\(\int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx\) [66]

Optimal result

Integrand size = 21, antiderivative size = 99 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {x}{16 a}-\frac {\cos (c+d x) \sin (c+d x)}{16 a d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{8 a d}+\frac {\cos ^3(c+d x) \sin ^3(c+d x)}{6 a d}+\frac {\sin ^5(c+d x)}{5 a d} \]

[Out]

-1/16*x/a-1/16*cos(d*x+c)*sin(d*x+c)/a/d+1/8*cos(d*x+c)^3*sin(d*x+c)/a/d+1/6*cos(d*x+c)^3*sin(d*x+c)^3/a/d+1/5
*sin(d*x+c)^5/a/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2918, 2644, 30, 2648, 2715, 8} \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\sin ^5(c+d x)}{5 a d}+\frac {\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac {\sin (c+d x) \cos (c+d x)}{16 a d}-\frac {x}{16 a} \]

[In]

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

[Out]

-1/16*x/a - (Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin
[c + d*x]^3)/(6*a*d) + Sin[c + d*x]^5/(5*a*d)

Rule 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2648

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(b*Cos[e
+ f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Cos[e + f*x
])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]

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 2918

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

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.13 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (75 c-60 d x+120 \sin (c+d x)+15 \sin (2 (c+d x))-60 \sin (3 (c+d x))+15 \sin (4 (c+d x))+12 \sin (5 (c+d x))-5 \sin (6 (c+d x))-75 \tan \left (\frac {c}{2}\right )\right )}{480 a d (1+\sec (c+d x))} \]

[In]

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

[Out]

(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(75*c - 60*d*x + 120*Sin[c + d*x] + 15*Sin[2*(c + d*x)] - 60*Sin[3*(c + d*x)]
+ 15*Sin[4*(c + d*x)] + 12*Sin[5*(c + d*x)] - 5*Sin[6*(c + d*x)] - 75*Tan[c/2]))/(480*a*d*(1 + Sec[c + d*x]))

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.78

method	result	size
parallelrisch	\(\frac {-60 d x +120 \sin \left (d x +c \right )+12 \sin \left (5 d x +5 c \right )-60 \sin \left (3 d x +3 c \right )-5 \sin \left (6 d x +6 c \right )+15 \sin \left (4 d x +4 c \right )+15 \sin \left (2 d x +2 c \right )}{960 d a}\)	\(77\)
risch	\(-\frac {x}{16 a}+\frac {\sin \left (d x +c \right )}{8 a d}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}+\frac {\sin \left (5 d x +5 c \right )}{80 d a}+\frac {\sin \left (4 d x +4 c \right )}{64 d a}-\frac {\sin \left (3 d x +3 c \right )}{16 d a}+\frac {\sin \left (2 d x +2 c \right )}{64 d a}\)	\(107\)
derivativedivides	\(\frac {-\frac {64 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{512}+\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1536}-\frac {223 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1280}-\frac {33 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1280}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1536}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\)	\(116\)
default	\(\frac {-\frac {64 \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{512}+\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{1536}-\frac {223 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{1280}-\frac {33 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1280}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{1536}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\)	\(116\)
norman	\(\frac {-\frac {x}{16 a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 a d}+\frac {33 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 a d}+\frac {223 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 a d}-\frac {17 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 a d}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16 a}-\frac {5 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}-\frac {15 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16 a}-\frac {3 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a}-\frac {x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\)	\(238\)

[In]

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

[Out]

1/960*(-60*d*x+120*sin(d*x+c)+12*sin(5*d*x+5*c)-60*sin(3*d*x+3*c)-5*sin(6*d*x+6*c)+15*sin(4*d*x+4*c)+15*sin(2*
d*x+2*c))/d/a

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.71 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {15 \, d x + {\left (40 \, \cos \left (d x + c\right )^{5} - 48 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{3} + 96 \, \cos \left (d x + c\right )^{2} + 15 \, \cos \left (d x + c\right ) - 48\right )} \sin \left (d x + c\right )}{240 \, a d} \]

[In]

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

[Out]

-1/240*(15*d*x + (40*cos(d*x + c)^5 - 48*cos(d*x + c)^4 - 70*cos(d*x + c)^3 + 96*cos(d*x + c)^2 + 15*cos(d*x +
c) - 48)*sin(d*x + c))/(a*d)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (89) = 178\).

Time = 0.30 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.81 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {85 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {198 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1338 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {85 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {15 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a + \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \]

[In]

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

[Out]

1/120*((15*sin(d*x + c)/(cos(d*x + c) + 1) + 85*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 198*sin(d*x + c)^5/(cos(
d*x + c) + 1)^5 + 1338*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 85*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 15*sin(d
*x + c)^11/(cos(d*x + c) + 1)^11)/(a + 6*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a*sin(d*x + c)^4/(cos(d*x
+ c) + 1)^4 + 20*a*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a*sin(d*
x + c)^10/(cos(d*x + c) + 1)^10 + a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) - 15*arctan(sin(d*x + c)/(cos(d*x +
c) + 1))/a)/d

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {15 \, {\left (d x + c\right )}}{a} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 85 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1338 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 198 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 85 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a}}{240 \, d} \]

[In]

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

[Out]

-1/240*(15*(d*x + c)/a + 2*(15*tan(1/2*d*x + 1/2*c)^11 + 85*tan(1/2*d*x + 1/2*c)^9 - 1338*tan(1/2*d*x + 1/2*c)
^7 - 198*tan(1/2*d*x + 1/2*c)^5 - 85*tan(1/2*d*x + 1/2*c)^3 - 15*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^
2 + 1)^6*a))/d

Mupad [B] (verification not implemented)

Time = 16.46 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {223\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{a\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}-\frac {x}{16\,a} \]

[In]

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

[Out]

(tan(c/2 + (d*x)/2)/8 + (17*tan(c/2 + (d*x)/2)^3)/24 + (33*tan(c/2 + (d*x)/2)^5)/20 + (223*tan(c/2 + (d*x)/2)^
7)/20 - (17*tan(c/2 + (d*x)/2)^9)/24 - tan(c/2 + (d*x)/2)^11/8)/(a*d*(tan(c/2 + (d*x)/2)^2 + 1)^6) - x/(16*a)