\(\int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [779]

Optimal result

Integrand size = 29, antiderivative size = 120 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2 x}{a^2}+\frac {\cos (c+d x)}{a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2954, 2952, 2686, 200, 3554, 8, 2670, 276} \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\cos (c+d x)}{a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}+\frac {2 x}{a^2} \]

[In]

Int[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]

[Out]

(2*x)/a^2 + Cos[c + d*x]/(a^2*d) + (4*Sec[c + d*x])/(a^2*d) - (5*Sec[c + d*x]^3)/(3*a^2*d) + (2*Sec[c + d*x]^5
)/(5*a^2*d) - (2*Tan[c + d*x])/(a^2*d) + (2*Tan[c + d*x]^3)/(3*a^2*d) - (2*Tan[c + d*x]^5)/(5*a^2*d)

Rule 8

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 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 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 2686

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

Rule 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^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] && 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 \sec (c+d x) (a-a \sin (c+d x))^2 \tan ^5(c+d x) \, dx}{a^4} \\ & = \frac {\int \left (a^2 \sec (c+d x) \tan ^5(c+d x)-2 a^2 \tan ^6(c+d x)+a^2 \sin (c+d x) \tan ^6(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \sec (c+d x) \tan ^5(c+d x) \, dx}{a^2}+\frac {\int \sin (c+d x) \tan ^6(c+d x) \, dx}{a^2}-\frac {2 \int \tan ^6(c+d x) \, dx}{a^2} \\ & = -\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \int \tan ^4(c+d x) \, dx}{a^2}-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,\cos (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}-\frac {2 \int \tan ^2(c+d x) \, dx}{a^2}-\frac {\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 {\text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d} \\ & = \frac {\cos (c+d x)}{a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \int 1 \, dx}{a^2} \\ & = \frac {2 x}{a^2}+\frac {\cos (c+d x)}{a^2 d}+\frac {4 \sec (c+d x)}{a^2 d}-\frac {5 \sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \tan (c+d x)}{a^2 d}+\frac {2 \tan ^3(c+d x)}{3 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.23 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\sec (c+d x) (550+(-995+600 c+600 d x) \cos (c+d x)+376 \cos (2 (c+d x))+199 \cos (3 (c+d x))-120 c \cos (3 (c+d x))-120 d x \cos (3 (c+d x))-30 \cos (4 (c+d x))+400 \sin (c+d x)-796 \sin (2 (c+d x))+480 c \sin (2 (c+d x))+480 d x \sin (2 (c+d x))+304 \sin (3 (c+d x)))}{240 a^2 d (1+\sin (c+d x))^2} \]

[In]

Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^2,x]

[Out]

(Sec[c + d*x]*(550 + (-995 + 600*c + 600*d*x)*Cos[c + d*x] + 376*Cos[2*(c + d*x)] + 199*Cos[3*(c + d*x)] - 120
*c*Cos[3*(c + d*x)] - 120*d*x*Cos[3*(c + d*x)] - 30*Cos[4*(c + d*x)] + 400*Sin[c + d*x] - 796*Sin[2*(c + d*x)]
+ 480*c*Sin[2*(c + d*x)] + 480*d*x*Sin[2*(c + d*x)] + 304*Sin[3*(c + d*x)]))/(240*a^2*d*(1 + Sin[c + d*x])^2)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {64}{32+32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {17}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{2}}\)	\(129\)
default	\(\frac {-\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {64}{32+32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {1}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {17}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{2}}\)	\(129\)
risch	\(\frac {2 x}{a^{2}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{2}}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{2}}+\frac {20 i {\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{5 i \left (d x +c \right )}+\frac {32 i {\mathrm e}^{2 i \left (d x +c \right )}}{3}-\frac {40 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}-\frac {92 i}{15}-\frac {248 \,{\mathrm e}^{i \left (d x +c \right )}}{15}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5} d \,a^{2}}\)	\(138\)
parallelrisch	\(\frac {600 d x \cos \left (d x +c \right )+60 d x \cos \left (5 d x +5 c \right )+300 d x \cos \left (3 d x +3 c \right )-92 \sin \left (5 d x +5 c \right )-100 \sin \left (3 d x +3 c \right )-200 \sin \left (d x +c \right )+1120 \cos \left (d x +c \right )+112 \cos \left (5 d x +5 c \right )+330 \cos \left (4 d x +4 c \right )+785 \cos \left (2 d x +2 c \right )+15 \cos \left (6 d x +6 c \right )+560 \cos \left (3 d x +3 c \right )+662}{30 d \,a^{2} \left (\cos \left (5 d x +5 c \right )+5 \cos \left (3 d x +3 c \right )+10 \cos \left (d x +c \right )\right )}\)	\(171\)
norman	\(\frac {-\frac {2 x}{a}+\frac {128 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {8 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-\frac {16 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {24 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {26 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {16 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {16 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {26 x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {24 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {16 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {8 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {2 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {112}{15 a d}+\frac {4 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {16 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {88 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {92 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}+\frac {632 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {656 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {388 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 d a}+\frac {128 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {884 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {272 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}}{a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\)	\(453\)

