3.489 \(\int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx\)

Optimal. Leaf size=89 \[ -\frac {a \sin ^2(e+f x)^{\frac {1-m}{2}} \sec ^{n-1}(e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]

[Out]

-a*hypergeom([1/2-1/2*m, 1/2-1/2*n],[3/2-1/2*n],cos(f*x+e)^2)*sec(f*x+e)^(-1+n)*(a*sin(f*x+e))^(-1+m)*(sin(f*x
+e)^2)^(1/2-1/2*m)/f/(1-n)

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Rubi [A]  time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2587, 2576} \[ -\frac {a \sin ^2(e+f x)^{\frac {1-m}{2}} \sec ^{n-1}(e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left (\frac {1-m}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^n*(a*Sin[e + f*x])^m,x]

[Out]

-((a*Hypergeometric2F1[(1 - m)/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^(-1 + n)*(a*Sin[e + f*x])
^(-1 + m)*(Sin[e + f*x]^2)^((1 - m)/2))/(f*(1 - n)))

Rule 2576

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b^(2*IntPar
t[(n - 1)/2] + 1)*(b*Sin[e + f*x])^(2*FracPart[(n - 1)/2])*(a*Cos[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/
2, (1 - n)/2, (3 + m)/2, Cos[e + f*x]^2])/(a*f*(m + 1)*(Sin[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a,
b, e, f, m, n}, x] && SimplerQ[n, m]

Rule 2587

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[b^2*(b*Cos[e
+ f*x])^(n - 1)*(b*Sec[e + f*x])^(n - 1), Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e,
f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx &=\left (\cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) (a \sin (e+f x))^m \, dx\\ &=-\frac {a \, _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) (a \sin (e+f x))^{-1+m} \sin ^2(e+f x)^{\frac {1-m}{2}}}{f (1-n)}\\ \end {align*}

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Mathematica [C]  time = 0.15, size = 287, normalized size = 3.22 \[ \frac {4 (m+3) \sin \left (\frac {1}{2} (e+f x)\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \sec ^n(e+f x) (a \sin (e+f x))^m F_1\left (\frac {m+1}{2};n,m-n+1;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )}{f (m+1) \left ((m+3) (\cos (e+f x)+1) F_1\left (\frac {m+1}{2};n,m-n+1;\frac {m+3}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 \sin ^2\left (\frac {1}{2} (e+f x)\right ) \left ((m-n+1) F_1\left (\frac {m+3}{2};n,m-n+2;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n F_1\left (\frac {m+3}{2};n+1,m-n+1;\frac {m+5}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[e + f*x]^n*(a*Sin[e + f*x])^m,x]

[Out]

(4*(3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)
/2]^3*Sec[e + f*x]^n*Sin[(e + f*x)/2]*(a*Sin[e + f*x])^m)/(f*(1 + m)*((3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n
, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]) - 4*((1 + m - n)*AppellF1[(3 + m)/2,
n, 2 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + m)/2, 1 + n, 1 + m - n, (5
+ m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^2))

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fricas [F]  time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \sin \left (f x + e\right )\right )^{m} \sec \left (f x + e\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^n*(a*sin(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((a*sin(f*x + e))^m*sec(f*x + e)^n, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \sec \left (f x + e\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^n*(a*sin(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e))^m*sec(f*x + e)^n, x)

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maple [F]  time = 0.61, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{n}\left (f x +e \right )\right ) \left (a \sin \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^n*(a*sin(f*x+e))^m,x)

[Out]

int(sec(f*x+e)^n*(a*sin(f*x+e))^m,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \sec \left (f x + e\right )^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^n*(a*sin(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e))^m*sec(f*x + e)^n, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(e + f*x))^m*(1/cos(e + f*x))^n,x)

[Out]

int((a*sin(e + f*x))^m*(1/cos(e + f*x))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \sec ^{n}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**n*(a*sin(f*x+e))**m,x)

[Out]

Integral((a*sin(e + f*x))**m*sec(e + f*x)**n, x)

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