3.374 \(\int (b \csc (e+f x))^m \tan ^3(e+f x) \, dx\)

Optimal. Leaf size=40 \[ -\frac {(b \csc (e+f x))^m \, _2F_1\left (2,\frac {m}{2};\frac {m+2}{2};\csc ^2(e+f x)\right )}{f m} \]

[Out]

-(b*csc(f*x+e))^m*hypergeom([2, 1/2*m],[1+1/2*m],csc(f*x+e)^2)/f/m

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Rubi [A]  time = 0.05, antiderivative size = 40, 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 = {2606, 364} \[ -\frac {(b \csc (e+f x))^m \, _2F_1\left (2,\frac {m}{2};\frac {m+2}{2};\csc ^2(e+f x)\right )}{f m} \]

Antiderivative was successfully verified.

[In]

Int[(b*Csc[e + f*x])^m*Tan[e + f*x]^3,x]

[Out]

-(((b*Csc[e + f*x])^m*Hypergeometric2F1[2, m/2, (2 + m)/2, Csc[e + f*x]^2])/(f*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 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

\begin {align*} \int (b \csc (e+f x))^m \tan ^3(e+f x) \, dx &=-\frac {b \operatorname {Subst}\left (\int \frac {(b x)^{-1+m}}{\left (-1+x^2\right )^2} \, dx,x,\csc (e+f x)\right )}{f}\\ &=-\frac {(b \csc (e+f x))^m \, _2F_1\left (2,\frac {m}{2};\frac {2+m}{2};\csc ^2(e+f x)\right )}{f m}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 52, normalized size = 1.30 \[ -\frac {\sin ^4(e+f x) (b \csc (e+f x))^m \, _2F_1\left (2,2-\frac {m}{2};3-\frac {m}{2};\sin ^2(e+f x)\right )}{f (m-4)} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Csc[e + f*x])^m*Tan[e + f*x]^3,x]

[Out]

-(((b*Csc[e + f*x])^m*Hypergeometric2F1[2, 2 - m/2, 3 - m/2, Sin[e + f*x]^2]*Sin[e + f*x]^4)/(f*(-4 + m)))

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^m*tan(f*x+e)^3,x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e))^m*tan(f*x + e)^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^m*tan(f*x+e)^3,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^m*tan(f*x + e)^3, x)

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maple [F]  time = 0.49, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x +e \right )\right )^{m} \left (\tan ^{3}\left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*csc(f*x+e))^m*tan(f*x+e)^3,x)

[Out]

int((b*csc(f*x+e))^m*tan(f*x+e)^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{m} \tan \left (f x + e\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))^m*tan(f*x+e)^3,x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e))^m*tan(f*x + e)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^3\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^3*(b/sin(e + f*x))^m,x)

[Out]

int(tan(e + f*x)^3*(b/sin(e + f*x))^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc {\left (e + f x \right )}\right )^{m} \tan ^{3}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*csc(f*x+e))**m*tan(f*x+e)**3,x)

[Out]

Integral((b*csc(e + f*x))**m*tan(e + f*x)**3, x)

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