\(\int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [778]

Optimal result

Integrand size = 29, antiderivative size = 149 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {9 x}{2 a^2}-\frac {2 \cos (c+d x)}{a^2 d}-\frac {6 \sec (c+d x)}{a^2 d}+\frac {2 \sec ^3(c+d x)}{a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {9 \tan (c+d x)}{2 a^2 d}-\frac {3 \tan ^3(c+d x)}{2 a^2 d}+\frac {9 \tan ^5(c+d x)}{10 a^2 d}-\frac {\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d} \]

[Out]

-9/2*x/a^2-2*cos(d*x+c)/a^2/d-6*sec(d*x+c)/a^2/d+2*sec(d*x+c)^3/a^2/d-2/5*sec(d*x+c)^5/a^2/d+9/2*tan(d*x+c)/a^
2/d-3/2*tan(d*x+c)^3/a^2/d+9/10*tan(d*x+c)^5/a^2/d-1/2*sin(d*x+c)^2*tan(d*x+c)^5/a^2/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2954, 2789, 3554, 8, 2670, 276, 2671, 294, 308, 209} \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {2 \cos (c+d x)}{a^2 d}+\frac {9 \tan ^5(c+d x)}{10 a^2 d}-\frac {3 \tan ^3(c+d x)}{2 a^2 d}+\frac {9 \tan (c+d x)}{2 a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^3(c+d x)}{a^2 d}-\frac {6 \sec (c+d x)}{a^2 d}-\frac {\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac {9 x}{2 a^2} \]

[In]

Int[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]

[Out]

(-9*x)/(2*a^2) - (2*Cos[c + d*x])/(a^2*d) - (6*Sec[c + d*x])/(a^2*d) + (2*Sec[c + d*x]^3)/(a^2*d) - (2*Sec[c +
d*x]^5)/(5*a^2*d) + (9*Tan[c + d*x])/(2*a^2*d) - (3*Tan[c + d*x]^3)/(2*a^2*d) + (9*Tan[c + d*x]^5)/(10*a^2*d)
- (Sin[c + d*x]^2*Tan[c + d*x]^5)/(2*a^2*d)

Rule 8

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2670

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 2671

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[b*(ff/f), Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff
)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rule 2789

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Int[Expan
dIntegrand[(g*Tan[e + f*x])^p, (a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2
, 0] && IGtQ[m, 0]

Rule 2954

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

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a-a \sin (c+d x))^2 \tan ^6(c+d x) \, dx}{a^4} \\ & = \frac {\int \left (a^2 \tan ^6(c+d x)-2 a^2 \sin (c+d x) \tan ^6(c+d x)+a^2 \sin ^2(c+d x) \tan ^6(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \tan ^6(c+d x) \, dx}{a^2}+\frac {\int \sin ^2(c+d x) \tan ^6(c+d x) \, dx}{a^2}-\frac {2 \int \sin (c+d x) \tan ^6(c+d x) \, dx}{a^2} \\ & = \frac {\tan ^5(c+d x)}{5 a^2 d}-\frac {\int \tan ^4(c+d x) \, dx}{a^2}+\frac {\text {Subst}\left (\int \frac {x^8}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}+\frac {2 \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cos (c+d x)\right )}{a^2 d} \\ & = -\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan ^5(c+d x)}{5 a^2 d}-\frac {\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}+\frac {\int \tan ^2(c+d x) \, dx}{a^2}+\frac {2 \text {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac {7 \text {Subst}\left (\int \frac {x^6}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d} \\ & = -\frac {2 \cos (c+d x)}{a^2 d}-\frac {6 \sec (c+d x)}{a^2 d}+\frac {2 \sec ^3(c+d x)}{a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {\tan (c+d x)}{a^2 d}-\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan ^5(c+d x)}{5 a^2 d}-\frac {\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac {\int 1 \, dx}{a^2}+\frac {7 \text {Subst}\left (\int \left (1-x^2+x^4-\frac {1}{1+x^2}\right ) \, dx,x,\tan (c+d x)\right )}{2 a^2 d} \\ & = -\frac {x}{a^2}-\frac {2 \cos (c+d x)}{a^2 d}-\frac {6 \sec (c+d x)}{a^2 d}+\frac {2 \sec ^3(c+d x)}{a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {9 \tan (c+d x)}{2 a^2 d}-\frac {3 \tan ^3(c+d x)}{2 a^2 d}+\frac {9 \tan ^5(c+d x)}{10 a^2 d}-\frac {\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d}-\frac {7 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 a^2 d} \\ & = -\frac {9 x}{2 a^2}-\frac {2 \cos (c+d x)}{a^2 d}-\frac {6 \sec (c+d x)}{a^2 d}+\frac {2 \sec ^3(c+d x)}{a^2 d}-\frac {2 \sec ^5(c+d x)}{5 a^2 d}+\frac {9 \tan (c+d x)}{2 a^2 d}-\frac {3 \tan ^3(c+d x)}{2 a^2 d}+\frac {9 \tan ^5(c+d x)}{10 a^2 d}-\frac {\sin ^2(c+d x) \tan ^5(c+d x)}{2 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.28 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {500+10 (-103+90 c+90 d x) \cos (c+d x)+544 \cos (2 (c+d x))+206 \cos (3 (c+d x))-180 c \cos (3 (c+d x))-180 d x \cos (3 (c+d x))-20 \cos (4 (c+d x))+250 \sin (c+d x)-824 \sin (2 (c+d x))+720 c \sin (2 (c+d x))+720 d x \sin (2 (c+d x))+351 \sin (3 (c+d x))+5 \sin (5 (c+d x))}{160 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \]

