Integrand size = 25, antiderivative size = 112 \[ \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx=\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}+\frac {16 (c \sin (a+b x))^{3/2}}{77 b c d^3 (d \cos (a+b x))^{7/2}}+\frac {64 (c \sin (a+b x))^{3/2}}{231 b c d^5 (d \cos (a+b x))^{3/2}} \]
2/11*(c*sin(b*x+a))^(3/2)/b/c/d/(d*cos(b*x+a))^(11/2)+16/77*(c*sin(b*x+a)) ^(3/2)/b/c/d^3/(d*cos(b*x+a))^(7/2)+64/231*(c*sin(b*x+a))^(3/2)/b/c/d^5/(d *cos(b*x+a))^(3/2)
Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx=\frac {2 \sqrt {d \cos (a+b x)} (45+28 \cos (2 (a+b x))+4 \cos (4 (a+b x))) \sec ^6(a+b x) (c \sin (a+b x))^{3/2}}{231 b c d^7} \]
(2*Sqrt[d*Cos[a + b*x]]*(45 + 28*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Se c[a + b*x]^6*(c*Sin[a + b*x])^(3/2))/(231*b*c*d^7)
Time = 0.49 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}}dx\) |
\(\Big \downarrow \) 3051 |
\(\displaystyle \frac {8 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}}dx}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}}dx}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\) |
\(\Big \downarrow \) 3051 |
\(\displaystyle \frac {8 \left (\frac {4 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}}dx}{7 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}\right )}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {8 \left (\frac {4 \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}}dx}{7 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}\right )}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\) |
\(\Big \downarrow \) 3043 |
\(\displaystyle \frac {8 \left (\frac {8 (c \sin (a+b x))^{3/2}}{21 b c d^3 (d \cos (a+b x))^{3/2}}+\frac {2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}\right )}{11 d^2}+\frac {2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}\) |
(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)
3.3.66.3.1 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_)]*(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 = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.58
method | result | size |
default | \(\frac {2 \left (32 \left (\cos ^{4}\left (b x +a \right )\right )+24 \left (\cos ^{2}\left (b x +a \right )\right )+21\right ) \sqrt {c \sin \left (b x +a \right )}\, \tan \left (b x +a \right ) \left (\sec ^{4}\left (b x +a \right )\right )}{231 b \,d^{6} \sqrt {d \cos \left (b x +a \right )}}\) | \(65\) |
2/231/b*(32*cos(b*x+a)^4+24*cos(b*x+a)^2+21)*(c*sin(b*x+a))^(1/2)/d^6/(d*c os(b*x+a))^(1/2)*tan(b*x+a)*sec(b*x+a)^4
Time = 0.38 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.57 \[ \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx=\frac {2 \, {\left (32 \, \cos \left (b x + a\right )^{4} + 24 \, \cos \left (b x + a\right )^{2} + 21\right )} \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{231 \, b d^{7} \cos \left (b x + a\right )^{6}} \]
2/231*(32*cos(b*x + a)^4 + 24*cos(b*x + a)^2 + 21)*sqrt(d*cos(b*x + a))*sq rt(c*sin(b*x + a))*sin(b*x + a)/(b*d^7*cos(b*x + a)^6)
Timed out. \[ \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx=\int { \frac {\sqrt {c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {13}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx=\int { \frac {\sqrt {c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {13}{2}}} \,d x } \]
Time = 6.25 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.93 \[ \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx=-\frac {\sqrt {c\,\sin \left (a+b\,x\right )}\,\left (2\,{\sin \left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2-1\right )\,\left (2\,{\sin \left (\frac {5\,a}{2}+\frac {5\,b\,x}{2}\right )}^2+\sin \left (5\,a+5\,b\,x\right )\,1{}\mathrm {i}-1\right )\,\left (\frac {1984\,\sin \left (a+b\,x\right )\,\left (-2\,{\sin \left (\frac {5\,a}{2}+\frac {5\,b\,x}{2}\right )}^2+\sin \left (5\,a+5\,b\,x\right )\,1{}\mathrm {i}+1\right )}{231\,b\,d^6}+\frac {256\,\sin \left (3\,a+3\,b\,x\right )\,\left (-2\,{\sin \left (\frac {5\,a}{2}+\frac {5\,b\,x}{2}\right )}^2+\sin \left (5\,a+5\,b\,x\right )\,1{}\mathrm {i}+1\right )}{77\,b\,d^6}+\frac {128\,\sin \left (5\,a+5\,b\,x\right )\,\left (-2\,{\sin \left (\frac {5\,a}{2}+\frac {5\,b\,x}{2}\right )}^2+\sin \left (5\,a+5\,b\,x\right )\,1{}\mathrm {i}+1\right )}{231\,b\,d^6}\right )}{32\,{\left ({\sin \left (a+b\,x\right )}^2-1\right )}^3\,\sqrt {-d\,\left (2\,{\sin \left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2-1\right )}} \]
-((c*sin(a + b*x))^(1/2)*(2*sin(a/2 + (b*x)/2)^2 - 1)*(sin(5*a + 5*b*x)*1i + 2*sin((5*a)/2 + (5*b*x)/2)^2 - 1)*((1984*sin(a + b*x)*(sin(5*a + 5*b*x) *1i - 2*sin((5*a)/2 + (5*b*x)/2)^2 + 1))/(231*b*d^6) + (256*sin(3*a + 3*b* x)*(sin(5*a + 5*b*x)*1i - 2*sin((5*a)/2 + (5*b*x)/2)^2 + 1))/(77*b*d^6) + (128*sin(5*a + 5*b*x)*(sin(5*a + 5*b*x)*1i - 2*sin((5*a)/2 + (5*b*x)/2)^2 + 1))/(231*b*d^6)))/(32*(sin(a + b*x)^2 - 1)^3*(-d*(2*sin(a/2 + (b*x)/2)^2 - 1))^(1/2))