Integrand size = 28, antiderivative size = 83 \[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=-\frac {2 x \cos (e+f x)}{5 f \sin ^{\frac {5}{2}}(e+f x)}-\frac {4}{15 f^2 \sin ^{\frac {3}{2}}(e+f x)}-\frac {6 x \cos (e+f x)}{5 f \sqrt {\sin (e+f x)}}+\frac {12 \sqrt {\sin (e+f x)}}{5 f^2} \] Output:
-2/5*x*cos(f*x+e)/f/sin(f*x+e)^(5/2)-4/15/f^2/sin(f*x+e)^(3/2)-6/5*x*cos(f *x+e)/f/sin(f*x+e)^(1/2)+12/5*sin(f*x+e)^(1/2)/f^2
Time = 1.53 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70 \[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=\frac {-21 f x \cos (e+f x)+9 f x \cos (3 (e+f x))+46 \sin (e+f x)-18 \sin (3 (e+f x))}{30 f^2 \sin ^{\frac {5}{2}}(e+f x)} \] Input:
Integrate[x/Sin[e + f*x]^(7/2) + (3*x*Sqrt[Sin[e + f*x]])/5,x]
Output:
(-21*f*x*Cos[e + f*x] + 9*f*x*Cos[3*(e + f*x)] + 46*Sin[e + f*x] - 18*Sin[ 3*(e + f*x)])/(30*f^2*Sin[e + f*x]^(5/2))
Time = 0.25 (sec) , antiderivative size = 83, 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.
\(\displaystyle \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4}{15 f^2 \sin ^{\frac {3}{2}}(e+f x)}+\frac {12 \sqrt {\sin (e+f x)}}{5 f^2}-\frac {2 x \cos (e+f x)}{5 f \sin ^{\frac {5}{2}}(e+f x)}-\frac {6 x \cos (e+f x)}{5 f \sqrt {\sin (e+f x)}}\) |
Input:
Int[x/Sin[e + f*x]^(7/2) + (3*x*Sqrt[Sin[e + f*x]])/5,x]
Output:
(-2*x*Cos[e + f*x])/(5*f*Sin[e + f*x]^(5/2)) - 4/(15*f^2*Sin[e + f*x]^(3/2 )) - (6*x*Cos[e + f*x])/(5*f*Sqrt[Sin[e + f*x]]) + (12*Sqrt[Sin[e + f*x]]) /(5*f^2)
\[\int \left (\frac {x}{\sin \left (f x +e \right )^{\frac {7}{2}}}+\frac {3 x \sqrt {\sin \left (f x +e \right )}}{5}\right )d x\]
Input:
int(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x)
Output:
int(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x)
Exception generated. \[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=\frac {\int \frac {5 x}{\sin ^{\frac {7}{2}}{\left (e + f x \right )}}\, dx + \int 3 x \sqrt {\sin {\left (e + f x \right )}}\, dx}{5} \] Input:
integrate(x/sin(f*x+e)**(7/2)+3/5*x*sin(f*x+e)**(1/2),x)
Output:
(Integral(5*x/sin(e + f*x)**(7/2), x) + Integral(3*x*sqrt(sin(e + f*x)), x ))/5
\[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x, algorithm="maxima")
Output:
integrate(3/5*x*sqrt(sin(f*x + e)) + x/sin(f*x + e)^(7/2), x)
\[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\sin \left (f x + e\right )} + \frac {x}{\sin \left (f x + e\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x, algorithm="giac")
Output:
integrate(3/5*x*sqrt(sin(f*x + e)) + x/sin(f*x + e)^(7/2), x)
Time = 41.02 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.05 \[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=\left (\frac {12}{5\,f^2}+\frac {x\,6{}\mathrm {i}}{5\,f}\right )\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,\left (\frac {x\,3{}\mathrm {i}}{5\,f}-\frac {32+f\,x\,66{}\mathrm {i}}{30\,f^2}\right )}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^2}-\frac {x\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,12{}\mathrm {i}}{5\,f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}+\frac {x\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,16{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3} \] Input:
int((3*x*sin(e + f*x)^(1/2))/5 + x/sin(e + f*x)^(7/2),x)
Output:
((x*6i)/(5*f) + 12/(5*f^2))*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x *1i)*1i)/2)^(1/2) - (exp(e*2i + f*x*2i)*((exp(- e*1i - f*x*1i)*1i)/2 - (ex p(e*1i + f*x*1i)*1i)/2)^(1/2)*((x*3i)/(5*f) - (f*x*66i + 32)/(30*f^2)))/(e xp(e*2i + f*x*2i) - 1)^2 - (x*exp(e*2i + f*x*2i)*((exp(- e*1i - f*x*1i)*1i )/2 - (exp(e*1i + f*x*1i)*1i)/2)^(1/2)*12i)/(5*f*(exp(e*2i + f*x*2i) - 1)) + (x*exp(e*2i + f*x*2i)*((exp(- e*1i - f*x*1i)*1i)/2 - (exp(e*1i + f*x*1i )*1i)/2)^(1/2)*16i)/(5*f*(exp(e*2i + f*x*2i) - 1)^3)
\[ \int \left (\frac {x}{\sin ^{\frac {7}{2}}(e+f x)}+\frac {3}{5} x \sqrt {\sin (e+f x)}\right ) \, dx=\int \frac {\sqrt {\sin \left (f x +e \right )}\, x}{\sin \left (f x +e \right )^{4}}d x +\frac {3 \left (\int \sqrt {\sin \left (f x +e \right )}\, x d x \right )}{5} \] Input:
int(x/sin(f*x+e)^(7/2)+3/5*x*sin(f*x+e)^(1/2),x)
Output:
(5*int((sqrt(sin(e + f*x))*x)/sin(e + f*x)**4,x) + 3*int(sqrt(sin(e + f*x) )*x,x))/5