Optimal. Leaf size=78 \[ -\frac{(a \cos (e+f x))^{m+1} \, _2F_1\left (-\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )}{a b f (m+1) \sqrt [4]{\sin ^2(e+f x)} \sqrt{b \csc (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11167, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2586, 2576} \[ -\frac{(a \cos (e+f x))^{m+1} \, _2F_1\left (-\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right )}{a b f (m+1) \sqrt [4]{\sin ^2(e+f x)} \sqrt{b \csc (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2586
Rule 2576
Rubi steps
\begin{align*} \int \frac{(a \cos (e+f x))^m}{(b \csc (e+f x))^{3/2}} \, dx &=\frac{\int (a \cos (e+f x))^m (b \sin (e+f x))^{3/2} \, dx}{b^2 \sqrt{b \csc (e+f x)} \sqrt{b \sin (e+f x)}}\\ &=-\frac{(a \cos (e+f x))^{1+m} \, _2F_1\left (-\frac{1}{4},\frac{1+m}{2};\frac{3+m}{2};\cos ^2(e+f x)\right )}{a b f (1+m) \sqrt{b \csc (e+f x)} \sqrt [4]{\sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.76986, size = 116, normalized size = 1.49 \[ \frac{2 a \cos (2 (e+f x)) \left (-\cot ^2(e+f x)\right )^{\frac{1-m}{2}} (a \cos (e+f x))^{m-1} \, _2F_1\left (\frac{1}{4} (-2 m-3),\frac{1-m}{2};\frac{1}{4} (1-2 m);\csc ^2(e+f x)\right )}{b f (2 m+3) \left (\csc ^2(e+f x)-2\right ) \sqrt{b \csc (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.309, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a\cos \left ( fx+e \right ) \right ) ^{m} \left ( b\csc \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac{3}{2}}}\,{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 (\frac{\sqrt{b \csc \left (f x + e\right )} \left (a \cos \left (f x + e\right )\right )^{m}}{b^{2} \csc \left (f x + e\right )^{2}}, 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 \frac{\left (a \cos \left (f x + e\right )\right )^{m}}{\left (b \csc \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]