\(\int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [849]

Optimal result

Integrand size = 27, antiderivative size = 145 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\sec ^5(c+d x)}{5 a^4 d}+\frac {9 \sec ^7(c+d x)}{7 a^4 d}-\frac {16 \sec ^9(c+d x)}{9 a^4 d}+\frac {8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac {4 \tan ^5(c+d x)}{5 a^4 d}-\frac {16 \tan ^7(c+d x)}{7 a^4 d}-\frac {20 \tan ^9(c+d x)}{9 a^4 d}-\frac {8 \tan ^{11}(c+d x)}{11 a^4 d} \]

[Out]

-1/5*sec(d*x+c)^5/a^4/d+9/7*sec(d*x+c)^7/a^4/d-16/9*sec(d*x+c)^9/a^4/d+8/11*sec(d*x+c)^11/a^4/d-4/5*tan(d*x+c)
^5/a^4/d-16/7*tan(d*x+c)^7/a^4/d-20/9*tan(d*x+c)^9/a^4/d-8/11*tan(d*x+c)^11/a^4/d

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2954, 2952, 2686, 14, 2687, 276} \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {8 \tan ^{11}(c+d x)}{11 a^4 d}-\frac {20 \tan ^9(c+d x)}{9 a^4 d}-\frac {16 \tan ^7(c+d x)}{7 a^4 d}-\frac {4 \tan ^5(c+d x)}{5 a^4 d}+\frac {8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac {16 \sec ^9(c+d x)}{9 a^4 d}+\frac {9 \sec ^7(c+d x)}{7 a^4 d}-\frac {\sec ^5(c+d x)}{5 a^4 d} \]

[In]

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

[Out]

-1/5*Sec[c + d*x]^5/(a^4*d) + (9*Sec[c + d*x]^7)/(7*a^4*d) - (16*Sec[c + d*x]^9)/(9*a^4*d) + (8*Sec[c + d*x]^1
1)/(11*a^4*d) - (4*Tan[c + d*x]^5)/(5*a^4*d) - (16*Tan[c + d*x]^7)/(7*a^4*d) - (20*Tan[c + d*x]^9)/(9*a^4*d) -
(8*Tan[c + d*x]^11)/(11*a^4*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^9(c+d x) (a-a \sin (c+d x))^4 \tan ^3(c+d x) \, dx}{a^8} \\ & = \frac {\int \left (a^4 \sec ^9(c+d x) \tan ^3(c+d x)-4 a^4 \sec ^8(c+d x) \tan ^4(c+d x)+6 a^4 \sec ^7(c+d x) \tan ^5(c+d x)-4 a^4 \sec ^6(c+d x) \tan ^6(c+d x)+a^4 \sec ^5(c+d x) \tan ^7(c+d x)\right ) \, dx}{a^8} \\ & = \frac {\int \sec ^9(c+d x) \tan ^3(c+d x) \, dx}{a^4}+\frac {\int \sec ^5(c+d x) \tan ^7(c+d x) \, dx}{a^4}-\frac {4 \int \sec ^8(c+d x) \tan ^4(c+d x) \, dx}{a^4}-\frac {4 \int \sec ^6(c+d x) \tan ^6(c+d x) \, dx}{a^4}+\frac {6 \int \sec ^7(c+d x) \tan ^5(c+d x) \, dx}{a^4} \\ & = \frac {\text {Subst}\left (\int x^8 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}+\frac {\text {Subst}\left (\int x^4 \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int x^6 \left (1+x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int x^4 \left (1+x^2\right )^3 \, dx,x,\tan (c+d x)\right )}{a^4 d}+\frac {6 \text {Subst}\left (\int x^6 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^4 d} \\ & = \frac {\text {Subst}\left (\int \left (-x^4+3 x^6-3 x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}+\frac {\text {Subst}\left (\int \left (-x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int \left (x^6+2 x^8+x^{10}\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int \left (x^4+3 x^6+3 x^8+x^{10}\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}+\frac {6 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d} \\ & = -\frac {\sec ^5(c+d x)}{5 a^4 d}+\frac {9 \sec ^7(c+d x)}{7 a^4 d}-\frac {16 \sec ^9(c+d x)}{9 a^4 d}+\frac {8 \sec ^{11}(c+d x)}{11 a^4 d}-\frac {4 \tan ^5(c+d x)}{5 a^4 d}-\frac {16 \tan ^7(c+d x)}{7 a^4 d}-\frac {20 \tan ^9(c+d x)}{9 a^4 d}-\frac {8 \tan ^{11}(c+d x)}{11 a^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.14 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\sec ^3(c+d x) (844800-215721 \cos (c+d x)-619520 \cos (2 (c+d x))+23969 \cos (3 (c+d x))+32768 \cos (4 (c+d x))+54475 \cos (5 (c+d x))-8192 \cos (6 (c+d x))-2179 \cos (7 (c+d x))+844800 \sin (c+d x)-191752 \sin (2 (c+d x))+11264 \sin (3 (c+d x))-69728 \sin (4 (c+d x))+25600 \sin (5 (c+d x))+17432 \sin (6 (c+d x))-1024 \sin (7 (c+d x)))}{7096320 a^4 d (1+\sin (c+d x))^4} \]

