\(\int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [850]

Optimal result

Integrand size = 29, antiderivative size = 143 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {4 \sec ^7(c+d x)}{7 a^4 d}+\frac {4 \sec ^9(c+d x)}{3 a^4 d}-\frac {8 \sec ^{11}(c+d x)}{11 a^4 d}+\frac {\tan ^3(c+d x)}{3 a^4 d}+\frac {2 \tan ^5(c+d x)}{a^4 d}+\frac {25 \tan ^7(c+d x)}{7 a^4 d}+\frac {8 \tan ^9(c+d x)}{3 a^4 d}+\frac {8 \tan ^{11}(c+d x)}{11 a^4 d} \]

[Out]

-4/7*sec(d*x+c)^7/a^4/d+4/3*sec(d*x+c)^9/a^4/d-8/11*sec(d*x+c)^11/a^4/d+1/3*tan(d*x+c)^3/a^4/d+2*tan(d*x+c)^5/
a^4/d+25/7*tan(d*x+c)^7/a^4/d+8/3*tan(d*x+c)^9/a^4/d+8/11*tan(d*x+c)^11/a^4/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2949, 2751, 3852, 8} \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {8 \tan (c+d x)}{231 a^4 d}-\frac {4 \sec (c+d x)}{231 d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {4 \sec (c+d x)}{231 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {\sec ^3(c+d x)}{6 a d (a \sin (c+d x)+a)^3}-\frac {5 \sec (c+d x)}{231 a d (a \sin (c+d x)+a)^3}-\frac {\sec (c+d x)}{33 d (a \sin (c+d x)+a)^4}-\frac {a \sec (c+d x)}{22 d (a \sin (c+d x)+a)^5} \]

[In]

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

[Out]

-1/22*(a*Sec[c + d*x])/(d*(a + a*Sin[c + d*x])^5) - Sec[c + d*x]/(33*d*(a + a*Sin[c + d*x])^4) - (5*Sec[c + d*
x])/(231*a*d*(a + a*Sin[c + d*x])^3) + Sec[c + d*x]^3/(6*a*d*(a + a*Sin[c + d*x])^3) - (4*Sec[c + d*x])/(231*d
*(a^2 + a^2*Sin[c + d*x])^2) - (4*Sec[c + d*x])/(231*d*(a^4 + a^4*Sin[c + d*x])) + (8*Tan[c + d*x])/(231*a^4*d
)

Rule 8

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

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 2949

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*((a + b*Sin[e + f*x])^(m + 1)/(2*b*f*g*(m + 1))), x] + Dist[a/
(2*g^2), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x]
&& EqQ[a^2 - b^2, 0] && EqQ[m - p, 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}+\frac {1}{2} a \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^5} \, dx \\ & = -\frac {a \sec (c+d x)}{22 d (a+a \sin (c+d x))^5}+\frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}+\frac {3}{11} \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx \\ & = -\frac {a \sec (c+d x)}{22 d (a+a \sin (c+d x))^5}-\frac {\sec (c+d x)}{33 d (a+a \sin (c+d x))^4}+\frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}+\frac {5 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx}{33 a} \\ & = -\frac {a \sec (c+d x)}{22 d (a+a \sin (c+d x))^5}-\frac {\sec (c+d x)}{33 d (a+a \sin (c+d x))^4}-\frac {5 \sec (c+d x)}{231 a d (a+a \sin (c+d x))^3}+\frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}+\frac {20 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{231 a^2} \\ & = -\frac {a \sec (c+d x)}{22 d (a+a \sin (c+d x))^5}-\frac {\sec (c+d x)}{33 d (a+a \sin (c+d x))^4}-\frac {5 \sec (c+d x)}{231 a d (a+a \sin (c+d x))^3}+\frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{231 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {4 \int \frac {\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx}{77 a^3} \\ & = -\frac {a \sec (c+d x)}{22 d (a+a \sin (c+d x))^5}-\frac {\sec (c+d x)}{33 d (a+a \sin (c+d x))^4}-\frac {5 \sec (c+d x)}{231 a d (a+a \sin (c+d x))^3}+\frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{231 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {4 \sec (c+d x)}{231 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac {8 \int \sec ^2(c+d x) \, dx}{231 a^4} \\ & = -\frac {a \sec (c+d x)}{22 d (a+a \sin (c+d x))^5}-\frac {\sec (c+d x)}{33 d (a+a \sin (c+d x))^4}-\frac {5 \sec (c+d x)}{231 a d (a+a \sin (c+d x))^3}+\frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{231 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {4 \sec (c+d x)}{231 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {8 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{231 a^4 d} \\ & = -\frac {a \sec (c+d x)}{22 d (a+a \sin (c+d x))^5}-\frac {\sec (c+d x)}{33 d (a+a \sin (c+d x))^4}-\frac {5 \sec (c+d x)}{231 a d (a+a \sin (c+d x))^3}+\frac {\sec ^3(c+d x)}{6 a d (a+a \sin (c+d x))^3}-\frac {4 \sec (c+d x)}{231 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {4 \sec (c+d x)}{231 d \left (a^4+a^4 \sin (c+d x)\right )}+\frac {8 \tan (c+d x)}{231 a^4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.16 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\sec ^3(c+d x) (11264-1287 \cos (c+d x)-5632 \cos (2 (c+d x))+143 \cos (3 (c+d x))-2048 \cos (4 (c+d x))+325 \cos (5 (c+d x))+512 \cos (6 (c+d x))-13 \cos (7 (c+d x))+26048 \sin (c+d x)-1144 \sin (2 (c+d x))-704 \sin (3 (c+d x))-416 \sin (4 (c+d x))-1600 \sin (5 (c+d x))+104 \sin (6 (c+d x))+64 \sin (7 (c+d x)))}{118272 a^4 d (1+\sin (c+d x))^4} \]

