3.3.89 \(\int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx\) [289]

Optimal. Leaf size=230 \[ \frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}-\frac {a^3 (1+4 n) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{f \left (1-n^2\right ) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (7+4 n) \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(e+f x)\right ) \sec ^n(e+f x) \sin (e+f x)}{f n (2+n) \sqrt {\sin ^2(e+f x)}} \]

[Out]

a^3*(5+2*n)*sec(f*x+e)^(1+n)*sin(f*x+e)/f/(1+n)/(2+n)+sec(f*x+e)^(1+n)*(a^3+a^3*sec(f*x+e))*sin(f*x+e)/f/(2+n)
-a^3*(1+4*n)*hypergeom([1/2, 1/2-1/2*n],[3/2-1/2*n],cos(f*x+e)^2)*sec(f*x+e)^(-1+n)*sin(f*x+e)/f/(-n^2+1)/(sin
(f*x+e)^2)^(1/2)+a^3*(7+4*n)*hypergeom([1/2, -1/2*n],[1-1/2*n],cos(f*x+e)^2)*sec(f*x+e)^n*sin(f*x+e)/f/n/(2+n)
/(sin(f*x+e)^2)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3899, 4082, 3872, 3857, 2722} \begin {gather*} -\frac {a^3 (4 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right )}{f \left (1-n^2\right ) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (4 n+7) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(e+f x)\right )}{f n (n+2) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (2 n+5) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2)}+\frac {\sin (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) \sec ^{n+1}(e+f x)}{f (n+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^3,x]

[Out]

(a^3*(5 + 2*n)*Sec[e + f*x]^(1 + n)*Sin[e + f*x])/(f*(1 + n)*(2 + n)) + (Sec[e + f*x]^(1 + n)*(a^3 + a^3*Sec[e
+ f*x])*Sin[e + f*x])/(f*(2 + n)) - (a^3*(1 + 4*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[e + f*x]^
2]*Sec[e + f*x]^(-1 + n)*Sin[e + f*x])/(f*(1 - n^2)*Sqrt[Sin[e + f*x]^2]) + (a^3*(7 + 4*n)*Hypergeometric2F1[1
/2, -1/2*n, (2 - n)/2, Cos[e + f*x]^2]*Sec[e + f*x]^n*Sin[e + f*x])/(f*n*(2 + n)*Sqrt[Sin[e + f*x]^2])

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rule 3857

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 3872

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

Rule 3899

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-b^2)
*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m - 2)*((d*Csc[e + f*x])^n/(f*(m + n - 1))), x] + Dist[b/(m + n - 1), Int[
(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^n*(b*(m + 2*n - 1) + a*(3*m + 2*n - 4)*Csc[e + f*x]), x], x] /;
FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + n - 1, 0] && IntegerQ[2*m]

Rule 4082

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.
) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Dist[1/(n + 1), Int[(d
*Csc[e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e,
f, A, B}, x] && NeQ[A*b - a*B, 0] &&  !LeQ[n, -1]

