3.5.41 \(\int (d \cos (e+f x))^n (a+a \sec (e+f x))^3 \, dx\) [441]

Optimal. Leaf size=244 \[ -\frac {a^3 (7-4 n) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {2+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f (2-n) n \sqrt {\sin ^2(e+f x)}}-\frac {a^3 (1-4 n) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin (e+f x)}{f (1-n) (1+n) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (5-2 n) (d \cos (e+f x))^n \tan (e+f x)}{f (1-n) (2-n)}+\frac {(d \cos (e+f x))^n \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{f (2-n)} \]

[Out]

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

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Rubi [A]
time = 0.28, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4349, 3899, 4082, 3872, 3857, 2722} \begin {gather*} -\frac {a^3 (7-4 n) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n}{2};\frac {n+2}{2};\cos ^2(e+f x)\right )}{f (2-n) n \sqrt {\sin ^2(e+f x)}}-\frac {a^3 (1-4 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left (\frac {1}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{f (1-n) (n+1) \sqrt {\sin ^2(e+f x)}}+\frac {a^3 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac {\tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) (d \cos (e+f x))^n}{f (2-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rule 4349

Int[(u_)*((c_.)*sin[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Csc[a + b*x])^m*(c*Sin[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Csc[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[
u, x]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.65, size = 308, normalized size = 1.26 \begin {gather*} \frac {i 2^{-3-n} a^3 \left (e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right )\right )^n \cos ^{3-n}(e+f x) (d \cos (e+f x))^n \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 {12 e^{2 i (e+f x)} \, _2F_1\left (1,\frac {n}{2};2-\frac {n}{2};-e^{2 i (e+f x)}\right )}{\left (1+e^{2 i (e+f x)}\right ) (-2+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,\frac {2+n}{2};1-\frac {n}{2};-e^{2 i (e+f x)}\right )}{n}\right ) \sec ^6\left (\frac {1}{2} (e+f x)\right ) (1+\sec (e+f x))^3}{f} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (d \cos \left (f x +e \right )\right )^{n} \left (a +a \sec \left (f x +e \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((a*sec(f*x + e) + a)^3*(d*cos(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((d*cos(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)*(d*cos(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 \left (d \cos {\left (e + f x \right )}\right )^{n}\, dx + \int 3 \left (d \cos {\left (e + f x \right )}\right )^{n} \sec {\left (e + f x \right )}\, dx + \int 3 \left (d \cos {\left (e + f x \right )}\right )^{n} \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (d \cos {\left (e + f x \right )}\right )^{n} \sec ^{3}{\left (e + f x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*(Integral((d*cos(e + f*x))**n, x) + Integral(3*(d*cos(e + f*x))**n*sec(e + f*x), x) + Integral(3*(d*cos(e
+ f*x))**n*sec(e + f*x)**2, x) + Integral((d*cos(e + f*x))**n*sec(e + f*x)**3, 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((d*cos(f*x+e))^n*(a+a*sec(f*x+e))^3,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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