[In]

Integrate[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]

[Out]

-1/160*(500 + 10*(-103 + 90*c + 90*d*x)*Cos[c + d*x] + 544*Cos[2*(c + d*x)] + 206*Cos[3*(c + d*x)] - 180*c*Cos
[3*(c + d*x)] - 180*d*x*Cos[3*(c + d*x)] - 20*Cos[4*(c + d*x)] + 250*Sin[c + d*x] - 824*Sin[2*(c + d*x)] + 720
*c*Sin[2*(c + d*x)] + 720*d*x*Sin[2*(c + d*x)] + 351*Sin[3*(c + d*x)] + 5*Sin[5*(c + d*x)])/(a^2*d*(Cos[(c + d
*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.03

method	result	size
parallelrisch	\(\frac {\left (-720 d x +664\right ) \sin \left (2 d x +2 c \right )-900 d x \cos \left (d x +c \right )+180 d x \cos \left (3 d x +3 c \right )-351 \sin \left (3 d x +3 c \right )-5 \sin \left (5 d x +5 c \right )+830 \cos \left (d x +c \right )-544 \cos \left (2 d x +2 c \right )-166 \cos \left (3 d x +3 c \right )+20 \cos \left (4 d x +4 c \right )-250 \sin \left (d x +c \right )-500}{40 d \,a^{2} \left (5 \cos \left (d x +c \right )-\cos \left (3 d x +3 c \right )+4 \sin \left (2 d x +2 c \right )\right )}\)	\(154\)
derivativedivides	\(\frac {-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {1}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {31}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{2}}\)	\(168\)
default	\(\frac {-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8 \left (\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}+\frac {1}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-9 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {7}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {31}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{2}}\)	\(168\)
risch	\(-\frac {9 x}{2 a^{2}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{2}}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{d \,a^{2}}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{2}}-\frac {2 \left (-40 \,{\mathrm e}^{3 i \left (d x +c \right )}+75 i {\mathrm e}^{4 i \left (d x +c \right )}+30 \,{\mathrm e}^{5 i \left (d x +c \right )}-78 \,{\mathrm e}^{i \left (d x +c \right )}+60 i {\mathrm e}^{2 i \left (d x +c \right )}-27 i\right )}{5 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} d \,a^{2}}\)	\(174\)
norman	\(\frac {\frac {90 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {72 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {81 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {117 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}+\frac {189 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {117 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {90 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {189 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {72 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {64}{5 a d}-\frac {81 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {18 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {9 x}{2 a}-\frac {78 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {396 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}-\frac {9 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {9 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {211 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5 d a}-\frac {436 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {18 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-\frac {819 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {634 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}-\frac {696 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {461 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {136 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {684 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}-\frac {132 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {36 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\)	\(525\)

[In]

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

[Out]

