3.93 \(\int \frac{\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=73 \[ \frac{\cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{2 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d} \]

[Out]

-Cos[c + d*x]^4/(4*a^3*d) + (3*Cos[c + d*x]^5)/(5*a^3*d) - Cos[c + d*x]^6/(2*a^3*d) + Cos[c + d*x]^7/(7*a^3*d)

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Rubi [A]  time = 0.164819, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2836, 12, 43} \[ \frac{\cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{2 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]

[Out]

-Cos[c + d*x]^4/(4*a^3*d) + (3*Cos[c + d*x]^5)/(5*a^3*d) - Cos[c + d*x]^6/(2*a^3*d) + Cos[c + d*x]^7/(7*a^3*d)

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]

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 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])

Rubi steps

\begin{align*} \int \frac{\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac{\cos ^3(c+d x) \sin ^7(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-a-x)^3 x^3}{a^3} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int (-a-x)^3 x^3 \, dx,x,-a \cos (c+d x)\right )}{a^{10} d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^3 x^3-3 a^2 x^4-3 a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{10} d}\\ &=-\frac{\cos ^4(c+d x)}{4 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^6(c+d x)}{2 a^3 d}+\frac{\cos ^7(c+d x)}{7 a^3 d}\\ \end{align*}

Mathematica [A]  time = 1.66604, size = 80, normalized size = 1.1 \[ \frac{4060 \cos (c+d x)-3220 \cos (2 (c+d x))+2100 \cos (3 (c+d x))-1120 \cos (4 (c+d x))+476 \cos (5 (c+d x))-140 \cos (6 (c+d x))+20 \cos (7 (c+d x))-2421}{8960 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]

[Out]

(-2421 + 4060*Cos[c + d*x] - 3220*Cos[2*(c + d*x)] + 2100*Cos[3*(c + d*x)] - 1120*Cos[4*(c + d*x)] + 476*Cos[5
*(c + d*x)] - 140*Cos[6*(c + d*x)] + 20*Cos[7*(c + d*x)])/(8960*a^3*d)

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Maple [A]  time = 0.087, size = 50, normalized size = 0.7 \begin{align*} -{\frac{1}{d{a}^{3}} \left ({\frac{1}{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}}+{\frac{1}{2\, \left ( \sec \left ( dx+c \right ) \right ) ^{6}}}-{\frac{3}{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{5}}}-{\frac{1}{7\, \left ( \sec \left ( dx+c \right ) \right ) ^{7}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^7/(a+a*sec(d*x+c))^3,x)

[Out]

-1/d/a^3*(1/4/sec(d*x+c)^4+1/2/sec(d*x+c)^6-3/5/sec(d*x+c)^5-1/7/sec(d*x+c)^7)

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Maxima [A]  time = 0.985246, size = 66, normalized size = 0.9 \begin{align*} \frac{20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/140*(20*cos(d*x + c)^7 - 70*cos(d*x + c)^6 + 84*cos(d*x + c)^5 - 35*cos(d*x + c)^4)/(a^3*d)

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Fricas [A]  time = 1.7563, size = 128, normalized size = 1.75 \begin{align*} \frac{20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/140*(20*cos(d*x + c)^7 - 70*cos(d*x + c)^6 + 84*cos(d*x + c)^5 - 35*cos(d*x + c)^4)/(a^3*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**7/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.38915, size = 220, normalized size = 3.01 \begin{align*} \frac{4 \,{\left (\frac{91 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{273 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{455 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{490 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{210 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{140 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 13\right )}}{35 \, a^{3} d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^7/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

4/35*(91*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 273*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 455*(cos(d*x
+ c) - 1)^3/(cos(d*x + c) + 1)^3 - 490*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 210*(cos(d*x + c) - 1)^5/(c
os(d*x + c) + 1)^5 - 140*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 13)/(a^3*d*((cos(d*x + c) - 1)/(cos(d*x +
c) + 1) - 1)^7)