3.172 \(\int \csc (e+f x) (b (c \tan (e+f x))^n)^p \, dx\)

Optimal. Leaf size=81 \[ \frac{\sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \text{Hypergeometric2F1}\left (\frac{n p}{2},\frac{1}{2} (n p+1),\frac{1}{2} (n p+2),\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]

[Out]

((Cos[e + f*x]^2)^((1 + n*p)/2)*Hypergeometric2F1[(n*p)/2, (1 + n*p)/2, (2 + n*p)/2, Sin[e + f*x]^2]*Sec[e + f
*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)

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Rubi [A]  time = 0.123923, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3659, 2601, 2577} \[ \frac{\sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left (\frac{n p}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+2);\sin ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]

[Out]

((Cos[e + f*x]^2)^((1 + n*p)/2)*Hypergeometric2F1[(n*p)/2, (1 + n*p)/2, (2 + n*p)/2, Sin[e + f*x]^2]*Sec[e + f
*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)

Rule 3659

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Tan[e + f*x
])^n)^FracPart[p])/(c*Tan[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

Rule 2601

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(Cos[e + f*x
]^n*(b*Tan[e + f*x])^n)/(a*Sin[e + f*x])^n, Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x], x] /; FreeQ[{a, b
, e, f, m, n}, x] &&  !IntegerQ[n] && (ILtQ[m, 0] || (EqQ[m, 1] && EqQ[n, -2^(-1)]) || IntegersQ[m - 1/2, n -
1/2])

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]

Rubi steps

\begin{align*} \int \csc (e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \csc (e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\left (\cos ^{n p}(e+f x) \sin ^{-n p}(e+f x) \left (b (c \tan (e+f x))^n\right )^p\right ) \int \cos ^{-n p}(e+f x) \sin ^{-1+n p}(e+f x) \, dx\\ &=\frac{\cos ^2(e+f x)^{\frac{1}{2} (1+n p)} \, _2F_1\left (\frac{n p}{2},\frac{1}{2} (1+n p);\frac{1}{2} (2+n p);\sin ^2(e+f x)\right ) \sec (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f n p}\\ \end{align*}

Mathematica [A]  time = 0.201881, size = 77, normalized size = 0.95 \[ \frac{\left (\cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )\right )^{n p} \text{Hypergeometric2F1}\left (\frac{n p}{2},n p,\frac{n p}{2}+1,\tan ^2\left (\frac{1}{2} (e+f x)\right )\right ) \left (b (c \tan (e+f x))^n\right )^p}{f n p} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]

[Out]

(Hypergeometric2F1[(n*p)/2, n*p, 1 + (n*p)/2, Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(n*p)*(b*(
c*Tan[e + f*x])^n)^p)/(f*n*p)

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Maple [F]  time = 3.082, size = 0, normalized size = 0. \begin{align*} \int \csc \left ( fx+e \right ) \left ( b \left ( c\tan \left ( fx+e \right ) \right ) ^{n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x)

[Out]

int(csc(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

integrate(((c*tan(f*x + e))^n*b)^p*csc(f*x + e), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

integral(((c*tan(f*x + e))^n*b)^p*csc(f*x + e), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \left (c \tan{\left (e + f x \right )}\right )^{n}\right )^{p} \csc{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(b*(c*tan(f*x+e))**n)**p,x)

[Out]

Integral((b*(c*tan(e + f*x))**n)**p*csc(e + f*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \csc \left (f x + e\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)*(b*(c*tan(f*x+e))^n)^p,x, algorithm="giac")

[Out]

integrate(((c*tan(f*x + e))^n*b)^p*csc(f*x + e), x)