3.166 \(\int (a \sin (e+f x))^m \tan ^4(e+f x) \, dx\)

Optimal. Leaf size=68 \[ \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+5} \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {m+7}{2};\sin ^2(e+f x)\right )}{a^5 f (m+5)} \]

[Out]

hypergeom([5/2, 5/2+1/2*m],[7/2+1/2*m],sin(f*x+e)^2)*sec(f*x+e)*(a*sin(f*x+e))^(5+m)*(cos(f*x+e)^2)^(1/2)/a^5/
f/(5+m)

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Rubi [A]  time = 0.09, antiderivative size = 68, 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 = {2600, 2577} \[ \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) (a \sin (e+f x))^{m+5} \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {m+7}{2};\sin ^2(e+f x)\right )}{a^5 f (m+5)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sin[e + f*x])^m*Tan[e + f*x]^4,x]

[Out]

(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[5/2, (5 + m)/2, (7 + m)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(a*Sin[e + f*x
])^(5 + m))/(a^5*f*(5 + m))

Rule 2577

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

Rule 2600

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

Rubi steps

\begin {align*} \int (a \sin (e+f x))^m \tan ^4(e+f x) \, dx &=\frac {\int \sec ^4(e+f x) (a \sin (e+f x))^{4+m} \, dx}{a^4}\\ &=\frac {\sqrt {\cos ^2(e+f x)} \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7+m}{2};\sin ^2(e+f x)\right ) \sec (e+f x) (a \sin (e+f x))^{5+m}}{a^5 f (5+m)}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 71, normalized size = 1.04 \[ \frac {\sin ^4(e+f x) \sqrt {\cos ^2(e+f x)} \tan (e+f x) (a \sin (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {m+7}{2};\sin ^2(e+f x)\right )}{f (m+5)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sin[e + f*x])^m*Tan[e + f*x]^4,x]

[Out]

(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[5/2, (5 + m)/2, (7 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^4*(a*Sin[e + f
*x])^m*Tan[e + f*x])/(f*(5 + m))

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(f*x+e))^m*tan(f*x+e)^4,x, algorithm="fricas")

[Out]

integral((a*sin(f*x + e))^m*tan(f*x + e)^4, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(f*x+e))^m*tan(f*x+e)^4,x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e))^m*tan(f*x + e)^4, x)

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maple [F]  time = 0.41, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x +e \right )\right )^{m} \left (\tan ^{4}\left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(f*x+e))^m*tan(f*x+e)^4,x)

[Out]

int((a*sin(f*x+e))^m*tan(f*x+e)^4,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(f*x+e))^m*tan(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e))^m*tan(f*x + e)^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^4\,{\left (a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^4*(a*sin(e + f*x))^m,x)

[Out]

int(tan(e + f*x)^4*(a*sin(e + f*x))^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (e + f x \right )}\right )^{m} \tan ^{4}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sin(f*x+e))**m*tan(f*x+e)**4,x)

[Out]

Integral((a*sin(e + f*x))**m*tan(e + f*x)**4, x)

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