Integrand size = 17, antiderivative size = 25 \[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=-\frac {b (b \sec (e+f x))^{-1+n}}{f (1-n)} \] Output:
-b*(b*sec(f*x+e))^(-1+n)/f/(1-n)
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=\frac {b (b \sec (e+f x))^{-1+n}}{f (-1+n)} \] Input:
Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x],x]
Output:
(b*(b*Sec[e + f*x])^(-1 + n))/(f*(-1 + n))
Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3042, 3102, 15}
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.
\(\displaystyle \int \sin (e+f x) (b \sec (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(b \sec (e+f x))^n}{\csc (e+f x)}dx\) |
\(\Big \downarrow \) 3102 |
\(\displaystyle \frac {b \int (b \sec (e+f x))^{n-2}d(b \sec (e+f x))}{f}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {b (b \sec (e+f x))^{n-1}}{f (1-n)}\) |
Input:
Int[(b*Sec[e + f*x])^n*Sin[e + f*x],x]
Output:
-((b*(b*Sec[e + f*x])^(-1 + n))/(f*(1 - n)))
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_S ymbol] :> Simp[1/(f*a^n) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/ 2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1 )/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Time = 0.82 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {\cos \left (f x +e \right ) \left (b \sec \left (f x +e \right )\right )^{n}}{f \left (-1+n \right )}\) | \(26\) |
derivativedivides | \(\frac {{\mathrm e}^{n \ln \left (b \sec \left (f x +e \right )\right )}}{f \left (-1+n \right ) \sec \left (f x +e \right )}\) | \(30\) |
default | \(\frac {{\mathrm e}^{n \ln \left (b \sec \left (f x +e \right )\right )}}{f \left (-1+n \right ) \sec \left (f x +e \right )}\) | \(30\) |
norman | \(\frac {\frac {{\mathrm e}^{n \ln \left (\frac {b \left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}{1-\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}\right )}}{f \left (-1+n \right )}-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} {\mathrm e}^{n \ln \left (\frac {b \left (1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}\right )}{1-\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}\right )}}{f \left (-1+n \right )}}{1+\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}\) | \(120\) |
risch | \(\frac {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{-n} \cos \left (f x +e \right ) 2^{n} \left ({\mathrm e}^{i \left (f x +e \right )}\right )^{n} b^{n} {\mathrm e}^{\frac {i \pi n \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{3}+\operatorname {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{i \left (f x +e \right )}\right )+\operatorname {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )-\operatorname {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \operatorname {csgn}\left (i {\mathrm e}^{i \left (f x +e \right )}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )+\operatorname {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \operatorname {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2}-\operatorname {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \operatorname {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \operatorname {csgn}\left (i b \right )-\operatorname {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{3}+\operatorname {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \operatorname {csgn}\left (i b \right )\right )}{2}}}{\left (-1+n \right ) f}\) | \(413\) |
Input:
int((b*sec(f*x+e))^n*sin(f*x+e),x,method=_RETURNVERBOSE)
Output:
1/f/(-1+n)*cos(f*x+e)*(b*sec(f*x+e))^n
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=\frac {\left (\frac {b}{\cos \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{f n - f} \] Input:
integrate((b*sec(f*x+e))^n*sin(f*x+e),x, algorithm="fricas")
Output:
(b/cos(f*x + e))^n*cos(f*x + e)/(f*n - f)
\[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{n} \sin {\left (e + f x \right )}\, dx \] Input:
integrate((b*sec(f*x+e))**n*sin(f*x+e),x)
Output:
Integral((b*sec(e + f*x))**n*sin(e + f*x), x)
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=\frac {b^{n} \cos \left (f x + e\right )^{-n} \cos \left (f x + e\right )}{f {\left (n - 1\right )}} \] Input:
integrate((b*sec(f*x+e))^n*sin(f*x+e),x, algorithm="maxima")
Output:
b^n*cos(f*x + e)^(-n)*cos(f*x + e)/(f*(n - 1))
\[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{n} \sin \left (f x + e\right ) \,d x } \] Input:
integrate((b*sec(f*x+e))^n*sin(f*x+e),x, algorithm="giac")
Output:
integrate((b*sec(f*x + e))^n*sin(f*x + e), x)
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=\frac {\cos \left (e+f\,x\right )\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^n}{f\,\left (n-1\right )} \] Input:
int(sin(e + f*x)*(b/cos(e + f*x))^n,x)
Output:
(cos(e + f*x)*(b/cos(e + f*x))^n)/(f*(n - 1))
\[ \int (b \sec (e+f x))^n \sin (e+f x) \, dx=b^{n} \left (\int \sec \left (f x +e \right )^{n} \sin \left (f x +e \right )d x \right ) \] Input:
int((b*sec(f*x+e))^n*sin(f*x+e),x)
Output:
b**n*int(sec(e + f*x)**n*sin(e + f*x),x)