3.77 \(\int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=55 \[ -\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d} \]

[Out]

-1/3*cos(d*x+c)^3/a^2/d+1/2*cos(d*x+c)^4/a^2/d-1/5*cos(d*x+c)^5/a^2/d

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2836, 12, 43} \[ -\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]

[Out]

-Cos[c + d*x]^3/(3*a^2*d) + Cos[c + d*x]^4/(2*a^2*d) - Cos[c + d*x]^5/(5*a^2*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2836

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

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^5(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^2 x^2}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int (-a-x)^2 x^2 \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2 x^2+2 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=-\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.60, size = 42, normalized size = 0.76 \[ \frac {4 \sin ^6\left (\frac {1}{2} (c+d x)\right ) (3 \cos (c+d x)+3 \cos (2 (c+d x))+4)}{15 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]

[Out]

(4*(4 + 3*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]^6)/(15*a^2*d)

________________________________________________________________________________________

fricas [A]  time = 1.46, size = 39, normalized size = 0.71 \[ -\frac {6 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3}}{30 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/30*(6*cos(d*x + c)^5 - 15*cos(d*x + c)^4 + 10*cos(d*x + c)^3)/(a^2*d)

________________________________________________________________________________________

giac [B]  time = 0.25, size = 119, normalized size = 2.16 \[ -\frac {8 \, {\left (\frac {10 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {20 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {15 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 2\right )}}{15 \, a^{2} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

-8/15*(10*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 20*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 15*(cos(d*x +
c) - 1)^3/(cos(d*x + c) + 1)^3 - 15*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 2)/(a^2*d*((cos(d*x + c) - 1)
/(cos(d*x + c) + 1) - 1)^5)

________________________________________________________________________________________

maple [A]  time = 0.61, size = 39, normalized size = 0.71 \[ \frac {-\frac {1}{3 \sec \left (d x +c \right )^{3}}+\frac {1}{2 \sec \left (d x +c \right )^{4}}-\frac {1}{5 \sec \left (d x +c \right )^{5}}}{d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^5/(a+a*sec(d*x+c))^2,x)

[Out]

1/d/a^2*(-1/3/sec(d*x+c)^3+1/2/sec(d*x+c)^4-1/5/sec(d*x+c)^5)

________________________________________________________________________________________

maxima [A]  time = 0.32, size = 39, normalized size = 0.71 \[ -\frac {6 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3}}{30 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^5/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/30*(6*cos(d*x + c)^5 - 15*cos(d*x + c)^4 + 10*cos(d*x + c)^3)/(a^2*d)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 36, normalized size = 0.65 \[ -\frac {{\cos \left (c+d\,x\right )}^3\,\left (6\,{\cos \left (c+d\,x\right )}^2-15\,\cos \left (c+d\,x\right )+10\right )}{30\,a^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^5/(a + a/cos(c + d*x))^2,x)

[Out]

-(cos(c + d*x)^3*(6*cos(c + d*x)^2 - 15*cos(c + d*x) + 10))/(30*a^2*d)

________________________________________________________________________________________

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(sin(d*x+c)**5/(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

________________________________________________________________________________________