Integrand size = 28, antiderivative size = 42 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=-\frac {2 x \cos (e+f x)}{3 f \sin ^{\frac {3}{2}}(e+f x)}-\frac {4}{3 f^2 \sqrt {\sin (e+f x)}} \] Output:
-2/3*x*cos(f*x+e)/f/sin(f*x+e)^(3/2)-4/3/f^2/sin(f*x+e)^(1/2)
Time = 1.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=-\frac {2 (f x \cos (e+f x)+2 \sin (e+f x))}{3 f^2 \sin ^{\frac {3}{2}}(e+f x)} \] Input:
Integrate[x/Sin[e + f*x]^(5/2) - x/(3*Sqrt[Sin[e + f*x]]),x]
Output:
(-2*(f*x*Cos[e + f*x] + 2*Sin[e + f*x]))/(3*f^2*Sin[e + f*x]^(3/2))
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {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.
\(\displaystyle \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{3 f^2 \sqrt {\sin (e+f x)}}-\frac {2 x \cos (e+f x)}{3 f \sin ^{\frac {3}{2}}(e+f x)}\) |
Input:
Int[x/Sin[e + f*x]^(5/2) - x/(3*Sqrt[Sin[e + f*x]]),x]
Output:
(-2*x*Cos[e + f*x])/(3*f*Sin[e + f*x]^(3/2)) - 4/(3*f^2*Sqrt[Sin[e + f*x]] )
\[\int \left (\frac {x}{\sin \left (f x +e \right )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\sin \left (f x +e \right )}}\right )d x\]
Input:
int(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x)
Output:
int(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x)
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=\frac {2 \, {\left (f x \cos \left (f x + e\right ) + 2 \, \sin \left (f x + e\right )\right )} \sqrt {\sin \left (f x + e\right )}}{3 \, {\left (f^{2} \cos \left (f x + e\right )^{2} - f^{2}\right )}} \] Input:
integrate(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x, algorithm="fricas")
Output:
2/3*(f*x*cos(f*x + e) + 2*sin(f*x + e))*sqrt(sin(f*x + e))/(f^2*cos(f*x + e)^2 - f^2)
\[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=- \frac {\int \left (- \frac {3 x}{\sin ^{\frac {5}{2}}{\left (e + f x \right )}}\right )\, dx + \int \frac {x}{\sqrt {\sin {\left (e + f x \right )}}}\, dx}{3} \] Input:
integrate(x/sin(f*x+e)**(5/2)-1/3*x/sin(f*x+e)**(1/2),x)
Output:
-(Integral(-3*x/sin(e + f*x)**(5/2), x) + Integral(x/sqrt(sin(e + f*x)), x ))/3
\[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=\int { -\frac {x}{3 \, \sqrt {\sin \left (f x + e\right )}} + \frac {x}{\sin \left (f x + e\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x, algorithm="maxima")
Output:
integrate(-1/3*x/sqrt(sin(f*x + e)) + x/sin(f*x + e)^(5/2), x)
\[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=\int { -\frac {x}{3 \, \sqrt {\sin \left (f x + e\right )}} + \frac {x}{\sin \left (f x + e\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x, algorithm="giac")
Output:
integrate(-1/3*x/sqrt(sin(f*x + e)) + x/sin(f*x + e)^(5/2), x)
Time = 37.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.33 \[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=-\frac {4\,\sqrt {\sin \left (e+f\,x\right )}\,\left (20\,\sin \left (e+f\,x\right )-10\,\sin \left (3\,e+3\,f\,x\right )+2\,\sin \left (5\,e+5\,f\,x\right )-2\,f\,x\,\left (2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )+3\,f\,x\,\left (2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2-1\right )-f\,x\,\left (2\,{\sin \left (\frac {5\,e}{2}+\frac {5\,f\,x}{2}\right )}^2-1\right )\right )}{3\,f^2\,\left (30\,{\sin \left (e+f\,x\right )}^2-12\,{\sin \left (2\,e+2\,f\,x\right )}^2+2\,{\sin \left (3\,e+3\,f\,x\right )}^2\right )} \] Input:
int(x/sin(e + f*x)^(5/2) - x/(3*sin(e + f*x)^(1/2)),x)
Output:
-(4*sin(e + f*x)^(1/2)*(20*sin(e + f*x) - 10*sin(3*e + 3*f*x) + 2*sin(5*e + 5*f*x) - 2*f*x*(2*sin(e/2 + (f*x)/2)^2 - 1) + 3*f*x*(2*sin((3*e)/2 + (3* f*x)/2)^2 - 1) - f*x*(2*sin((5*e)/2 + (5*f*x)/2)^2 - 1)))/(3*f^2*(2*sin(3* e + 3*f*x)^2 - 12*sin(2*e + 2*f*x)^2 + 30*sin(e + f*x)^2))
\[ \int \left (\frac {x}{\sin ^{\frac {5}{2}}(e+f x)}-\frac {x}{3 \sqrt {\sin (e+f x)}}\right ) \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )}\, x}{\sin \left (f x +e \right )^{3}}d x -\frac {\left (\int \frac {\sqrt {\sin \left (f x +e \right )}\, x}{\sin \left (f x +e \right )}d x \right )}{3} \] Input:
int(x/sin(f*x+e)^(5/2)-1/3*x/sin(f*x+e)^(1/2),x)
Output:
(3*int((sqrt(sin(e + f*x))*x)/sin(e + f*x)**3,x) - int((sqrt(sin(e + f*x)) *x)/sin(e + f*x),x))/3