\(\int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [101]

Optimal result

Integrand size = 21, antiderivative size = 129 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {23 x}{16 a^3}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {23 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d} \]

[Out]

-23/16*x/a^3+4*sin(d*x+c)/a^3/d-23/16*cos(d*x+c)*sin(d*x+c)/a^3/d-23/24*cos(d*x+c)^3*sin(d*x+c)/a^3/d-1/6*cos(
d*x+c)^5*sin(d*x+c)/a^3/d-7/3*sin(d*x+c)^3/a^3/d+3/5*sin(d*x+c)^5/a^3/d

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2948, 2836, 2713, 2715, 8} \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}-\frac {23 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac {23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {23 x}{16 a^3} \]

[In]

Int[Sin[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]

[Out]

(-23*x)/(16*a^3) + (4*Sin[c + d*x])/(a^3*d) - (23*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) - (23*Cos[c + d*x]^3*S
in[c + d*x])/(24*a^3*d) - (Cos[c + d*x]^5*Sin[c + d*x])/(6*a^3*d) - (7*Sin[c + d*x]^3)/(3*a^3*d) + (3*Sin[c +
d*x]^5)/(5*a^3*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2836

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rule 2948

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Dist[a^(2*m), Int[(d*Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f,
n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 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 ^6(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = -\frac {\int \cos ^3(c+d x) (-a+a \cos (c+d x))^3 \, dx}{a^6} \\ & = -\frac {\int \left (-a^3 \cos ^3(c+d x)+3 a^3 \cos ^4(c+d x)-3 a^3 \cos ^5(c+d x)+a^3 \cos ^6(c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \cos ^3(c+d x) \, dx}{a^3}-\frac {\int \cos ^6(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^4(c+d x) \, dx}{a^3}+\frac {3 \int \cos ^5(c+d x) \, dx}{a^3} \\ & = -\frac {3 \cos ^3(c+d x) \sin (c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {5 \int \cos ^4(c+d x) \, dx}{6 a^3}-\frac {9 \int \cos ^2(c+d x) \, dx}{4 a^3}-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{a^3 d} \\ & = \frac {4 \sin (c+d x)}{a^3 d}-\frac {9 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {23 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {5 \int \cos ^2(c+d x) \, dx}{8 a^3}-\frac {9 \int 1 \, dx}{8 a^3} \\ & = -\frac {9 x}{8 a^3}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {23 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {5 \int 1 \, dx}{16 a^3} \\ & = -\frac {23 x}{16 a^3}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {23 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.73 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (-2760 d x+5040 \sin (c+d x)-1890 \sin (2 (c+d x))+760 \sin (3 (c+d x))-270 \sin (4 (c+d x))+72 \sin (5 (c+d x))-10 \sin (6 (c+d x))+9 \tan \left (\frac {c}{2}\right )\right )}{240 a^3 d (1+\sec (c+d x))^3} \]

[In]

Integrate[Sin[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]

[Out]

(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(-2760*d*x + 5040*Sin[c + d*x] - 1890*Sin[2*(c + d*x)] + 760*Sin[3*(c + d*x
)] - 270*Sin[4*(c + d*x)] + 72*Sin[5*(c + d*x)] - 10*Sin[6*(c + d*x)] + 9*Tan[c/2]))/(240*a^3*d*(1 + Sec[c + d
*x])^3)

