3.146 \(\int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx\)

Optimal. Leaf size=123 \[ \frac {\left (n^2-13 n+32\right ) (a \sec (c+d x)+a)^{n+3} \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac {\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac {(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d} \]

[Out]

1/20*(12-n)*cos(d*x+c)^4*(a+a*sec(d*x+c))^(3+n)/a^3/d-1/5*cos(d*x+c)^5*(a+a*sec(d*x+c))^(3+n)/a^3/d+1/20*(n^2-
13*n+32)*hypergeom([4, 3+n],[4+n],1+sec(d*x+c))*(a+a*sec(d*x+c))^(3+n)/a^3/d/(3+n)

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Rubi [A]  time = 0.11, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3873, 89, 78, 65} \[ \frac {\left (n^2-13 n+32\right ) (a \sec (c+d x)+a)^{n+3} \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac {\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac {(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^5,x]

[Out]

((12 - n)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3 + n))/(20*a^3*d) - (Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3 +
n))/(5*a^3*d) + ((32 - 13*n + n^2)*Hypergeometric2F1[4, 3 + n, 4 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(
3 + n))/(20*a^3*d*(3 + n))

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 3873

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Dist[(f*b^(p - 1)
)^(-1), Subst[Int[((-a + b*x)^((p - 1)/2)*(a + b*x)^(m + (p - 1)/2))/x^(p + 1), x], x, Csc[e + f*x]], x] /; Fr
eeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(-a-a x)^2 (a-a x)^{2+n}}{x^6} \, dx,x,-\sec (c+d x)\right )}{a^4 d}\\ &=-\frac {\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^{2+n} \left (a^3 (12-n)+5 a^3 x\right )}{x^5} \, dx,x,-\sec (c+d x)\right )}{5 a^5 d}\\ &=\frac {(12-n) \cos ^4(c+d x) (a+a \sec (c+d x))^{3+n}}{20 a^3 d}-\frac {\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}-\frac {\left (32-13 n+n^2\right ) \operatorname {Subst}\left (\int \frac {(a-a x)^{2+n}}{x^4} \, dx,x,-\sec (c+d x)\right )}{20 a^2 d}\\ &=\frac {(12-n) \cos ^4(c+d x) (a+a \sec (c+d x))^{3+n}}{20 a^3 d}-\frac {\cos ^5(c+d x) (a+a \sec (c+d x))^{3+n}}{5 a^3 d}+\frac {\left (32-13 n+n^2\right ) \, _2F_1(4,3+n;4+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{3+n}}{20 a^3 d (3+n)}\\ \end {align*}

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Mathematica [A]  time = 0.52, size = 84, normalized size = 0.68 \[ -\frac {(\sec (c+d x)+1)^3 (a (\sec (c+d x)+1))^n \left ((n+3) \cos ^4(c+d x) (4 \cos (c+d x)+n-12)-\left (n^2-13 n+32\right ) \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)\right )}{20 d (n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^5,x]

[Out]

-1/20*(((3 + n)*Cos[c + d*x]^4*(-12 + n + 4*Cos[c + d*x]) - (32 - 13*n + n^2)*Hypergeometric2F1[4, 3 + n, 4 +
n, 1 + Sec[c + d*x]])*(1 + Sec[c + d*x])^3*(a*(1 + Sec[c + d*x]))^n)/(d*(3 + n))

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fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^n*sin(d*x+c)^5,x, algorithm="fricas")

[Out]

integral((cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*(a*sec(d*x + c) + a)^n*sin(d*x + c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^n*sin(d*x+c)^5,x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^n*sin(d*x + c)^5, x)

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maple [F]  time = 2.79, size = 0, normalized size = 0.00 \[ \int \left (a +a \sec \left (d x +c \right )\right )^{n} \left (\sin ^{5}\left (d x +c \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^n*sin(d*x+c)^5,x)

[Out]

int((a+a*sec(d*x+c))^n*sin(d*x+c)^5,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{5}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^n*sin(d*x+c)^5,x, algorithm="maxima")

[Out]

integrate((a*sec(d*x + c) + a)^n*sin(d*x + c)^5, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\sin \left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^5*(a + a/cos(c + d*x))^n,x)

[Out]

int(sin(c + d*x)^5*(a + a/cos(c + d*x))^n, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**n*sin(d*x+c)**5,x)

[Out]

Timed out

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