\(\int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx\) [787]

Optimal result

Integrand size = 29, antiderivative size = 151 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3 x}{a^3}+\frac {\cos (c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d} \]

[Out]

3*x/a^3+cos(d*x+c)/a^3/d+7*sec(d*x+c)/a^3/d-5*sec(d*x+c)^3/a^3/d+13/5*sec(d*x+c)^5/a^3/d-4/7*sec(d*x+c)^7/a^3/
d-3*tan(d*x+c)/a^3/d+tan(d*x+c)^3/a^3/d-3/5*tan(d*x+c)^5/a^3/d+4/7*tan(d*x+c)^7/a^3/d

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {2954, 2952, 2687, 30, 2686, 200, 3554, 8, 2670, 276} \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\cos (c+d x)}{a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}+\frac {3 x}{a^3} \]

[In]

Int[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]

[Out]

(3*x)/a^3 + Cos[c + d*x]/(a^3*d) + (7*Sec[c + d*x])/(a^3*d) - (5*Sec[c + d*x]^3)/(a^3*d) + (13*Sec[c + d*x]^5)
/(5*a^3*d) - (4*Sec[c + d*x]^7)/(7*a^3*d) - (3*Tan[c + d*x])/(a^3*d) + Tan[c + d*x]^3/(a^3*d) - (3*Tan[c + d*x
]^5)/(5*a^3*d) + (4*Tan[c + d*x]^7)/(7*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 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 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 ^2(c+d x) (a-a \sin (c+d x))^3 \tan ^6(c+d x) \, dx}{a^6} \\ & = \frac {\int \left (a^3 \sec ^2(c+d x) \tan ^6(c+d x)-3 a^3 \sec (c+d x) \tan ^7(c+d x)+3 a^3 \tan ^8(c+d x)-a^3 \sin (c+d x) \tan ^8(c+d x)\right ) \, dx}{a^6} \\ & = \frac {\int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^3}-\frac {\int \sin (c+d x) \tan ^8(c+d x) \, dx}{a^3}-\frac {3 \int \sec (c+d x) \tan ^7(c+d x) \, dx}{a^3}+\frac {3 \int \tan ^8(c+d x) \, dx}{a^3} \\ & = \frac {3 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \int \tan ^6(c+d x) \, dx}{a^3}+\frac {\text {Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^3 d}+\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{x^8} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = -\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {3 \int \tan ^4(c+d x) \, dx}{a^3}+\frac {\text {Subst}\left (\int \left (1+\frac {1}{x^8}-\frac {4}{x^6}+\frac {6}{x^4}-\frac {4}{x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac {3 \text {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d} \\ & = \frac {\cos (c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}-\frac {3 \int \tan ^2(c+d x) \, dx}{a^3} \\ & = \frac {\cos (c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d}+\frac {3 \int 1 \, dx}{a^3} \\ & = \frac {3 x}{a^3}+\frac {\cos (c+d x)}{a^3 d}+\frac {7 \sec (c+d x)}{a^3 d}-\frac {5 \sec ^3(c+d x)}{a^3 d}+\frac {13 \sec ^5(c+d x)}{5 a^3 d}-\frac {4 \sec ^7(c+d x)}{7 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {\tan ^3(c+d x)}{a^3 d}-\frac {3 \tan ^5(c+d x)}{5 a^3 d}+\frac {4 \tan ^7(c+d x)}{7 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.48 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {8400+14 (-1483+840 c+840 d x) \cos (c+d x)+5152 \cos (2 (c+d x))+8898 \cos (3 (c+d x))-5040 c \cos (3 (c+d x))-5040 d x \cos (3 (c+d x))-2288 \cos (4 (c+d x))+8008 \sin (c+d x)-20762 \sin (2 (c+d x))+11760 c \sin (2 (c+d x))+11760 d x \sin (2 (c+d x))+6588 \sin (3 (c+d x))+1483 \sin (4 (c+d x))-840 c \sin (4 (c+d x))-840 d x \sin (4 (c+d x))-140 \sin (5 (c+d x))}{2240 a^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^7} \]

[In]

