Integrand size = 18, antiderivative size = 103 \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}+\frac {20 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right ) \sqrt {\sec (a+b x)}}{147 b^2}+\frac {4 \sin (a+b x)}{49 b^2 \sec ^{\frac {5}{2}}(a+b x)}+\frac {20 \sin (a+b x)}{147 b^2 \sqrt {\sec (a+b x)}} \]
-2/7*x/b/sec(b*x+a)^(7/2)+4/49*sin(b*x+a)/b^2/sec(b*x+a)^(5/2)+20/147*sin( b*x+a)/b^2/sec(b*x+a)^(1/2)+20/147*(cos(1/2*a+1/2*b*x)^2)^(1/2)/cos(1/2*a+ 1/2*b*x)*EllipticF(sin(1/2*a+1/2*b*x),2^(1/2))*cos(b*x+a)^(1/2)*sec(b*x+a) ^(1/2)/b^2
Time = 0.42 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {\sec (a+b x)} \left (-63 b x-84 b x \cos (2 (a+b x))-21 b x \cos (4 (a+b x))+80 \sqrt {\cos (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+52 \sin (2 (a+b x))+6 \sin (4 (a+b x))\right )}{588 b^2} \]
(Sqrt[Sec[a + b*x]]*(-63*b*x - 84*b*x*Cos[2*(a + b*x)] - 21*b*x*Cos[4*(a + b*x)] + 80*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 52*Sin[2*(a + b *x)] + 6*Sin[4*(a + b*x)]))/(588*b^2)
Time = 0.49 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4700, 3042, 4256, 3042, 4256, 3042, 4258, 3042, 3120}
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.
\(\displaystyle \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx\) |
\(\Big \downarrow \) 4700 |
\(\displaystyle \frac {2 \int \frac {1}{\sec ^{\frac {7}{2}}(a+b x)}dx}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \frac {1}{\csc \left (a+b x+\frac {\pi }{2}\right )^{7/2}}dx}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \int \frac {1}{\sec ^{\frac {3}{2}}(a+b x)}dx+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\right )}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \int \frac {1}{\csc \left (a+b x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\right )}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 4256 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\sec (a+b x)}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\right )}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\csc \left (a+b x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\right )}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \frac {1}{\sqrt {\cos (a+b x)}}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\right )}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \int \frac {1}{\sqrt {\sin \left (a+b x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}\right )+\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}\right )}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {2 \left (\frac {2 \sin (a+b x)}{7 b \sec ^{\frac {5}{2}}(a+b x)}+\frac {5}{7} \left (\frac {2 \sin (a+b x)}{3 b \sqrt {\sec (a+b x)}}+\frac {2 \sqrt {\cos (a+b x)} \sqrt {\sec (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b}\right )\right )}{7 b}-\frac {2 x}{7 b \sec ^{\frac {7}{2}}(a+b x)}\) |
(-2*x)/(7*b*Sec[a + b*x]^(7/2)) + (2*((2*Sin[a + b*x])/(7*b*Sec[a + b*x]^( 5/2)) + (5*((2*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2]*Sqrt[Sec[a + b *x]])/(3*b) + (2*Sin[a + b*x])/(3*b*Sqrt[Sec[a + b*x]])))/7))/(7*b)
3.4.43.3.1 Defintions of rubi rules used
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(x_)^(m_.)*Sec[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*Sin[(a_.) + (b_.)*(x_)^(n _.)], x_Symbol] :> Simp[x^(m - n + 1)*(Sec[a + b*x^n]^(p - 1)/(b*n*(p - 1)) ), x] - Simp[(m - n + 1)/(b*n*(p - 1)) Int[x^(m - n)*Sec[a + b*x^n]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
\[\int \frac {x \sin \left (x b +a \right )}{\sec \left (x b +a \right )^{\frac {5}{2}}}d x\]
Exception generated. \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x \sin {\left (a + b x \right )}}{\sec ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
\[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\sec \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \sin \left (b x + a\right )}{\sec \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \sin (a+b x)}{\sec ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\sin \left (a+b\,x\right )}{{\left (\frac {1}{\cos \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]