\(\int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx\) [30]

Optimal result

Integrand size = 21, antiderivative size = 157 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {25 a^2 x}{16}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {7 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d} \]

[Out]

-25/16*a^2*x+2*a^2*arctanh(sin(d*x+c))/d-2*a^2*sin(d*x+c)/d+7/16*a^2*cos(d*x+c)*sin(d*x+c)/d+7/24*a^2*cos(d*x+
c)^3*sin(d*x+c)/d-1/6*a^2*cos(d*x+c)^5*sin(d*x+c)/d-2/3*a^2*sin(d*x+c)^3/d-2/5*a^2*sin(d*x+c)^5/d+a^2*tan(d*x+
c)/d

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2951, 2717, 2713, 2715, 8, 3855, 3852} \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \tan (c+d x)}{d}-\frac {a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}-\frac {25 a^2 x}{16} \]

[In]

Int[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^6,x]

[Out]

(-25*a^2*x)/16 + (2*a^2*ArcTanh[Sin[c + d*x]])/d - (2*a^2*Sin[c + d*x])/d + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/
(16*d) + (7*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a^2*Sin[c +
d*x]^3)/(3*d) - (2*a^2*Sin[c + d*x]^5)/(5*d) + (a^2*Tan[c + d*x])/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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2951

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

Rule 3852

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

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 (-a-a \cos (c+d x))^2 \sin ^4(c+d x) \tan ^2(c+d x) \, dx \\ & = \frac {\int \left (-2 a^8-6 a^8 \cos (c+d x)+6 a^8 \cos ^3(c+d x)+2 a^8 \cos ^4(c+d x)-2 a^8 \cos ^5(c+d x)-a^8 \cos ^6(c+d x)+2 a^8 \sec (c+d x)+a^8 \sec ^2(c+d x)\right ) \, dx}{a^6} \\ & = -2 a^2 x-a^2 \int \cos ^6(c+d x) \, dx+a^2 \int \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \, dx-\left (2 a^2\right ) \int \cos ^5(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \, dx-\left (6 a^2\right ) \int \cos (c+d x) \, dx+\left (6 a^2\right ) \int \cos ^3(c+d x) \, dx \\ & = -2 a^2 x+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {6 a^2 \sin (c+d x)}{d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {1}{6} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{2} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac {a^2 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (6 a^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = -2 a^2 x+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d}-\frac {1}{8} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{4} \left (3 a^2\right ) \int 1 \, dx \\ & = -\frac {5 a^2 x}{4}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {7 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d}-\frac {1}{16} \left (5 a^2\right ) \int 1 \, dx \\ & = -\frac {25 a^2 x}{16}+\frac {2 a^2 \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a^2 \sin (c+d x)}{d}+\frac {7 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {7 a^2 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \tan (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.79 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (1080 c+1080 d x+420 \arctan (\tan (c+d x))-1920 \text {arctanh}(\sin (c+d x))+1920 \sin (c+d x)+640 \sin ^3(c+d x)+384 \sin ^5(c+d x)-255 \sin (2 (c+d x))-15 \sin (4 (c+d x))+5 \sin (6 (c+d x))-960 \tan (c+d x)\right )}{3840 d} \]

[In]

Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^6,x]

[Out]

-1/3840*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(1080*c + 1080*d*x + 420*ArcTan[Tan[c + d*x]] - 1920*ArcT
anh[Sin[c + d*x]] + 1920*Sin[c + d*x] + 640*Sin[c + d*x]^3 + 384*Sin[c + d*x]^5 - 255*Sin[2*(c + d*x)] - 15*Si
n[4*(c + d*x)] + 5*Sin[6*(c + d*x)] - 960*Tan[c + d*x]))/d

Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(\frac {a^{2} \left (-3000 d x \cos \left (d x +c \right )-3840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (d x +c \right )+3840 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (d x +c \right )+2175 \sin \left (d x +c \right )-2360 \sin \left (2 d x +2 c \right )-24 \sin \left (6 d x +6 c \right )+256 \sin \left (4 d x +4 c \right )-5 \sin \left (7 d x +7 c \right )+10 \sin \left (5 d x +5 c \right )+270 \sin \left (3 d x +3 c \right )\right )}{1920 d \cos \left (d x +c \right )}\)	\(142\)
derivativedivides	\(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(164\)
default	\(\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\)	\(164\)
parts	\(\frac {a^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}+\frac {2 a^{2} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}\)	\(169\)
risch	\(-\frac {25 a^{2} x}{16}+\frac {17 i a^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {17 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}-\frac {11 i a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {11 i a^{2} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {2 i a^{2}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{192 d}-\frac {a^{2} \sin \left (5 d x +5 c \right )}{40 d}+\frac {a^{2} \sin \left (4 d x +4 c \right )}{64 d}+\frac {7 a^{2} \sin \left (3 d x +3 c \right )}{24 d}\)	\(212\)
norman	\(\frac {\frac {25 a^{2} x}{16}+\frac {7 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {27 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 d}+\frac {797 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 d}-\frac {91 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d}-\frac {8041 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}-\frac {431 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 d}+\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{16}+\frac {225 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16}-\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}-\frac {225 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{16}-\frac {125 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}-\frac {25 a^{2} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{16}-\frac {57 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\)	\(330\)