Integrate[(Sin[c + d*x]^4*Tan[c + d*x]^2)/(a + a*Sin[c + d*x])^3,x]

[Out]

(8400 + 14*(-1483 + 840*c + 840*d*x)*Cos[c + d*x] + 5152*Cos[2*(c + d*x)] + 8898*Cos[3*(c + d*x)] - 5040*c*Cos
[3*(c + d*x)] - 5040*d*x*Cos[3*(c + d*x)] - 2288*Cos[4*(c + d*x)] + 8008*Sin[c + d*x] - 20762*Sin[2*(c + d*x)]
+ 11760*c*Sin[2*(c + d*x)] + 11760*d*x*Sin[2*(c + d*x)] + 6588*Sin[3*(c + d*x)] + 1483*Sin[4*(c + d*x)] - 840
*c*Sin[4*(c + d*x)] - 840*d*x*Sin[4*(c + d*x)] - 140*Sin[5*(c + d*x)])/(2240*a^3*d*(Cos[(c + d*x)/2] - Sin[(c
+ d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.05

method	result	size
derivativedivides	\(\frac {-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {128}{64+64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {14}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {17}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {49}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\)	\(159\)
default	\(\frac {-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {128}{64+64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+6 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {8}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {14}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {17}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {49}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d \,a^{3}}\)	\(159\)
risch	\(\frac {3 x}{a^{3}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,a^{3}}+\frac {-46 i {\mathrm e}^{4 i \left (d x +c \right )}-94 \,{\mathrm e}^{5 i \left (d x +c \right )}-\frac {254 \,{\mathrm e}^{3 i \left (d x +c \right )}}{5}+58 i {\mathrm e}^{6 i \left (d x +c \right )}+14 \,{\mathrm e}^{7 i \left (d x +c \right )}-\frac {434 i {\mathrm e}^{2 i \left (d x +c \right )}}{5}+\frac {1682 \,{\mathrm e}^{i \left (d x +c \right )}}{35}+\frac {362 i}{35}}{\left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{7} d \,a^{3}}\)	\(161\)
parallelrisch	\(\frac {105 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +\left (630 d x +210\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1575 d x +1260\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2100 d x +3080\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1470 d x +3780\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1932 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1470 d x -1148\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2100 d x -3960\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1575 d x -4020\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-630 d x -1902\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-105 d x -352}{35 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}\)	\(226\)
norman	\(\frac {-\frac {234 x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {114 x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {138 x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {54 x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {222 x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {186 x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {138 x \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {222 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {234 x \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {352}{35 a d}+\frac {186 x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {114 x \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {3 x}{a}+\frac {1686 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}-\frac {5076 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}+\frac {54 x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {18 x \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {106 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {1902 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{35 d a}+\frac {6 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {4906 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}-\frac {18 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}+\frac {11286 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}-\frac {9666 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}+\frac {5136 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}-\frac {15654 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}-\frac {12076 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}-\frac {14264 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d a}+\frac {36 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {1996 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d a}+\frac {216 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7} a^{2} \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 )^{4}}\)	\(597\)

[In]

int(sec(d*x+c)^2*sin(d*x+c)^6/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

128/d/a^3*(-1/1024/(tan(1/2*d*x+1/2*c)-1)+1/64/(1+tan(1/2*d*x+1/2*c)^2)+3/64*arctan(tan(1/2*d*x+1/2*c))-1/112/
(tan(1/2*d*x+1/2*c)+1)^7+1/32/(tan(1/2*d*x+1/2*c)+1)^6-7/320/(tan(1/2*d*x+1/2*c)+1)^5-3/128/(tan(1/2*d*x+1/2*c
)+1)^4+1/256/(tan(1/2*d*x+1/2*c)+1)^3+17/512/(tan(1/2*d*x+1/2*c)+1)^2+49/1024/(tan(1/2*d*x+1/2*c)+1))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.05 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {315 \, d x \cos \left (d x + c\right )^{3} + 286 \, \cos \left (d x + c\right )^{4} - 420 \, d x \cos \left (d x + c\right ) - 447 \, \cos \left (d x + c\right )^{2} + {\left (105 \, d x \cos \left (d x + c\right )^{3} + 35 \, \cos \left (d x + c\right )^{4} - 420 \, d x \cos \left (d x + c\right ) - 438 \, \cos \left (d x + c\right )^{2} - 20\right )} \sin \left (d x + c\right ) - 15}{35 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)^6/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/35*(315*d*x*cos(d*x + c)^3 + 286*cos(d*x + c)^4 - 420*d*x*cos(d*x + c) - 447*cos(d*x + c)^2 + (105*d*x*cos(d
*x + c)^3 + 35*cos(d*x + c)^4 - 420*d*x*cos(d*x + c) - 438*cos(d*x + c)^2 - 20)*sin(d*x + c) - 15)/(3*a^3*d*co
s(d*x + c)^3 - 4*a^3*d*cos(d*x + c) + (a^3*d*cos(d*x + c)^3 - 4*a^3*d*cos(d*x + c))*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**2*sin(d*x+c)**6/(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. 421 vs. \(2 (143) = 286\).

