Optimal. Leaf size=24 \[ \frac {b (b \csc (e+f x))^{n-1}}{f (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2621, 30} \[ \frac {b (b \csc (e+f x))^{n-1}}{f (1-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2621
Rubi steps
\begin {align*} \int \cos (e+f x) (b \csc (e+f x))^n \, dx &=-\frac {b \operatorname {Subst}\left (\int x^{-2+n} \, dx,x,b \csc (e+f x)\right )}{f}\\ &=\frac {b (b \csc (e+f x))^{-1+n}}{f (1-n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 23, normalized size = 0.96 \[ -\frac {b (b \csc (e+f x))^{n-1}}{f (n-1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.86, size = 29, normalized size = 1.21 \[ -\frac {\left (\frac {b}{\sin \left (f x + e\right )}\right )^{n} \sin \left (f x + e\right )}{f n - f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{n} \cos \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.30, size = 66, normalized size = 2.75 \[ -\frac {2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) {\mathrm e}^{n \ln \left (\frac {b \left (1+\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}}{f \left (-1+n \right ) \left (1+\tan ^{2}\left (\frac {e}{2}+\frac {f x}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.71, size = 29, normalized size = 1.21 \[ -\frac {b^{n} \sin \left (f x + e\right )^{-n} \sin \left (f x + e\right )}{f {\left (n - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.32, size = 28, normalized size = 1.17 \[ -\frac {\sin \left (e+f\,x\right )\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^n}{f\,\left (n-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc {\left (e + f x \right )}\right )^{n} \cos {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________