[In]

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

[Out]

(Sec[c + d*x]^3*(844800 - 215721*Cos[c + d*x] - 619520*Cos[2*(c + d*x)] + 23969*Cos[3*(c + d*x)] + 32768*Cos[4
*(c + d*x)] + 54475*Cos[5*(c + d*x)] - 8192*Cos[6*(c + d*x)] - 2179*Cos[7*(c + d*x)] + 844800*Sin[c + d*x] - 1
91752*Sin[2*(c + d*x)] + 11264*Sin[3*(c + d*x)] - 69728*Sin[4*(c + d*x)] + 25600*Sin[5*(c + d*x)] + 17432*Sin[
6*(c + d*x)] - 1024*Sin[7*(c + d*x)]))/(7096320*a^4*d*(1 + Sin[c + d*x])^4)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.89 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {32 \left (924 i {\mathrm e}^{8 i \left (d x +c \right )}+693 \,{\mathrm e}^{9 i \left (d x +c \right )}-726 i {\mathrm e}^{6 i \left (d x +c \right )}-1650 \,{\mathrm e}^{7 i \left (d x +c \right )}-22 i {\mathrm e}^{4 i \left (d x +c \right )}+517 \,{\mathrm e}^{5 i \left (d x +c \right )}-50 i {\mathrm e}^{2 i \left (d x +c \right )}-64 \,{\mathrm e}^{3 i \left (d x +c \right )}+2 i+16 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3465 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{11} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d \,a^{4}}\)	\(143\)
parallelrisch	\(\frac {-\frac {244}{3465}-\frac {584 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105}-\frac {7808 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3465}-\frac {1016 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315}-\frac {64 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315}-\frac {1220 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{693}-\frac {1952 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3465}-\frac {188 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15}-4 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {32 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {128 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}\)	\(165\)
derivativedivides	\(\frac {-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {16}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}+\frac {184}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {32}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {235}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {145}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {58}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {13}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {13}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {16}{512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+512}}{d \,a^{4}}\)	\(220\)
default	\(\frac {-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {16}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}-\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}+\frac {184}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {32}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {235}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {145}{6 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {58}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {13}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {13}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {16}{512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+512}}{d \,a^{4}}\)	\(220\)
norman	\(\frac {-\frac {128 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}-\frac {244}{3465 a d}-\frac {4 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {32 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}-\frac {188 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d a}-\frac {1016 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d a}-\frac {64 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d a}-\frac {1952 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3465 d a}-\frac {1220 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{693 d a}-\frac {7808 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3465 d a}-\frac {584 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} a^{3}}\)	\(228\)

[In]

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

[Out]

-32/3465*(924*I*exp(8*I*(d*x+c))+693*exp(9*I*(d*x+c))-726*I*exp(6*I*(d*x+c))-1650*exp(7*I*(d*x+c))-22*I*exp(4*
I*(d*x+c))+517*exp(5*I*(d*x+c))-50*I*exp(2*I*(d*x+c))-64*exp(3*I*(d*x+c))+2*I+16*exp(I*(d*x+c)))/(exp(I*(d*x+c
))+I)^11/(exp(I*(d*x+c))-I)^3/d/a^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.06 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {128 \, \cos \left (d x + c\right )^{6} - 320 \, \cos \left (d x + c\right )^{4} + 805 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (8 \, \cos \left (d x + c\right )^{6} - 60 \, \cos \left (d x + c\right )^{4} + 35 \, \cos \left (d x + c\right )^{2} - 105\right )} \sin \left (d x + c\right ) - 735}{3465 \, {\left (a^{4} d \cos \left (d x + c\right )^{7} - 8 \, a^{4} d \cos \left (d x + c\right )^{5} + 8 \, a^{4} d \cos \left (d x + c\right )^{3} - 4 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} - 2 \, a^{4} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

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

[Out]

-1/3465*(128*cos(d*x + c)^6 - 320*cos(d*x + c)^4 + 805*cos(d*x + c)^2 + 4*(8*cos(d*x + c)^6 - 60*cos(d*x + c)^
4 + 35*cos(d*x + c)^2 - 105)*sin(d*x + c) - 735)/(a^4*d*cos(d*x + c)^7 - 8*a^4*d*cos(d*x + c)^5 + 8*a^4*d*cos(
d*x + c)^3 - 4*(a^4*d*cos(d*x + c)^5 - 2*a^4*d*cos(d*x + c)^3)*sin(d*x + c))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (129) = 258\).

