Optimal. Leaf size=89 \[ \frac{4 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+3} F_1\left (m+\frac{5}{2};-\frac{3}{2},4;m+\frac{7}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{a^3 f (2 m+5)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0993711, antiderivative size = 89, normalized size of antiderivative = 1., 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 = {2719, 137, 136} \[ \frac{4 \sqrt{2} \sqrt{1-\sin (e+f x)} \sec (e+f x) (a \sin (e+f x)+a)^{m+3} F_1\left (m+\frac{5}{2};-\frac{3}{2},4;m+\frac{7}{2};\frac{1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )}{a^3 f (2 m+5)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2719
Rule 137
Rule 136
Rubi steps
\begin{align*} \int \cot ^4(e+f x) (a+a \sin (e+f x))^m \, dx &=\frac{\left (\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{(a-x)^{3/2} (a+x)^{\frac{3}{2}+m}}{x^4} \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac{\left (2 \sqrt{2} \sec (e+f x) (a-a \sin (e+f x)) \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{\frac{3}{2}+m} \left (\frac{1}{2}-\frac{x}{2 a}\right )^{3/2}}{x^4} \, dx,x,a \sin (e+f x)\right )}{f \sqrt{\frac{a-a \sin (e+f x)}{a}}}\\ &=\frac{4 \sqrt{2} F_1\left (\frac{5}{2}+m;-\frac{3}{2},4;\frac{7}{2}+m;\frac{1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sec (e+f x) \sqrt{1-\sin (e+f x)} (a+a \sin (e+f x))^{3+m}}{a^3 f (5+2 m)}\\ \end{align*}
Mathematica [F] time = 0.740835, size = 0, normalized size = 0. \[ \int \cot ^4(e+f x) (a+a \sin (e+f x))^m \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.294, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{4} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )}^{m} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]