\(\int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [839]

Optimal result

Integrand size = 29, antiderivative size = 178 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {x}{a^3}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d} \]

[Out]

x/a^3+3*sec(d*x+c)/a^3/d-13/3*sec(d*x+c)^3/a^3/d+21/5*sec(d*x+c)^5/a^3/d-15/7*sec(d*x+c)^7/a^3/d+4/9*sec(d*x+c
)^9/a^3/d-tan(d*x+c)/a^3/d+1/3*tan(d*x+c)^3/a^3/d-1/5*tan(d*x+c)^5/a^3/d+1/7*tan(d*x+c)^7/a^3/d-4/9*tan(d*x+c)
^9/a^3/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2954, 2952, 2686, 276, 2687, 30, 200, 3554, 8} \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}+\frac {x}{a^3} \]

[In]

Int[(Sin[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]

[Out]

x/a^3 + (3*Sec[c + d*x])/(a^3*d) - (13*Sec[c + d*x]^3)/(3*a^3*d) + (21*Sec[c + d*x]^5)/(5*a^3*d) - (15*Sec[c +
d*x]^7)/(7*a^3*d) + (4*Sec[c + d*x]^9)/(9*a^3*d) - Tan[c + d*x]/(a^3*d) + Tan[c + d*x]^3/(3*a^3*d) - Tan[c +
d*x]^5/(5*a^3*d) + Tan[c + d*x]^7/(7*a^3*d) - (4*Tan[c + d*x]^9)/(9*a^3*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 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 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 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 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 ^3(c+d x) (a-a \sin (c+d x))^3 \tan ^7(c+d x) \, dx}{a^6} \\ & = \frac {\int \left (a^3 \sec ^3(c+d x) \tan ^7(c+d x)-3 a^3 \sec ^2(c+d x) \tan ^8(c+d x)+3 a^3 \sec (c+d x) \tan ^9(c+d x)-a^3 \tan ^{10}(c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \sec ^3(c+d x) \tan ^7(c+d x) \, dx}{a^3}-\frac {\int \tan ^{10}(c+d x) \, dx}{a^3}-\frac {3 \int \sec ^2(c+d x) \tan ^8(c+d x) \, dx}{a^3}+\frac {3 \int \sec (c+d x) \tan ^9(c+d x) \, dx}{a^3} \\ & = -\frac {\tan ^9(c+d x)}{9 a^3 d}+\frac {\int \tan ^8(c+d x) \, dx}{a^3}+\frac {\text {Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int x^8 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac {3 \text {Subst}\left (\int \left (-1+x^2\right )^4 \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = \frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}-\frac {\int \tan ^6(c+d x) \, dx}{a^3}+\frac {\text {Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {3 \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = \frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\int \tan ^4(c+d x) \, dx}{a^3} \\ & = \frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}-\frac {\int \tan ^2(c+d x) \, dx}{a^3} \\ & = \frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {\int 1 \, dx}{a^3} \\ & = \frac {x}{a^3}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {13 \sec ^3(c+d x)}{3 a^3 d}+\frac {21 \sec ^5(c+d x)}{5 a^3 d}-\frac {15 \sec ^7(c+d x)}{7 a^3 d}+\frac {4 \sec ^9(c+d x)}{9 a^3 d}-\frac {\tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {\tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^7(c+d x)}{7 a^3 d}-\frac {4 \tan ^9(c+d x)}{9 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.53 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {169344-675036 \cos (c+d x)+362880 (c+d x) \cos (c+d x)+173952 \cos (2 (c+d x))-37502 \cos (3 (c+d x))+20160 (c+d x) \cos (3 (c+d x))+54912 \cos (4 (c+d x))+112506 \cos (5 (c+d x))-60480 (c+d x) \cos (5 (c+d x))-21376 \cos (6 (c+d x))+93312 \sin (c+d x)-506277 \sin (2 (c+d x))+272160 (c+d x) \sin (2 (c+d x))+125248 \sin (3 (c+d x))-225012 \sin (4 (c+d x))+120960 (c+d x) \sin (4 (c+d x))+67776 \sin (5 (c+d x))+18751 \sin (6 (c+d x))-10080 (c+d x) \sin (6 (c+d x))}{322560 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+a \sin (c+d x))^3} \]

