d x +4 c \right )+36\right )}\) | \(341\) |
1/d/a^3*(-1/40/(1+sin(d*x+c))^5-3/64/(1+sin(d*x+c))^4-1/16/(1+sin(d*x+c))^ 3-5/64/(1+sin(d*x+c))^2-15/128/(1+sin(d*x+c))+21/256*ln(1+sin(d*x+c))+1/12 8/(sin(d*x+c)-1)^2-3/64/(sin(d*x+c)-1)-21/256*ln(sin(d*x+c)-1))
Time = 0.37 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.45 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {210 \, \cos \left (d x + c\right )^{6} - 910 \, \cos \left (d x + c\right )^{4} + 252 \, \cos \left (d x + c\right )^{2} - 105 \, {\left (3 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} + {\left (\cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, {\left (3 \, \cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4} + {\left (\cos \left (d x + c\right )^{6} - 4 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 14 \, {\left (45 \, \cos \left (d x + c\right )^{4} - 30 \, \cos \left (d x + c\right )^{2} - 16\right )} \sin \left (d x + c\right ) + 96}{1280 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{6} - 4 \, a^{3} d \cos \left (d x + c\right )^{4} + {\left (a^{3} d \cos \left (d x + c\right )^{6} - 4 \, a^{3} d \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )}} \]
-1/1280*(210*cos(d*x + c)^6 - 910*cos(d*x + c)^4 + 252*cos(d*x + c)^2 - 10 5*(3*cos(d*x + c)^6 - 4*cos(d*x + c)^4 + (cos(d*x + c)^6 - 4*cos(d*x + c)^ 4)*sin(d*x + c))*log(sin(d*x + c) + 1) + 105*(3*cos(d*x + c)^6 - 4*cos(d*x + c)^4 + (cos(d*x + c)^6 - 4*cos(d*x + c)^4)*sin(d*x + c))*log(-sin(d*x + c) + 1) - 14*(45*cos(d*x + c)^4 - 30*cos(d*x + c)^2 - 16)*sin(d*x + c) + 96)/(3*a^3*d*cos(d*x + c)^6 - 4*a^3*d*cos(d*x + c)^4 + (a^3*d*cos(d*x + c) ^6 - 4*a^3*d*cos(d*x + c)^4)*sin(d*x + c))
\[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\sec ^{5}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Time = 0.20 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.10 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (105 \, \sin \left (d x + c\right )^{6} + 315 \, \sin \left (d x + c\right )^{5} + 140 \, \sin \left (d x + c\right )^{4} - 420 \, \sin \left (d x + c\right )^{3} - 469 \, \sin \left (d x + c\right )^{2} - 7 \, \sin \left (d x + c\right ) + 176\right )}}{a^{3} \sin \left (d x + c\right )^{7} + 3 \, a^{3} \sin \left (d x + c\right )^{6} + a^{3} \sin \left (d x + c\right )^{5} - 5 \, a^{3} \sin \left (d x + c\right )^{4} - 5 \, a^{3} \sin \left (d x + c\right )^{3} + a^{3} \sin \left (d x + c\right )^{2} + 3 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac {105 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}} + \frac {105 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3}}}{1280 \, d} \]
-1/1280*(2*(105*sin(d*x + c)^6 + 315*sin(d*x + c)^5 + 140*sin(d*x + c)^4 - 420*sin(d*x + c)^3 - 469*sin(d*x + c)^2 - 7*sin(d*x + c) + 176)/(a^3*sin( d*x + c)^7 + 3*a^3*sin(d*x + c)^6 + a^3*sin(d*x + c)^5 - 5*a^3*sin(d*x + c )^4 - 5*a^3*sin(d*x + c)^3 + a^3*sin(d*x + c)^2 + 3*a^3*sin(d*x + c) + a^3 ) - 105*log(sin(d*x + c) + 1)/a^3 + 105*log(sin(d*x + c) - 1)/a^3)/d
Time = 0.34 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.80 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} - \frac {420 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3}} + \frac {10 \, {\left (63 \, \sin \left (d x + c\right )^{2} - 150 \, \sin \left (d x + c\right ) + 91\right )}}{a^{3} {\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {959 \, \sin \left (d x + c\right )^{5} + 5395 \, \sin \left (d x + c\right )^{4} + 12390 \, \sin \left (d x + c\right )^{3} + 14710 \, \sin \left (d x + c\right )^{2} + 9275 \, \sin \left (d x + c\right ) + 2647}{a^{3} {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{5120 \, d} \]
1/5120*(420*log(abs(sin(d*x + c) + 1))/a^3 - 420*log(abs(sin(d*x + c) - 1) )/a^3 + 10*(63*sin(d*x + c)^2 - 150*sin(d*x + c) + 91)/(a^3*(sin(d*x + c) - 1)^2) - (959*sin(d*x + c)^5 + 5395*sin(d*x + c)^4 + 12390*sin(d*x + c)^3 + 14710*sin(d*x + c)^2 + 9275*sin(d*x + c) + 2647)/(a^3*(sin(d*x + c) + 1 )^5))/d
Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.01 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {21\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )}{128\,a^3\,d}-\frac {\frac {21\,{\sin \left (c+d\,x\right )}^6}{128}+\frac {63\,{\sin \left (c+d\,x\right )}^5}{128}+\frac {7\,{\sin \left (c+d\,x\right )}^4}{32}-\frac {21\,{\sin \left (c+d\,x\right )}^3}{32}-\frac {469\,{\sin \left (c+d\,x\right )}^2}{640}-\frac {7\,\sin \left (c+d\,x\right )}{640}+\frac {11}{40}}{d\,\left (a^3\,{\sin \left (c+d\,x\right )}^7+3\,a^3\,{\sin \left (c+d\,x\right )}^6+a^3\,{\sin \left (c+d\,x\right )}^5-5\,a^3\,{\sin \left (c+d\,x\right )}^4-5\,a^3\,{\sin \left (c+d\,x\right )}^3+a^3\,{\sin \left (c+d\,x\right )}^2+3\,a^3\,\sin \left (c+d\,x\right )+a^3\right )} \]
(21*atanh(sin(c + d*x)))/(128*a^3*d) - ((7*sin(c + d*x)^4)/32 - (469*sin(c + d*x)^2)/640 - (21*sin(c + d*x)^3)/32 - (7*sin(c + d*x))/640 + (63*sin(c + d*x)^5)/128 + (21*sin(c + d*x)^6)/128 + 11/40)/(d*(3*a^3*sin(c + d*x) + a^3 + a^3*sin(c + d*x)^2 - 5*a^3*sin(c + d*x)^3 - 5*a^3*sin(c + d*x)^4 + a^3*sin(c + d*x)^5 + 3*a^3*sin(c + d*x)^6 + a^3*sin(c + d*x)^7))