Time = 0.30 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.79 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {2 \, {\left (\frac {\frac {951 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2010 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1980 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {574 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {966 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1890 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1540 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {630 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {105 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 176}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {14 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {14 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {20 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{35 \, d} \]

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)^6/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

2/35*((951*sin(d*x + c)/(cos(d*x + c) + 1) + 2010*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1980*sin(d*x + c)^3/(c
os(d*x + c) + 1)^3 + 574*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 966*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 1890*
sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1540*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 630*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8 - 105*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + 176)/(a^3 + 6*a^3*sin(d*x + c)/(cos(d*x + c) + 1) + 15*
a^3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 20*a^3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 14*a^3*sin(d*x + c)^4/(
cos(d*x + c) + 1)^4 - 14*a^3*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 20*a^3*sin(d*x + c)^7/(cos(d*x + c) + 1)^7
- 15*a^3*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - 6*a^3*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - a^3*sin(d*x + c)^10
/(cos(d*x + c) + 1)^10) + 105*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.17 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {840 \, {\left (d x + c\right )}}{a^{3}} - \frac {35 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 17\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )} a^{3}} + \frac {1715 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 11480 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31815 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 45920 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35161 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 13832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2221}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{280 \, d} \]

[In]

integrate(sec(d*x+c)^2*sin(d*x+c)^6/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/280*(840*(d*x + c)/a^3 - 35*(tan(1/2*d*x + 1/2*c)^2 - 16*tan(1/2*d*x + 1/2*c) + 17)/((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^3) + (1715*tan(1/2*d*x + 1/2*c)^6 + 11480*tan(1/2*d*x
+ 1/2*c)^5 + 31815*tan(1/2*d*x + 1/2*c)^4 + 45920*tan(1/2*d*x + 1/2*c)^3 + 35161*tan(1/2*d*x + 1/2*c)^2 + 1383
2*tan(1/2*d*x + 1/2*c) + 2221)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^7))/d

Mupad [B] (verification not implemented)

Time = 20.22 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^4(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {3\,x}{a^3}-\frac {-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-88\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-108\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {276\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {164\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}+\frac {792\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{7}+\frac {804\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{7}+\frac {1902\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}+\frac {352}{35}}{a^3\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]

[In]

int(sin(c + d*x)^6/(cos(c + d*x)^2*(a + a*sin(c + d*x))^3),x)

[Out]

(3*x)/a^3 - ((1902*tan(c/2 + (d*x)/2))/35 + (804*tan(c/2 + (d*x)/2)^2)/7 + (792*tan(c/2 + (d*x)/2)^3)/7 + (164
*tan(c/2 + (d*x)/2)^4)/5 - (276*tan(c/2 + (d*x)/2)^5)/5 - 108*tan(c/2 + (d*x)/2)^6 - 88*tan(c/2 + (d*x)/2)^7 -
36*tan(c/2 + (d*x)/2)^8 - 6*tan(c/2 + (d*x)/2)^9 + 352/35)/(a^3*d*(tan(c/2 + (d*x)/2) + 1)^7*(tan(c/2 + (d*x)
/2) - tan(c/2 + (d*x)/2)^2 + tan(c/2 + (d*x)/2)^3 - 1))