[In]

Integrate[(Sin[c + d*x]^3*Tan[c + d*x]^4)/(a + a*Sin[c + d*x])^3,x]

[Out]

(169344 - 675036*Cos[c + d*x] + 362880*(c + d*x)*Cos[c + d*x] + 173952*Cos[2*(c + d*x)] - 37502*Cos[3*(c + d*x
)] + 20160*(c + d*x)*Cos[3*(c + d*x)] + 54912*Cos[4*(c + d*x)] + 112506*Cos[5*(c + d*x)] - 60480*(c + d*x)*Cos
[5*(c + d*x)] - 21376*Cos[6*(c + d*x)] + 93312*Sin[c + d*x] - 506277*Sin[2*(c + d*x)] + 272160*(c + d*x)*Sin[2
*(c + d*x)] + 125248*Sin[3*(c + d*x)] - 225012*Sin[4*(c + d*x)] + 120960*(c + d*x)*Sin[4*(c + d*x)] + 67776*Si
n[5*(c + d*x)] + 18751*Sin[6*(c + d*x)] - 10080*(c + d*x)*Sin[6*(c + d*x)])/(322560*d*(Cos[(c + d*x)/2] - Sin[
(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + a*Sin[c + d*x])^3)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.98 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {x}{a^{3}}+\frac {20 i {\mathrm e}^{10 i \left (d x +c \right )}+6 \,{\mathrm e}^{11 i \left (d x +c \right )}+40 i {\mathrm e}^{8 i \left (d x +c \right )}-\frac {50 \,{\mathrm e}^{9 i \left (d x +c \right )}}{3}-\frac {168 i {\mathrm e}^{6 i \left (d x +c \right )}}{5}-\frac {428 \,{\mathrm e}^{7 i \left (d x +c \right )}}{5}-\frac {2608 i {\mathrm e}^{4 i \left (d x +c \right )}}{35}-\frac {2348 \,{\mathrm e}^{5 i \left (d x +c \right )}}{35}-\frac {3244 i {\mathrm e}^{2 i \left (d x +c \right )}}{105}+\frac {2578 \,{\mathrm e}^{3 i \left (d x +c \right )}}{315}+\frac {1336 i}{315}+\frac {2042 \,{\mathrm e}^{i \left (d x +c \right )}}{105}}{\left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{3}}\)	\(172\)
derivativedivides	\(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {40}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {21}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\)	\(202\)
default	\(\frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {40}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {4}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {21}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {3}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {57}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\)	\(202\)
parallelrisch	\(\frac {315 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +\left (1890 d x +630\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3780 d x +3780\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (630 d x +7350\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8505 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +\left (-11340 d x -19404\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-22344 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (11340 d x +7092\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (8505 d x +19872\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-630 d x +5878\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3780 d x -5052\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1890 d x -3786\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-315 d x -736}{315 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}\)	\(244\)

[In]

int(sec(d*x+c)^4*sin(d*x+c)^7/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

x/a^3+2/315*(3150*I*exp(10*I*(d*x+c))+945*exp(11*I*(d*x+c))+6300*I*exp(8*I*(d*x+c))-2625*exp(9*I*(d*x+c))-5292
*I*exp(6*I*(d*x+c))-13482*exp(7*I*(d*x+c))-11736*I*exp(4*I*(d*x+c))-10566*exp(5*I*(d*x+c))-4866*I*exp(2*I*(d*x
+c))+1289*exp(3*I*(d*x+c))+668*I+3063*exp(I*(d*x+c)))/(exp(I*(d*x+c))+I)^9/(exp(I*(d*x+c))-I)^3/d/a^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {945 \, d x \cos \left (d x + c\right )^{5} + 668 \, \cos \left (d x + c\right )^{6} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1431 \, \cos \left (d x + c\right )^{4} + 465 \, \cos \left (d x + c\right )^{2} + {\left (315 \, d x \cos \left (d x + c\right )^{5} - 1260 \, d x \cos \left (d x + c\right )^{3} - 1059 \, \cos \left (d x + c\right )^{4} + 305 \, \cos \left (d x + c\right )^{2} - 35\right )} \sin \left (d x + c\right ) - 70}{315 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/315*(945*d*x*cos(d*x + c)^5 + 668*cos(d*x + c)^6 - 1260*d*x*cos(d*x + c)^3 - 1431*cos(d*x + c)^4 + 465*cos(d
*x + c)^2 + (315*d*x*cos(d*x + c)^5 - 1260*d*x*cos(d*x + c)^3 - 1059*cos(d*x + c)^4 + 305*cos(d*x + c)^2 - 35)
*sin(d*x + c) - 70)/(3*a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x + c)^3 + (a^3*d*cos(d*x + c)^5 - 4*a^3*d*cos(d*x
+ c)^3)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**4*sin(d*x+c)**7/(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (162) = 324\).

