Optimal. Leaf size=83 \[ -\frac {(2-m) (a \sin (e+f x)+a)^{m+2} \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)}{2 a^2 f (m+2)}-\frac {\csc ^2(e+f x) (a \sin (e+f x)+a)^{m+2}}{2 a^2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2707, 78, 65} \[ -\frac {(2-m) (a \sin (e+f x)+a)^{m+2} \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)}{2 a^2 f (m+2)}-\frac {\csc ^2(e+f x) (a \sin (e+f x)+a)^{m+2}}{2 a^2 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 78
Rule 2707
Rubi steps
\begin {align*} \int \cot ^3(e+f x) (a+a \sin (e+f x))^m \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x) (a+x)^{1+m}}{x^3} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=-\frac {\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac {(2-m) \operatorname {Subst}\left (\int \frac {(a+x)^{1+m}}{x^2} \, dx,x,a \sin (e+f x)\right )}{2 f}\\ &=-\frac {\csc ^2(e+f x) (a+a \sin (e+f x))^{2+m}}{2 a^2 f}-\frac {(2-m) \, _2F_1(2,2+m;3+m;1+\sin (e+f x)) (a+a \sin (e+f x))^{2+m}}{2 a^2 f (2+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 68, normalized size = 0.82 \[ -\frac {(\sin (e+f x)+1)^2 (a (\sin (e+f x)+1))^m \left ((m+2) \csc ^2(e+f x)-(m-2) \, _2F_1(2,m+2;m+3;\sin (e+f x)+1)\right )}{2 f (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{3}\left (f x +e \right )\right ) \left (a +a \sin \left (f x +e \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^3\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{m} \cot ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________