3.238 \(\int (c (d \sec (e+f x))^p)^n (a+b \sec (e+f x))^2 \, dx\)

Optimal. Leaf size=211 \[ -\frac{\left (a^2 (n p+1)+b^2 n p\right ) \sin (e+f x) \cos (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (1-n p),\frac{1}{2} (3-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f \left (1-n^2 p^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{n p}{2},\frac{1}{2} (2-n p),\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) \left (c (d \sec (e+f x))^p\right )^n}{f (n p+1)} \]

[Out]

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

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Rubi [A]  time = 0.242389, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {3948, 3788, 3772, 2643, 4046} \[ -\frac{\left (a^2 (n p+1)+b^2 n p\right ) \sin (e+f x) \cos (e+f x) \, _2F_1\left (\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f \left (1-n^2 p^2\right ) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) \, _2F_1\left (\frac{1}{2},-\frac{n p}{2};\frac{1}{2} (2-n p);\cos ^2(e+f x)\right ) \left (c (d \sec (e+f x))^p\right )^n}{f n p \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) \left (c (d \sec (e+f x))^p\right )^n}{f (n p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^2,x]

[Out]

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

Rule 3948

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

Rule 3788

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Dist[(2*a*b)/
d, Int[(d*Csc[e + f*x])^(n + 1), x], x] + Int[(d*Csc[e + f*x])^n*(a^2 + b^2*Csc[e + f*x]^2), x] /; FreeQ[{a, b
, d, e, f, n}, x]

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]

Rule 4046

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> -Simp[(C*Cot[
e + f*x]*(b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[(C*m + A*(m + 1))/(m + 1), Int[(b*Csc[e + f*x])^m, x], x]
/; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.492597, size = 200, normalized size = 0.95 \[ \frac{\sqrt{-\tan ^2(e+f x)} \csc (e+f x) \sec (e+f x) \left (c (d \sec (e+f x))^p\right )^n \left (a^2 \left (n^2 p^2+3 n p+2\right ) \cos ^2(e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n p}{2},\frac{n p}{2}+1,\sec ^2(e+f x)\right )+b n p \left (2 a (n p+2) \cos (e+f x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{1}{2} (n p+1),\frac{1}{2} (n p+3),\sec ^2(e+f x)\right )+b (n p+1) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{n p}{2}+1,\frac{n p}{2}+2,\sec ^2(e+f x)\right )\right )\right )}{f n p (n p+1) (n p+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*(d*Sec[e + f*x])^p)^n*(a + b*Sec[e + f*x])^2,x]

[Out]

(Csc[e + f*x]*(a^2*(2 + 3*n*p + n^2*p^2)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (n*p)/2, 1 + (n*p)/2, Sec[e + f
*x]^2] + b*n*p*(b*(1 + n*p)*Hypergeometric2F1[1/2, 1 + (n*p)/2, 2 + (n*p)/2, Sec[e + f*x]^2] + 2*a*(2 + n*p)*C
os[e + f*x]*Hypergeometric2F1[1/2, (1 + n*p)/2, (3 + n*p)/2, Sec[e + f*x]^2]))*Sec[e + f*x]*(c*(d*Sec[e + f*x]
)^p)^n*Sqrt[-Tan[e + f*x]^2])/(f*n*p*(1 + n*p)*(2 + n*p))

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Maple [F]  time = 0.15, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\sec \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+b\sec \left ( fx+e \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*(d*sec(f*x+e))^p)^n*(a+b*sec(f*x+e))^2,x)

[Out]

int((c*(d*sec(f*x+e))^p)^n*(a+b*sec(f*x+e))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sec \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sec(f*x+e))^p)^n*(a+b*sec(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^2*((d*sec(f*x + e))^p*c)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \sec \left (f x + e\right )^{2} + 2 \, a b \sec \left (f x + e\right ) + a^{2}\right )} \left (\left (d \sec \left (f x + e\right )\right )^{p} c\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sec(f*x+e))^p)^n*(a+b*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

integral((b^2*sec(f*x + e)^2 + 2*a*b*sec(f*x + e) + a^2)*((d*sec(f*x + e))^p*c)^n, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \left (d \sec{\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \sec{\left (e + f x \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sec(f*x+e))**p)**n*(a+b*sec(f*x+e))**2,x)

[Out]

Integral((c*(d*sec(e + f*x))**p)**n*(a + b*sec(e + f*x))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (f x + e\right ) + a\right )}^{2} \left (\left (d \sec \left (f x + e\right )\right )^{p} c\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*(d*sec(f*x+e))^p)^n*(a+b*sec(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^2*((d*sec(f*x + e))^p*c)^n, x)