[In]

int(sec(d*x+c)^2*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

64/d/a^2*(-1/256/(tan(1/2*d*x+1/2*c)-1)+1/32/(1+tan(1/2*d*x+1/2*c)^2)+1/16*arctan(tan(1/2*d*x+1/2*c))+1/80/(ta
n(1/2*d*x+1/2*c)+1)^5-1/32/(tan(1/2*d*x+1/2*c)+1)^4-1/192/(tan(1/2*d*x+1/2*c)+1)^3+5/128/(tan(1/2*d*x+1/2*c)+1
)^2+17/256/(tan(1/2*d*x+1/2*c)+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {30 \, d x \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{4} - 60 \, d x \cos \left (d x + c\right ) - 62 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (30 \, d x \cos \left (d x + c\right ) + 38 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) - 9}{15 \, {\left (a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )\right )}} \]

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/15*(30*d*x*cos(d*x + c)^3 + 15*cos(d*x + c)^4 - 60*d*x*cos(d*x + c) - 62*cos(d*x + c)^2 - 2*(30*d*x*cos(d*x
+ c) + 38*cos(d*x + c)^2 + 3)*sin(d*x + c) - 9)/(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. \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**2*sin(d*x+c)**5/(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. 335 vs. \(2 (112) = 224\).

Time = 0.30 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.79 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {4 \, {\left (\frac {\frac {97 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {108 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {27 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {40 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {85 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {60 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 28}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} + \frac {15 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{15 \, d} \]

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

4/15*((97*sin(d*x + c)/(cos(d*x + c) + 1) + 108*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 27*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 - 40*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 85*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 60*sin(d*x
+ c)^6/(cos(d*x + c) + 1)^6 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 28)/(a^2 + 4*a^2*sin(d*x + c)/(cos(d*x
+ c) + 1) + 6*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 4*a^2*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 4*a^2*sin(
d*x + c)^5/(cos(d*x + c) + 1)^5 - 6*a^2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 4*a^2*sin(d*x + c)^7/(cos(d*x +
c) + 1)^7 - a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8) + 15*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.26 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {120 \, {\left (d x + c\right )}}{a^{2}} - \frac {15 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )} a^{2}} + \frac {255 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1170 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1310 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 313}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}}}{60 \, d} \]

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)^5/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/60*(120*(d*x + c)/a^2 - 15*(tan(1/2*d*x + 1/2*c)^2 - 8*tan(1/2*d*x + 1/2*c) + 9)/((tan(1/2*d*x + 1/2*c)^3 -
tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c) - 1)*a^2) + (255*tan(1/2*d*x + 1/2*c)^4 + 1170*tan(1/2*d*x + 1/2
*c)^3 + 1960*tan(1/2*d*x + 1/2*c)^2 + 1310*tan(1/2*d*x + 1/2*c) + 313)/(a^2*(tan(1/2*d*x + 1/2*c) + 1)^5))/d

Mupad [B] (verification not implemented)

Time = 17.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.30 \[ \int \frac {\sin ^3(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {2\,x}{a^2}-\frac {-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {68\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{5}+\frac {144\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {388\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}+\frac {112}{15}}{a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^5\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]

[In]

int(sin(c + d*x)^5/(cos(c + d*x)^2*(a + a*sin(c + d*x))^2),x)

[Out]

(2*x)/a^2 - ((388*tan(c/2 + (d*x)/2))/15 + (144*tan(c/2 + (d*x)/2)^2)/5 + (36*tan(c/2 + (d*x)/2)^3)/5 - (32*ta
n(c/2 + (d*x)/2)^4)/3 - (68*tan(c/2 + (d*x)/2)^5)/3 - 16*tan(c/2 + (d*x)/2)^6 - 4*tan(c/2 + (d*x)/2)^7 + 112/1
5)/(a^2*d*(tan(c/2 + (d*x)/2) + 1)^5*(tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2 + tan(c/2 + (d*x)/2)^3 - 1))