Optimal. Leaf size=63 \[ -\frac {\cot ^3(e+f x) (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{2},\frac {3+m}{2};\frac {5}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {3+m}{2}}}{3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697}
\begin {gather*} -\frac {\cot ^3(e+f x) \sin ^2(e+f x)^{\frac {m+3}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{2},\frac {m+3}{2};\frac {5}{2};\cos ^2(e+f x)\right )}{3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2697
Rubi steps
\begin {align*} \int \cot ^2(e+f x) (b \csc (e+f x))^m \, dx &=-\frac {\cot ^3(e+f x) (b \csc (e+f x))^m \, _2F_1\left (\frac {3}{2},\frac {3+m}{2};\frac {5}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {3+m}{2}}}{3 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(186\) vs. \(2(63)=126\).
time = 1.25, size = 186, normalized size = 2.95 \begin {gather*} -\frac {(b \csc (e+f x))^m \left (-4 (1+m) \, _2F_1\left (1-m,\frac {1}{2}-\frac {m}{2};\frac {3}{2}-\frac {m}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(-1+m) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (-\frac {1}{2}-\frac {m}{2},-m;\frac {1}{2}-\frac {m}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+(1+m) \, _2F_1\left (\frac {1}{2}-\frac {m}{2},-m;\frac {3}{2}-\frac {m}{2};-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )^{-m} \tan \left (\frac {1}{2} (e+f x)\right )}{2 f \left (-1+m^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (\cot ^{2}\left (f x +e \right )\right ) \left (b \csc \left (f x +e \right )\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{2}{\left (e + f x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________