[In]

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

[Out]

(Sec[c + d*x]^3*(11264 - 1287*Cos[c + d*x] - 5632*Cos[2*(c + d*x)] + 143*Cos[3*(c + d*x)] - 2048*Cos[4*(c + d*
x)] + 325*Cos[5*(c + d*x)] + 512*Cos[6*(c + d*x)] - 13*Cos[7*(c + d*x)] + 26048*Sin[c + d*x] - 1144*Sin[2*(c +
d*x)] - 704*Sin[3*(c + d*x)] - 416*Sin[4*(c + d*x)] - 1600*Sin[5*(c + d*x)] + 104*Sin[6*(c + d*x)] + 64*Sin[7
*(c + d*x)]))/(118272*a^4*d*(1 + Sin[c + d*x])^4)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.92 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {16 i \left (176 i {\mathrm e}^{7 i \left (d x +c \right )}+154 \,{\mathrm e}^{8 i \left (d x +c \right )}-88 i {\mathrm e}^{5 i \left (d x +c \right )}-253 \,{\mathrm e}^{6 i \left (d x +c \right )}-32 i {\mathrm e}^{3 i \left (d x +c \right )}+11 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 i {\mathrm e}^{i \left (d x +c \right )}+25 \,{\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{231 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{11} d \,a^{4}}\)	\(132\)
parallelrisch	\(\frac {-\frac {16}{231}+\frac {32 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}-\frac {376 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77}-\frac {32 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {32 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {400 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231}-\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{231}-\frac {16 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {16 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {8 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {32 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {80 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21}}{d \,a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}\)	\(178\)
derivativedivides	\(\frac {-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {16}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {64}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {36}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {295}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {71}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {43}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {9}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {109}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128}}{d \,a^{4}}\)	\(218\)
default	\(\frac {-\frac {1}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {16}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{10}}-\frac {64}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {36}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {295}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {71}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {43}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {9}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {109}{48 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128}}{d \,a^{4}}\)	\(218\)
norman	\(\frac {-\frac {80 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}-\frac {16}{231 a d}-\frac {8 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {32 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {16 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {32 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}-\frac {32 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d a}-\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{231 d a}-\frac {400 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231 d a}-\frac {16 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d a}-\frac {376 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{77 d a}+\frac {32 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{11} a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\)	\(247\)

[In]

int(sec(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

-16/231*I*(176*I*exp(7*I*(d*x+c))+154*exp(8*I*(d*x+c))-88*I*exp(5*I*(d*x+c))-253*exp(6*I*(d*x+c))-32*I*exp(3*I
*(d*x+c))+11*exp(4*I*(d*x+c))+8*I*exp(I*(d*x+c))+25*exp(2*I*(d*x+c))-1)/(exp(I*(d*x+c))-I)^3/(exp(I*(d*x+c))+I
)^11/d/a^4

