\(\int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [93]

Optimal result

Integrand size = 21, antiderivative size = 73 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^4(c+d x)}{4 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^6(c+d x)}{2 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 45} \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\cos ^7(c+d x)}{7 a^3 d}-\frac {\cos ^6(c+d x)}{2 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^4(c+d x)}{4 a^3 d} \]

[In]

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

[Out]

-1/4*Cos[c + d*x]^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 12

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

Rule 45

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 2915

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/b)*x
)^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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^7(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^3 x^3}{a^3} \, dx,x,-a \cos (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int (-a-x)^3 x^3 \, dx,x,-a \cos (c+d x)\right )}{a^{10} d} \\ & = \frac {\text {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] (verified)

Time = 0.85 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {-2421+4060 \cos (c+d x)-3220 \cos (2 (c+d x))+2100 \cos (3 (c+d x))-1120 \cos (4 (c+d x))+476 \cos (5 (c+d x))-140 \cos (6 (c+d x))+20 \cos (7 (c+d x))}{8960 a^3 d} \]

[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)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\)	\(49\)
default	\(\frac {\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\)	\(49\)
parallelrisch	\(\frac {-805 \cos \left (2 d x +2 c \right )+2784-280 \cos \left (4 d x +4 c \right )-35 \cos \left (6 d x +6 c \right )+1015 \cos \left (d x +c \right )+525 \cos \left (3 d x +3 c \right )+119 \cos \left (5 d x +5 c \right )+5 \cos \left (7 d x +7 c \right )}{2240 a^{3} d}\)	\(85\)
risch	\(\frac {29 \cos \left (d x +c \right )}{64 a^{3} d}+\frac {\cos \left (7 d x +7 c \right )}{448 d \,a^{3}}-\frac {\cos \left (6 d x +6 c \right )}{64 d \,a^{3}}+\frac {17 \cos \left (5 d x +5 c \right )}{320 d \,a^{3}}-\frac {\cos \left (4 d x +4 c \right )}{8 d \,a^{3}}+\frac {15 \cos \left (3 d x +3 c \right )}{64 d \,a^{3}}-\frac {23 \cos \left (2 d x +2 c \right )}{64 d \,a^{3}}\)	\(118\)
norman	\(\frac {\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{a d}+\frac {52 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d a}+\frac {52}{35 a d}+\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d a}+\frac {24 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}+\frac {52 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d a}+\frac {156 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7} a^{2}}\)	\(143\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.67 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {20 \, \cos \left (d x + c\right )^{7} - 70 \, \cos \left (d x + c\right )^{6} + 84 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{4}}{140 \, a^{3} d} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (65) = 130\).

Time = 0.39 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.23 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {4 \, {\left (\frac {91 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {273 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {455 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {490 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {210 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {140 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 13\right )}}{35 \, a^{3} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \]

[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)

Mupad [B] (verification not implemented)

Time = 13.42 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6}{2\,a^3}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^3}}{d} \]

[In]

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

[Out]

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