Integrand size = 25, antiderivative size = 141 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx=\frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {2 c (c \sin (a+b x))^{3/2}}{55 b d^3 (d \cos (a+b x))^{11/2}}-\frac {16 c (c \sin (a+b x))^{3/2}}{385 b d^5 (d \cos (a+b x))^{7/2}}-\frac {64 c (c \sin (a+b x))^{3/2}}{1155 b d^7 (d \cos (a+b x))^{3/2}} \]
2/15*c*(c*sin(b*x+a))^(3/2)/b/d/(d*cos(b*x+a))^(15/2)-2/55*c*(c*sin(b*x+a) )^(3/2)/b/d^3/(d*cos(b*x+a))^(11/2)-16/385*c*(c*sin(b*x+a))^(3/2)/b/d^5/(d *cos(b*x+a))^(7/2)-64/1155*c*(c*sin(b*x+a))^(3/2)/b/d^7/(d*cos(b*x+a))^(3/ 2)
Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.48 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx=\frac {2 \sqrt {d \cos (a+b x)} (117+44 \cos (2 (a+b x))+4 \cos (4 (a+b x))) \sec ^8(a+b x) (c \sin (a+b x))^{7/2}}{1155 b c d^9} \]
(2*Sqrt[d*Cos[a + b*x]]*(117 + 44*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*S ec[a + b*x]^8*(c*Sin[a + b*x])^(7/2))/(1155*b*c*d^9)
Time = 0.62 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3046, 3042, 3051, 3042, 3051, 3042, 3043}
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.
\(\displaystyle \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}}dx\) |
\(\Big \downarrow \) 3046 |
\(\displaystyle \frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {c^2 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}}dx}{5 d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {c^2 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}}dx}{5 d^2}\) |
\(\Big \downarrow \) 3051 |
\(\displaystyle \frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {c^2 \left (\frac {8 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}}dx}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\right )}{5 d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {c^2 \left (\frac {8 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}}dx}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\right )}{5 d^2}\) |
\(\Big \downarrow \) 3051 |
\(\displaystyle \frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {c^2 \left (\frac {8 \left (\frac {4 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}}dx}{7 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}\right )}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\right )}{5 d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {c^2 \left (\frac {8 \left (\frac {4 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}}dx}{7 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}\right )}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\right )}{5 d^2}\) |
\(\Big \downarrow \) 3043 |
\(\displaystyle \frac {2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}-\frac {c^2 \left (\frac {8 \left (\frac {8 (c \sin (a+b x))^{3/2}}{21 b c d^3 (d \cos (a+b x))^{3/2}}+\frac {2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}\right )}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\right )}{5 d^2}\) |
(2*c*(c*Sin[a + b*x])^(3/2))/(15*b*d*(d*Cos[a + b*x])^(15/2)) - (c^2*((2*( c*Sin[a + b*x])^(3/2))/(11*b*c*d*(d*Cos[a + b*x])^(11/2)) + (8*((2*(c*Sin[ a + b*x])^(3/2))/(7*b*c*d*(d*Cos[a + b*x])^(7/2)) + (8*(c*Sin[a + b*x])^(3 /2))/(21*b*c*d^3*(d*Cos[a + b*x])^(3/2))))/(11*d^2)))/(5*d^2)
3.3.86.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^( m_.), x_Symbol] :> Simp[(a*Sin[e + f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/ (a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2, 0] & & NeQ[m, -1]