Time = 0.22 (sec) , antiderivative size = 508, normalized size of antiderivative = 3.50 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {4 \, {\left (\frac {488 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1525 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1952 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2794 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {176 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {4818 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5280 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {10857 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {5544 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {3465 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 61\right )}}{3465 \, {\left (a^{4} + \frac {8 \, a^{4} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {25 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {32 \, a^{4} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {11 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {88 \, a^{4} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {99 \, a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {99 \, a^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {88 \, a^{4} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {11 \, a^{4} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {32 \, a^{4} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {25 \, a^{4} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}} - \frac {8 \, a^{4} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {a^{4} \sin \left (d x + c\right )^{14}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{14}}\right )} d} \]

[In]

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

[Out]

4/3465*(488*sin(d*x + c)/(cos(d*x + c) + 1) + 1525*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1952*sin(d*x + c)^3/(
cos(d*x + c) + 1)^3 + 2794*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 176*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 481
8*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5280*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 10857*sin(d*x + c)^8/(cos(d
*x + c) + 1)^8 + 5544*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 3465*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 61)/(
(a^4 + 8*a^4*sin(d*x + c)/(cos(d*x + c) + 1) + 25*a^4*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 32*a^4*sin(d*x + c
)^3/(cos(d*x + c) + 1)^3 - 11*a^4*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 88*a^4*sin(d*x + c)^5/(cos(d*x + c) +
1)^5 - 99*a^4*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 99*a^4*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 88*a^4*sin(d*
x + c)^9/(cos(d*x + c) + 1)^9 + 11*a^4*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 32*a^4*sin(d*x + c)^11/(cos(d*x
+ c) + 1)^11 - 25*a^4*sin(d*x + c)^12/(cos(d*x + c) + 1)^12 - 8*a^4*sin(d*x + c)^13/(cos(d*x + c) + 1)^13 - a
^4*sin(d*x + c)^14/(cos(d*x + c) + 1)^14)*d)

Giac [A] (verification not implemented)

none

Time = 0.91 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.37 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\frac {1155 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {3465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 45045 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 279510 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 669900 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1205358 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1334718 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1144440 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 627660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 257345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 57013 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5498}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{11}}}{110880 \, d} \]

[In]

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

[Out]

-1/110880*(1155*(3*tan(1/2*d*x + 1/2*c)^2 - 3*tan(1/2*d*x + 1/2*c) + 2)/(a^4*(tan(1/2*d*x + 1/2*c) - 1)^3) - (
3465*tan(1/2*d*x + 1/2*c)^10 + 45045*tan(1/2*d*x + 1/2*c)^9 + 279510*tan(1/2*d*x + 1/2*c)^8 + 669900*tan(1/2*d
*x + 1/2*c)^7 + 1205358*tan(1/2*d*x + 1/2*c)^6 + 1334718*tan(1/2*d*x + 1/2*c)^5 + 1144440*tan(1/2*d*x + 1/2*c)
^4 + 627660*tan(1/2*d*x + 1/2*c)^3 + 257345*tan(1/2*d*x + 1/2*c)^2 + 57013*tan(1/2*d*x + 1/2*c) + 5498)/(a^4*(
tan(1/2*d*x + 1/2*c) + 1)^11))/d

Mupad [B] (verification not implemented)

Time = 16.96 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.09 \[ \int \frac {\sec (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {244\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3465}+\frac {1952\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3465}+\frac {1220\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{693}+\frac {7808\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3465}+\frac {1016\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{315}+\frac {64\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{315}+\frac {584\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{105}+\frac {128\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{21}+\frac {188\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{15}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5}+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{a^4\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^{11}} \]

[In]

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

[Out]

((244*cos(c/2 + (d*x)/2)^14)/3465 + (1952*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2))/3465 + 4*cos(c/2 + (d*x)/2
)^4*sin(c/2 + (d*x)/2)^10 + (32*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9)/5 + (188*cos(c/2 + (d*x)/2)^6*sin(c
/2 + (d*x)/2)^8)/15 + (128*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)/21 + (584*cos(c/2 + (d*x)/2)^8*sin(c/2 +
(d*x)/2)^6)/105 + (64*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5)/315 + (1016*cos(c/2 + (d*x)/2)^10*sin(c/2 +
(d*x)/2)^4)/315 + (7808*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3)/3465 + (1220*cos(c/2 + (d*x)/2)^12*sin(c/2
+ (d*x)/2)^2)/693)/(a^4*d*(cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2))^3*(cos(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2
))^11)