[In]

int((a+a*sec(d*x+c))^2*sin(d*x+c)^6,x,method=_RETURNVERBOSE)

[Out]

1/1920*a^2*(-3000*d*x*cos(d*x+c)-3840*ln(tan(1/2*d*x+1/2*c)-1)*cos(d*x+c)+3840*ln(tan(1/2*d*x+1/2*c)+1)*cos(d*
x+c)+2175*sin(d*x+c)-2360*sin(2*d*x+2*c)-24*sin(6*d*x+6*c)+256*sin(4*d*x+4*c)-5*sin(7*d*x+7*c)+10*sin(5*d*x+5*
c)+270*sin(3*d*x+3*c))/d/cos(d*x+c)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {375 \, a^{2} d x \cos \left (d x + c\right ) - 240 \, a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 240 \, a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (40 \, a^{2} \cos \left (d x + c\right )^{6} + 96 \, a^{2} \cos \left (d x + c\right )^{5} - 70 \, a^{2} \cos \left (d x + c\right )^{4} - 352 \, a^{2} \cos \left (d x + c\right )^{3} - 105 \, a^{2} \cos \left (d x + c\right )^{2} + 736 \, a^{2} \cos \left (d x + c\right ) - 240 \, a^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )} \]

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^6,x, algorithm="fricas")

[Out]

-1/240*(375*a^2*d*x*cos(d*x + c) - 240*a^2*cos(d*x + c)*log(sin(d*x + c) + 1) + 240*a^2*cos(d*x + c)*log(-sin(
d*x + c) + 1) + (40*a^2*cos(d*x + c)^6 + 96*a^2*cos(d*x + c)^5 - 70*a^2*cos(d*x + c)^4 - 352*a^2*cos(d*x + c)^
3 - 105*a^2*cos(d*x + c)^2 + 736*a^2*cos(d*x + c) - 240*a^2)*sin(d*x + c))/(d*cos(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))**2*sin(d*x+c)**6,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.11 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {64 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a^{2} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 120 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{2}}{960 \, d} \]

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^6,x, algorithm="maxima")

[Out]

-1/960*(64*(6*sin(d*x + c)^5 + 10*sin(d*x + c)^3 - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1) + 30*si
n(d*x + c))*a^2 - 5*(4*sin(2*d*x + 2*c)^3 + 60*d*x + 60*c + 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^2 + 12
0*(15*d*x + 15*c - (9*tan(d*x + c)^3 + 7*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1) - 8*tan(d*x + c
))*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.23 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=-\frac {375 \, {\left (d x + c\right )} a^{2} - 480 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 480 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {480 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {2 \, {\left (615 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3485 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7926 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8586 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2595 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 345 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]

[In]

integrate((a+a*sec(d*x+c))^2*sin(d*x+c)^6,x, algorithm="giac")

[Out]

-1/240*(375*(d*x + c)*a^2 - 480*a^2*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 480*a^2*log(abs(tan(1/2*d*x + 1/2*c)
- 1)) + 480*a^2*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) + 2*(615*a^2*tan(1/2*d*x + 1/2*c)^11 + 3485*
a^2*tan(1/2*d*x + 1/2*c)^9 + 7926*a^2*tan(1/2*d*x + 1/2*c)^7 + 8586*a^2*tan(1/2*d*x + 1/2*c)^5 + 2595*a^2*tan(
1/2*d*x + 1/2*c)^3 + 345*a^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d

Mupad [B] (verification not implemented)

Time = 14.71 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.50 \[ \int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx=\frac {\frac {57\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}+\frac {431\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{12}+\frac {8041\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{120}+\frac {91\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-\frac {797\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}-\frac {27\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}-\frac {7\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {25\,a^2\,x}{16}+\frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]

[In]

int(sin(c + d*x)^6*(a + a/cos(c + d*x))^2,x)

[Out]

((91*a^2*tan(c/2 + (d*x)/2)^7)/2 - (797*a^2*tan(c/2 + (d*x)/2)^5)/40 - (27*a^2*tan(c/2 + (d*x)/2)^3)/4 + (8041
*a^2*tan(c/2 + (d*x)/2)^9)/120 + (431*a^2*tan(c/2 + (d*x)/2)^11)/12 + (57*a^2*tan(c/2 + (d*x)/2)^13)/8 - (7*a^
2*tan(c/2 + (d*x)/2))/8)/(d*(5*tan(c/2 + (d*x)/2)^2 + 9*tan(c/2 + (d*x)/2)^4 + 5*tan(c/2 + (d*x)/2)^6 - 5*tan(
c/2 + (d*x)/2)^8 - 9*tan(c/2 + (d*x)/2)^10 - 5*tan(c/2 + (d*x)/2)^12 - tan(c/2 + (d*x)/2)^14 + 1)) - (25*a^2*x
)/16 + (4*a^2*atanh(tan(c/2 + (d*x)/2)))/d