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {32 \, \cos \left (d x + c\right )^{6} - 80 \, \cos \left (d x + c\right )^{4} + 28 \, \cos \left (d x + c\right )^{2} + {\left (8 \, \cos \left (d x + c\right )^{6} - 60 \, \cos \left (d x + c\right )^{4} + 35 \, \cos \left (d x + c\right )^{2} + 49\right )} \sin \left (d x + c\right ) + 28}{231 \, {\left (a^{4} d \cos \left (d x + c\right )^{7} - 8 \, a^{4} d \cos \left (d x + c\right )^{5} + 8 \, a^{4} d \cos \left (d x + c\right )^{3} - 4 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} - 2 \, a^{4} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/231*(32*cos(d*x + c)^6 - 80*cos(d*x + c)^4 + 28*cos(d*x + c)^2 + (8*cos(d*x + c)^6 - 60*cos(d*x + c)^4 + 35*
cos(d*x + c)^2 + 49)*sin(d*x + c) + 28)/(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]

\[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]

[In]

integrate(sec(d*x+c)**4*sin(d*x+c)**2/(a+a*sin(d*x+c))**4,x)

[Out]

Integral(sin(c + d*x)**2*sec(c + d*x)**4/(sin(c + d*x)**4 + 4*sin(c + d*x)**3 + 6*sin(c + d*x)**2 + 4*sin(c +
d*x) + 1), x)/a**4

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (129) = 258\).

Time = 0.28 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.69 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {8 \, {\left (\frac {16 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {50 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {141 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {132 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {132 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {44 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {110 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {154 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {308 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {154 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {77 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + 2\right )}}{231 \, {\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)^2/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

8/231*(16*sin(d*x + c)/(cos(d*x + c) + 1) + 50*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 141*sin(d*x + c)^3/(cos(d
*x + c) + 1)^3 + 132*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 132*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 44*sin(d*
x + c)^6/(cos(d*x + c) + 1)^6 + 110*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 154*sin(d*x + c)^8/(cos(d*x + c) + 1
)^8 + 308*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 154*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 77*sin(d*x + c)^11
/(cos(d*x + c) + 1)^11 + 2)/((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*si
n(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 = 0.45 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.38 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {\frac {77 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {462 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 5775 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 14399 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 29260 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30800 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 27874 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12650 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6556 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 935 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 127}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{11}}}{7392 \, d} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^2/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

-1/7392*(77*(6*tan(1/2*d*x + 1/2*c)^2 - 9*tan(1/2*d*x + 1/2*c) + 5)/(a^4*(tan(1/2*d*x + 1/2*c) - 1)^3) - (462*
tan(1/2*d*x + 1/2*c)^10 + 5775*tan(1/2*d*x + 1/2*c)^9 + 14399*tan(1/2*d*x + 1/2*c)^8 + 29260*tan(1/2*d*x + 1/2
*c)^7 + 30800*tan(1/2*d*x + 1/2*c)^6 + 27874*tan(1/2*d*x + 1/2*c)^5 + 12650*tan(1/2*d*x + 1/2*c)^4 + 6556*tan(
1/2*d*x + 1/2*c)^3 + 1210*tan(1/2*d*x + 1/2*c)^2 + 935*tan(1/2*d*x + 1/2*c) + 127)/(a^4*(tan(1/2*d*x + 1/2*c)
+ 1)^11))/d

Mupad [B] (verification not implemented)

Time = 16.89 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.29 \[ \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{231}+\frac {128\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{231}+\frac {400\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{231}+\frac {376\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{77}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{7}-\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{21}+\frac {80\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{21}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{3}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}}{a^4\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^{11}} \]

[In]

int(sin(c + d*x)^2/(cos(c + d*x)^4*(a + a*sin(c + d*x))^4),x)

[Out]

((16*cos(c/2 + (d*x)/2)^14)/231 + (128*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2))/231 + (8*cos(c/2 + (d*x)/2)^3
*sin(c/2 + (d*x)/2)^11)/3 + (16*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10)/3 + (32*cos(c/2 + (d*x)/2)^5*sin(c
/2 + (d*x)/2)^9)/3 + (16*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8)/3 + (80*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*
x)/2)^7)/21 - (32*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6)/21 + (32*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^
5)/7 + (32*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4)/7 + (376*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3)/77
+ (400*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2)/231)/(a^4*d*(cos(c/2 + (d*x)/2) - sin(c/2 + (d*x)/2))^3*(c
os(c/2 + (d*x)/2) + sin(c/2 + (d*x)/2))^11)