Time = 0.32 (sec) , antiderivative size = 487, normalized size of antiderivative = 2.74 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, {\left (\frac {\frac {1893 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2526 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2939 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {9936 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {3546 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {11172 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {9702 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3675 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {1890 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {315 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 368}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {315 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{315 \, d} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

2/315*((1893*sin(d*x + c)/(cos(d*x + c) + 1) + 2526*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2939*sin(d*x + c)^3/
(cos(d*x + c) + 1)^3 - 9936*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3546*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 1
1172*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 9702*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 3675*sin(d*x + c)^9/(cos
(d*x + c) + 1)^9 - 1890*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 315*sin(d*x + c)^11/(cos(d*x + c) + 1)^11 + 36
8)/(a^3 + 6*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 12*a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 2*a^3*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 - 27*a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 36*a^3*sin(d*x + c)^5/(cos(d*x + c)
+ 1)^5 + 36*a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 27*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 2*a^3*sin(d
*x + c)^9/(cos(d*x + c) + 1)^9 - 12*a^3*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 6*a^3*sin(d*x + c)^11/(cos(d*x
+ c) + 1)^11 - a^3*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) + 315*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/
d

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {10080 \, {\left (d x + c\right )}}{a^{3}} + \frac {105 \, {\left (21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 23\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {17955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 160020 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 624960 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1387260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1884582 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1556268 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 774792 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 215748 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25967}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/10080*(10080*(d*x + c)/a^3 + 105*(21*tan(1/2*d*x + 1/2*c)^2 - 48*tan(1/2*d*x + 1/2*c) + 23)/(a^3*(tan(1/2*d*
x + 1/2*c) - 1)^3) + (17955*tan(1/2*d*x + 1/2*c)^8 + 160020*tan(1/2*d*x + 1/2*c)^7 + 624960*tan(1/2*d*x + 1/2*
c)^6 + 1387260*tan(1/2*d*x + 1/2*c)^5 + 1884582*tan(1/2*d*x + 1/2*c)^4 + 1556268*tan(1/2*d*x + 1/2*c)^3 + 7747
92*tan(1/2*d*x + 1/2*c)^2 + 215748*tan(1/2*d*x + 1/2*c) + 25967)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^9))/d

Mupad [B] (verification not implemented)

Time = 20.07 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.95 \[ \int \frac {\sin ^3(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {x}{a^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}-\frac {308\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}-\frac {1064\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {788\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}+\frac {2208\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}+\frac {5878\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{315}-\frac {1684\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}-\frac {1262\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}-\frac {736}{315}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^9} \]

[In]

int(sin(c + d*x)^7/(cos(c + d*x)^4*(a + a*sin(c + d*x))^3),x)

[Out]

x/a^3 + ((5878*tan(c/2 + (d*x)/2)^3)/315 - (1684*tan(c/2 + (d*x)/2)^2)/105 - (1262*tan(c/2 + (d*x)/2))/105 + (
2208*tan(c/2 + (d*x)/2)^4)/35 + (788*tan(c/2 + (d*x)/2)^5)/35 - (1064*tan(c/2 + (d*x)/2)^6)/15 - (308*tan(c/2
+ (d*x)/2)^7)/5 + (70*tan(c/2 + (d*x)/2)^9)/3 + 12*tan(c/2 + (d*x)/2)^10 + 2*tan(c/2 + (d*x)/2)^11 - 736/315)/
(a^3*d*(tan(c/2 + (d*x)/2) - 1)^3*(tan(c/2 + (d*x)/2) + 1)^9)