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

Optimal. Leaf size=48 \[ \frac {(a \sin (e+f x))^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};\sin ^2(e+f x)\right )}{a^2 f (m+2)} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 = {2592, 364} \[ \frac {(a \sin (e+f x))^{m+2} \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};\sin ^2(e+f x)\right )}{a^2 f (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[e + f*x]^2]*(a*Sin[e + f*x])^(2 + m))/(a^2*f*(2 + m))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rubi steps

\begin {align*} \int (a \sin (e+f x))^m \tan (e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^{1+m}}{a^2-x^2} \, dx,x,a \sin (e+f x)\right )}{f}\\ &=\frac {\, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};\sin ^2(e+f x)\right ) (a \sin (e+f x))^{2+m}}{a^2 f (2+m)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 53, normalized size = 1.10 \[ \frac {\sin ^2(e+f x) (a \sin (e+f x))^m \, _2F_1\left (1,\frac {m+2}{2};\frac {m+2}{2}+1;\sin ^2(e+f x)\right )}{f (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Hypergeometric2F1[1, (2 + m)/2, 1 + (2 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^2*(a*Sin[e + f*x])^m)/(f*(2 + m))

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fricas [F]  time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (a \sin \left (f x + e\right )\right )^{m} \tan \left (f x + e\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin \left (f x +e \right )\right )^{m} \tan \left (f x +e \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {tan}\left (e+f\,x\right )\,{\left (a\,\sin \left (e+f\,x\right )\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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