3.68 \(\int ((b \sec (c+d x))^p)^n \, dx\)

Optimal. Leaf size=81 \[ -\frac{\sin (c+d x) \cos (c+d x) \left ((b \sec (c+d x))^p\right )^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (1-n p),\frac{1}{2} (3-n p),\cos ^2(c+d x)\right )}{d (1-n p) \sqrt{\sin ^2(c+d x)}} \]

[Out]

-((Cos[c + d*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[c + d*x]^2]*((b*Sec[c + d*x])^p)^n*Sin[c
+ d*x])/(d*(1 - n*p)*Sqrt[Sin[c + d*x]^2]))

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Rubi [A]  time = 0.0532608, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4123, 3772, 2643} \[ -\frac{\sin (c+d x) \cos (c+d x) \left ((b \sec (c+d x))^p\right )^n \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(c+d x)\right )}{d (1-n p) \sqrt{\sin ^2(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[((b*Sec[c + d*x])^p)^n,x]

[Out]

-((Cos[c + d*x]*Hypergeometric2F1[1/2, (1 - n*p)/2, (3 - n*p)/2, Cos[c + d*x]^2]*((b*Sec[c + d*x])^p)^n*Sin[c
+ d*x])/(d*(1 - n*p)*Sqrt[Sin[c + d*x]^2]))

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rule 3772

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x])^(n - 1)*((Sin[c + d*x]/b)^(n - 1)
*Int[1/(Sin[c + d*x]/b)^n, x]), x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int \left ((b \sec (c+d x))^p\right )^n \, dx &=\left ((b \sec (c+d x))^{-n p} \left ((b \sec (c+d x))^p\right )^n\right ) \int (b \sec (c+d x))^{n p} \, dx\\ &=\left (\left (\frac{\cos (c+d x)}{b}\right )^{n p} \left ((b \sec (c+d x))^p\right )^n\right ) \int \left (\frac{\cos (c+d x)}{b}\right )^{-n p} \, dx\\ &=-\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(c+d x)\right ) \left ((b \sec (c+d x))^p\right )^n \sin (c+d x)}{d (1-n p) \sqrt{\sin ^2(c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.0937918, size = 69, normalized size = 0.85 \[ \frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left ((b \sec (c+d x))^p\right )^n \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n p}{2},\frac{1}{2} (n p+2),\sec ^2(c+d x)\right )}{d n p} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((b*Sec[c + d*x])^p)^n,x]

[Out]

(Cot[c + d*x]*Hypergeometric2F1[1/2, (n*p)/2, (2 + n*p)/2, Sec[c + d*x]^2]*((b*Sec[c + d*x])^p)^n*Sqrt[-Tan[c
+ d*x]^2])/(d*n*p)

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Maple [F]  time = 0.444, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( b\sec \left ( dx+c \right ) \right ) ^{p} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*sec(d*x+c))^p)^n,x)

[Out]

int(((b*sec(d*x+c))^p)^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*sec(d*x+c))^p)^n,x, algorithm="maxima")

[Out]

integrate(((b*sec(d*x + c))^p)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*sec(d*x+c))^p)^n,x, algorithm="fricas")

[Out]

integral(((b*sec(d*x + c))^p)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*sec(d*x+c))**p)**n,x)

[Out]

Integral(((b*sec(c + d*x))**p)**n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((b*sec(d*x+c))^p)^n,x, algorithm="giac")

[Out]

integrate(((b*sec(d*x + c))^p)^n, x)