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(a*Sin[e + f*x])^(m - 1)*((b*Cos[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Simp[a^2*((m - 1)/(b^2*(n + 1))) Int[(a*Sin[e + f *x])^(m - 2)*(b*Cos[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (IntegersQ[2*m, 2*n] || EqQ[m + n, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-(b*Sin[e + f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1) /(a*b*f*(m + 1))), x] + Simp[(m + n + 2)/(a^2*(m + 1)) Int[(b*Sin[e + f*x ])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m , -1] && IntegersQ[2*m, 2*n]
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.50
method | result | size |
default | \(\frac {2 c^{2} \left (32 \left (\cos ^{4}\left (b x +a \right )\right )+56 \left (\cos ^{2}\left (b x +a \right )\right )+77\right ) \sqrt {c \sin \left (b x +a \right )}\, \left (\tan ^{3}\left (b x +a \right )\right ) \left (\sec ^{4}\left (b x +a \right )\right )}{1155 b \,d^{8} \sqrt {d \cos \left (b x +a \right )}}\) | \(70\) |
2/1155/b*c^2*(32*cos(b*x+a)^4+56*cos(b*x+a)^2+77)*(c*sin(b*x+a))^(1/2)/d^8 /(d*cos(b*x+a))^(1/2)*tan(b*x+a)^3*sec(b*x+a)^4
Time = 0.59 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.62 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx=-\frac {2 \, {\left (32 \, c^{2} \cos \left (b x + a\right )^{6} + 24 \, c^{2} \cos \left (b x + a\right )^{4} + 21 \, c^{2} \cos \left (b x + a\right )^{2} - 77 \, c^{2}\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{1155 \, b d^{9} \cos \left (b x + a\right )^{8}} \]
-2/1155*(32*c^2*cos(b*x + a)^6 + 24*c^2*cos(b*x + a)^4 + 21*c^2*cos(b*x + a)^2 - 77*c^2)*sqrt(d*cos(b*x + a))*sqrt(c*sin(b*x + a))*sin(b*x + a)/(b*d ^9*cos(b*x + a)^8)
Timed out. \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {17}{2}}} \,d x } \]
\[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx=\int { \frac {\left (c \sin \left (b x + a\right )\right )^{\frac {5}{2}}}{\left (d \cos \left (b x + a\right )\right )^{\frac {17}{2}}} \,d x } \]
Time = 6.51 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.47 \[ \int \frac {(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx=-\frac {{\mathrm {e}}^{-a\,7{}\mathrm {i}-b\,x\,7{}\mathrm {i}}\,\sqrt {c\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}\,\left (\frac {1216\,c^2\,{\mathrm {e}}^{a\,7{}\mathrm {i}+b\,x\,7{}\mathrm {i}}\,\sin \left (3\,a+3\,b\,x\right )}{385\,b\,d^8}+\frac {1024\,c^2\,{\mathrm {e}}^{a\,7{}\mathrm {i}+b\,x\,7{}\mathrm {i}}\,\sin \left (5\,a+5\,b\,x\right )}{1155\,b\,d^8}+\frac {128\,c^2\,{\mathrm {e}}^{a\,7{}\mathrm {i}+b\,x\,7{}\mathrm {i}}\,\sin \left (7\,a+7\,b\,x\right )}{1155\,b\,d^8}-\frac {3392\,c^2\,{\mathrm {e}}^{a\,7{}\mathrm {i}+b\,x\,7{}\mathrm {i}}\,\sin \left (a+b\,x\right )}{231\,b\,d^8}\right )}{128\,{\cos \left (a+b\,x\right )}^7\,\sqrt {d\,\left (\frac {{\mathrm {e}}^{-a\,1{}\mathrm {i}-b\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}}{2}\right )}} \]
-(exp(- a*7i - b*x*7i)*(c*((exp(- a*1i - b*x*1i)*1i)/2 - (exp(a*1i + b*x*1 i)*1i)/2))^(1/2)*((1216*c^2*exp(a*7i + b*x*7i)*sin(3*a + 3*b*x))/(385*b*d^ 8) + (1024*c^2*exp(a*7i + b*x*7i)*sin(5*a + 5*b*x))/(1155*b*d^8) + (128*c^ 2*exp(a*7i + b*x*7i)*sin(7*a + 7*b*x))/(1155*b*d^8) - (3392*c^2*exp(a*7i + b*x*7i)*sin(a + b*x))/(231*b*d^8)))/(128*cos(a + b*x)^7*(d*(exp(- a*1i - b*x*1i)/2 + exp(a*1i + b*x*1i)/2))^(1/2))