Rubi steps

\begin {align*} \int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx &=\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {a \int \sec ^n(e+f x) (a+a \sec (e+f x)) (a (2+2 n)+a (5+2 n) \sec (e+f x)) \, dx}{2+n}\\ &=\frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {a \int \sec ^n(e+f x) \left (a^2 (2+n) (1+4 n)+a^2 (1+n) (7+4 n) \sec (e+f x)\right ) \, dx}{2+3 n+n^2}\\ &=\frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {\left (a^3 (1+4 n)\right ) \int \sec ^n(e+f x) \, dx}{1+n}+\frac {\left (a^3 (7+4 n)\right ) \int \sec ^{1+n}(e+f x) \, dx}{2+n}\\ &=\frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}+\frac {\left (a^3 (1+4 n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-n}(e+f x) \, dx}{1+n}+\frac {\left (a^3 (7+4 n) \cos ^n(e+f x) \sec ^n(e+f x)\right ) \int \cos ^{-1-n}(e+f x) \, dx}{2+n}\\ &=\frac {a^3 (5+2 n) \sec ^{1+n}(e+f x) \sin (e+f x)}{f (1+n) (2+n)}+\frac {\sec ^{1+n}(e+f x) \left (a^3+a^3 \sec (e+f x)\right ) \sin (e+f x)}{f (2+n)}-\frac {a^3 (1+4 n) \, _2F_1\left (\frac {1}{2},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(e+f x)\right ) \sec ^{-1+n}(e+f x) \sin (e+f x)}{f \left (1-n^2\right ) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (7+4 n) \, _2F_1\left (\frac {1}{2},-\frac {n}{2};\frac {2-n}{2};\cos ^2(e+f x)\right ) \sec ^n(e+f x) \sin (e+f x)}{f n (2+n) \sqrt {\sin ^2(e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.52, size = 286, normalized size = 1.24 \begin {gather*} -\frac {i 2^{-3+n} a^3 \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^n (1+\cos (e+f x))^3 \left (\frac {8 e^{3 i (e+f x)} \, _2F_1\left (1,\frac {1}{2} (-1-n);\frac {5+n}{2};-e^{2 i (e+f x)}\right )}{\left (1+e^{2 i (e+f x)}\right )^2 (3+n)}+\frac {6 e^{i (e+f x)} \, _2F_1\left (1,\frac {1-n}{2};\frac {3+n}{2};-e^{2 i (e+f x)}\right )}{1+n}+\frac {\left (1+e^{2 i (e+f x)}\right ) \, _2F_1\left (1,1-\frac {n}{2};\frac {2+n}{2};-e^{2 i (e+f x)}\right )}{n}+\frac {12 e^{2 i (e+f x)} \, _2F_1\left (1,-\frac {n}{2};\frac {4+n}{2};-e^{2 i (e+f x)}\right )}{\left (1+e^{2 i (e+f x)}\right ) (2+n)}\right ) \sec ^6\left (\frac {1}{2} (e+f x)\right )}{f} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^3,x]

[Out]

((-I)*2^(-3 + n)*a^3*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^n*(1 + Cos[e + f*x])^3*((8*E^((3*I)*(e + f*x)
)*Hypergeometric2F1[1, (-1 - n)/2, (5 + n)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))^2*(3 + n)) + (
6*E^(I*(e + f*x))*Hypergeometric2F1[1, (1 - n)/2, (3 + n)/2, -E^((2*I)*(e + f*x))])/(1 + n) + ((1 + E^((2*I)*(
e + f*x)))*Hypergeometric2F1[1, 1 - n/2, (2 + n)/2, -E^((2*I)*(e + f*x))])/n + (12*E^((2*I)*(e + f*x))*Hyperge
ometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))*(2 + n)))*Sec[(e + f*x)/2]^
6)/f

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (\sec ^{n}\left (f x +e \right )\right ) \left (a +a \sec \left (f x +e \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^n*(a+a*sec(f*x+e))^3,x)

[Out]

int(sec(f*x+e)^n*(a+a*sec(f*x+e))^3,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^n*(a+a*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

integrate((a*sec(f*x + e) + a)^3*sec(f*x + e)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^n*(a+a*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

integral((a^3*sec(f*x + e)^3 + 3*a^3*sec(f*x + e)^2 + 3*a^3*sec(f*x + e) + a^3)*sec(f*x + e)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int 3 \sec {\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int 3 \sec ^{2}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int \sec ^{3}{\left (e + f x \right )} \sec ^{n}{\left (e + f x \right )}\, dx + \int \sec ^{n}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**n*(a+a*sec(f*x+e))**3,x)

[Out]

a**3*(Integral(3*sec(e + f*x)*sec(e + f*x)**n, x) + Integral(3*sec(e + f*x)**2*sec(e + f*x)**n, x) + Integral(
sec(e + f*x)**3*sec(e + f*x)**n, x) + Integral(sec(e + f*x)**n, x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^n*(a+a*sec(f*x+e))^3,x, algorithm="giac")

[Out]

integrate((a*sec(f*x + e) + a)^3*sec(f*x + e)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^3\,{\left (\frac {1}{\cos \left (e+f\,x\right )}\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(e + f*x))^3*(1/cos(e + f*x))^n,x)

[Out]

int((a + a/cos(e + f*x))^3*(1/cos(e + f*x))^n, x)

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