1/40*((-720*d*x+664)*sin(2*d*x+2*c)-900*d*x*cos(d*x+c)+180*d*x*cos(3*d*x+3*c)-351*sin(3*d*x+3*c)-5*sin(5*d*x+5
*c)+830*cos(d*x+c)-544*cos(2*d*x+2*c)-166*cos(3*d*x+3*c)+20*cos(4*d*x+4*c)-250*sin(d*x+c)-500)/d/a^2/(5*cos(d*
x+c)-cos(3*d*x+3*c)+4*sin(2*d*x+2*c))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {45 \, d x \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{4} - 90 \, d x \cos \left (d x + c\right ) - 78 \, \cos \left (d x + c\right )^{2} - {\left (5 \, \cos \left (d x + c\right )^{4} + 90 \, d x \cos \left (d x + c\right ) + 84 \, \cos \left (d x + c\right )^{2} - 6\right )} \sin \left (d x + c\right ) + 4}{10 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )\right )}} \]

[In]

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

[Out]

-1/10*(45*d*x*cos(d*x + c)^3 + 10*cos(d*x + c)^4 - 90*d*x*cos(d*x + c) - 78*cos(d*x + c)^2 - (5*cos(d*x + c)^4
+ 90*d*x*cos(d*x + c) + 84*cos(d*x + c)^2 - 6)*sin(d*x + c) + 4)/(a^2*d*cos(d*x + c)^3 - 2*a^2*d*cos(d*x + c)
*sin(d*x + c) - 2*a^2*d*cos(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (137) = 274\).

Time = 0.29 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.83 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {211 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {268 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {212 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {84 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {174 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {300 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {300 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {180 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {45 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 64}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {7 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{5 \, d} \]

[In]

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

[Out]

-1/5*((211*sin(d*x + c)/(cos(d*x + c) + 1) + 268*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 212*sin(d*x + c)^3/(cos
(d*x + c) + 1)^3 + 84*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 174*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 300*sin(
d*x + c)^6/(cos(d*x + c) + 1)^6 - 300*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 180*sin(d*x + c)^8/(cos(d*x + c) +
1)^8 - 45*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 64)/(a^2 + 4*a^2*sin(d*x + c)/(cos(d*x + c) + 1) + 7*a^2*sin(
d*x + c)^2/(cos(d*x + c) + 1)^2 + 8*a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 6*a^2*sin(d*x + c)^4/(cos(d*x +
c) + 1)^4 - 6*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 8*a^2*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 7*a^2*sin(
d*x + c)^8/(cos(d*x + c) + 1)^8 - 4*a^2*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - a^2*sin(d*x + c)^10/(cos(d*x + c
) + 1)^10) + 45*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {90 \, {\left (d x + c\right )}}{a^{2}} + \frac {20 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{2}} + \frac {5}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} + \frac {155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 690 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 750 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 181}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{20 \, d} \]

[In]

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

[Out]

-1/20*(90*(d*x + c)/a^2 + 20*(tan(1/2*d*x + 1/2*c)^3 + 4*tan(1/2*d*x + 1/2*c)^2 - tan(1/2*d*x + 1/2*c) + 4)/((
tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^2) + 5/(a^2*(tan(1/2*d*x + 1/2*c) - 1)) + (155*tan(1/2*d*x + 1/2*c)^4 + 690*ta
n(1/2*d*x + 1/2*c)^3 + 1120*tan(1/2*d*x + 1/2*c)^2 + 750*tan(1/2*d*x + 1/2*c) + 181)/(a^2*(tan(1/2*d*x + 1/2*c
) + 1)^5))/d

Mupad [B] (verification not implemented)

Time = 19.85 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.15 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {-9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-60\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-60\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {174\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {212\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {268\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {211\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {64}{5}}{a^2\,d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {9\,x}{2\,a^2} \]

[In]

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

[Out]

((211*tan(c/2 + (d*x)/2))/5 + (268*tan(c/2 + (d*x)/2)^2)/5 + (212*tan(c/2 + (d*x)/2)^3)/5 + (84*tan(c/2 + (d*x
)/2)^4)/5 - (174*tan(c/2 + (d*x)/2)^5)/5 - 60*tan(c/2 + (d*x)/2)^6 - 60*tan(c/2 + (d*x)/2)^7 - 36*tan(c/2 + (d
*x)/2)^8 - 9*tan(c/2 + (d*x)/2)^9 + 64/5)/(a^2*d*(tan(c/2 + (d*x)/2) - 1)*(tan(c/2 + (d*x)/2) + 1)^5*(tan(c/2
+ (d*x)/2)^2 + 1)^2) - (9*x)/(2*a^2)