Optimal. Leaf size=86 \[ -\frac{\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} \sec ^{n-1}(e+f x) \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0764943, antiderivative size = 86, normalized size of antiderivative = 1., 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 = {2587, 2576} \[ -\frac{\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} \sec ^{n-1}(e+f x) \, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2587
Rule 2576
Rubi steps
\begin{align*} \int \sec ^n(e+f x) \sin ^m(e+f x) \, dx &=\left (\cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \sin ^m(e+f x) \, dx\\ &=-\frac{\, _2F_1\left (\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin ^{-1+m}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}}}{f (1-n)}\\ \end{align*}
Mathematica [C] time = 1.38711, size = 285, normalized size = 3.31 \[ \frac{4 (m+3) \sin \left (\frac{1}{2} (e+f x)\right ) \cos ^3\left (\frac{1}{2} (e+f x)\right ) \sin ^m(e+f x) \sec ^n(e+f x) F_1\left (\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )}{f (m+1) \left ((m+3) (\cos (e+f x)+1) F_1\left (\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-4 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \left ((m-n+1) F_1\left (\frac{m+3}{2};n,m-n+2;\frac{m+5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-n F_1\left (\frac{m+3}{2};n+1,m-n+1;\frac{m+5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.516, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( fx+e \right ) \right ) ^{n} \left ( \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 \sec \left (f x + e\right )^{n} \sin \left (f x + e\right )^{m}\,{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 (\sec \left (f x + e\right )^{n} \sin \left (f x + e\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin ^{m}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx \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 \sec \left (f x + e\right )^{n} \sin \left (f x + e\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]