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.60

method	result	size
parallelrisch	\(\frac {-1380 d x +2520 \sin \left (d x +c \right )+36 \sin \left (5 d x +5 c \right )+380 \sin \left (3 d x +3 c \right )-5 \sin \left (6 d x +6 c \right )-135 \sin \left (4 d x +4 c \right )-945 \sin \left (2 d x +2 c \right )}{960 a^{3} d}\)	\(77\)
risch	\(-\frac {23 x}{16 a^{3}}+\frac {21 \sin \left (d x +c \right )}{8 a^{3} d}-\frac {\sin \left (6 d x +6 c \right )}{192 a^{3} d}+\frac {3 \sin \left (5 d x +5 c \right )}{80 a^{3} d}-\frac {9 \sin \left (4 d x +4 c \right )}{64 a^{3} d}+\frac {19 \sin \left (3 d x +3 c \right )}{48 a^{3} d}-\frac {63 \sin \left (2 d x +2 c \right )}{64 a^{3} d}\)	\(107\)
derivativedivides	\(\frac {-\frac {16 \left (-\frac {105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{128}-\frac {211 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{128}-\frac {969 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{320}-\frac {759 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{320}-\frac {391 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{384}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{3} d}\)	\(116\)
default	\(\frac {-\frac {16 \left (-\frac {105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{128}-\frac {211 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{128}-\frac {969 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{320}-\frac {759 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{320}-\frac {391 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{384}-\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{128}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{3} d}\)	\(116\)
norman	\(\frac {-\frac {23 x}{16 a}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {391 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}+\frac {759 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {969 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 a d}+\frac {211 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{8 a d}+\frac {105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 a d}-\frac {69 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {345 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 a}-\frac {115 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}-\frac {345 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a}-\frac {69 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a}-\frac {23 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} a^{2}}\)	\(241\)

[In]

int(sin(d*x+c)^6/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/960*(-1380*d*x+2520*sin(d*x+c)+36*sin(5*d*x+5*c)+380*sin(3*d*x+3*c)-5*sin(6*d*x+6*c)-135*sin(4*d*x+4*c)-945*
sin(2*d*x+2*c))/a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.54 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {345 \, d x + {\left (40 \, \cos \left (d x + c\right )^{5} - 144 \, \cos \left (d x + c\right )^{4} + 230 \, \cos \left (d x + c\right )^{3} - 272 \, \cos \left (d x + c\right )^{2} + 345 \, \cos \left (d x + c\right ) - 544\right )} \sin \left (d x + c\right )}{240 \, a^{3} d} \]

[In]

integrate(sin(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/240*(345*d*x + (40*cos(d*x + c)^5 - 144*cos(d*x + c)^4 + 230*cos(d*x + c)^3 - 272*cos(d*x + c)^2 + 345*cos(
d*x + c) - 544)*sin(d*x + c))/(a^3*d)

Sympy [F]

\[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\sin ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

[In]

integrate(sin(d*x+c)**6/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(sin(c + d*x)**6/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (117) = 234\).

Time = 0.28 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.26 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {\frac {345 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1955 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {4554 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5814 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3165 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {1575 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {345 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \]

[In]

integrate(sin(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/120*((345*sin(d*x + c)/(cos(d*x + c) + 1) + 1955*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 4554*sin(d*x + c)^5/(
cos(d*x + c) + 1)^5 + 5814*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 3165*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 15
75*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/(a^3 + 6*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^3*sin(d*x +
c)^4/(cos(d*x + c) + 1)^4 + 20*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a^3*sin(d*x + c)^8/(cos(d*x + c) +
1)^8 + 6*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 345*arctan(
sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.88 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {345 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, {\left (1575 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5814 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4554 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \]

[In]

integrate(sin(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/240*(345*(d*x + c)/a^3 - 2*(1575*tan(1/2*d*x + 1/2*c)^11 + 3165*tan(1/2*d*x + 1/2*c)^9 + 5814*tan(1/2*d*x +
1/2*c)^7 + 4554*tan(1/2*d*x + 1/2*c)^5 + 1955*tan(1/2*d*x + 1/2*c)^3 + 345*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*
x + 1/2*c)^2 + 1)^6*a^3))/d

Mupad [B] (verification not implemented)

Time = 16.40 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {211\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}+\frac {969\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {759\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {23\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}-\frac {23\,x}{16\,a^3} \]

[In]

int(sin(c + d*x)^6/(a + a/cos(c + d*x))^3,x)

[Out]

((23*tan(c/2 + (d*x)/2))/8 + (391*tan(c/2 + (d*x)/2)^3)/24 + (759*tan(c/2 + (d*x)/2)^5)/20 + (969*tan(c/2 + (d
*x)/2)^7)/20 + (211*tan(c/2 + (d*x)/2)^9)/8 + (105*tan(c/2 + (d*x)/2)^11)/8)/(a^3*d*(tan(c/2 + (d*x)/2)^2 + 1)
^6) - (23*x)/(16*a^3)