\(\int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx\) [903]

Optimal result

Integrand size = 29, antiderivative size = 192 \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{4 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d} \]

[Out]

3/256*arctanh(sin(d*x+c))/a/d-1/6*sec(d*x+c)^6/a/d+1/4*sec(d*x+c)^8/a/d-1/10*sec(d*x+c)^10/a/d+3/256*sec(d*x+c
)*tan(d*x+c)/a/d+1/128*sec(d*x+c)^3*tan(d*x+c)/a/d+1/160*sec(d*x+c)^5*tan(d*x+c)/a/d-3/80*sec(d*x+c)^7*tan(d*x
+c)/a/d+1/10*sec(d*x+c)^7*tan(d*x+c)^3/a/d

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2914, 2691, 3853, 3855, 2686, 272, 45} \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {\sec ^8(c+d x)}{4 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\tan ^3(c+d x) \sec ^7(c+d x)}{10 a d}-\frac {3 \tan (c+d x) \sec ^7(c+d x)}{80 a d}+\frac {\tan (c+d x) \sec ^5(c+d x)}{160 a d}+\frac {\tan (c+d x) \sec ^3(c+d x)}{128 a d}+\frac {3 \tan (c+d x) \sec (c+d x)}{256 a d} \]

[In]

Int[(Sec[c + d*x]^5*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(3*ArcTanh[Sin[c + d*x]])/(256*a*d) - Sec[c + d*x]^6/(6*a*d) + Sec[c + d*x]^8/(4*a*d) - Sec[c + d*x]^10/(10*a*
d) + (3*Sec[c + d*x]*Tan[c + d*x])/(256*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(128*a*d) + (Sec[c + d*x]^5*Tan[c
+ d*x])/(160*a*d) - (3*Sec[c + d*x]^7*Tan[c + d*x])/(80*a*d) + (Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*a*d)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sec ^7(c+d x) \tan ^4(c+d x) \, dx}{a}-\frac {\int \sec ^6(c+d x) \tan ^5(c+d x) \, dx}{a} \\ & = \frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}-\frac {3 \int \sec ^7(c+d x) \tan ^2(c+d x) \, dx}{10 a}-\frac {\text {Subst}\left (\int x^5 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a d} \\ & = -\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac {3 \int \sec ^7(c+d x) \, dx}{80 a}-\frac {\text {Subst}\left (\int (-1+x)^2 x^2 \, dx,x,\sec ^2(c+d x)\right )}{2 a d} \\ & = \frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac {\int \sec ^5(c+d x) \, dx}{32 a}-\frac {\text {Subst}\left (\int \left (x^2-2 x^3+x^4\right ) \, dx,x,\sec ^2(c+d x)\right )}{2 a d} \\ & = -\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{4 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac {3 \int \sec ^3(c+d x) \, dx}{128 a} \\ & = -\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{4 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d}+\frac {3 \int \sec (c+d x) \, dx}{256 a} \\ & = \frac {3 \text {arctanh}(\sin (c+d x))}{256 a d}-\frac {\sec ^6(c+d x)}{6 a d}+\frac {\sec ^8(c+d x)}{4 a d}-\frac {\sec ^{10}(c+d x)}{10 a d}+\frac {3 \sec (c+d x) \tan (c+d x)}{256 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^5(c+d x) \tan (c+d x)}{160 a d}-\frac {3 \sec ^7(c+d x) \tan (c+d x)}{80 a d}+\frac {\sec ^7(c+d x) \tan ^3(c+d x)}{10 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.60 \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {90 \text {arctanh}(\sin (c+d x))+\frac {30}{(-1+\sin (c+d x))^4}+\frac {40}{(-1+\sin (c+d x))^3}-\frac {45}{(-1+\sin (c+d x))^2}-\frac {48}{(1+\sin (c+d x))^5}+\frac {90}{(1+\sin (c+d x))^4}+\frac {20}{(1+\sin (c+d x))^3}-\frac {45}{(1+\sin (c+d x))^2}-\frac {90}{1+\sin (c+d x)}}{7680 a d} \]

[In]

Integrate[(Sec[c + d*x]^5*Tan[c + d*x]^4)/(a + a*Sin[c + d*x]),x]

[Out]

(90*ArcTanh[Sin[c + d*x]] + 30/(-1 + Sin[c + d*x])^4 + 40/(-1 + Sin[c + d*x])^3 - 45/(-1 + Sin[c + d*x])^2 - 4
8/(1 + Sin[c + d*x])^5 + 90/(1 + Sin[c + d*x])^4 + 20/(1 + Sin[c + d*x])^3 - 45/(1 + Sin[c + d*x])^2 - 90/(1 +
Sin[c + d*x]))/(7680*a*d)

Maple [A] (verified)

Time = 1.68 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {3}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\)	\(127\)
default	\(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{4}}+\frac {1}{192 \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {3}{512 \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{512}-\frac {1}{160 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{384 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {3}{512 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3}{256 \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{512}}{d a}\)	\(127\)
risch	\(-\frac {i \left (690 i {\mathrm e}^{14 i \left (d x +c \right )}+45 \,{\mathrm e}^{17 i \left (d x +c \right )}+90 i {\mathrm e}^{16 i \left (d x +c \right )}+300 \,{\mathrm e}^{15 i \left (d x +c \right )}-36514 i {\mathrm e}^{8 i \left (d x +c \right )}+804 \,{\mathrm e}^{13 i \left (d x +c \right )}+18182 i {\mathrm e}^{6 i \left (d x +c \right )}+6868 \,{\mathrm e}^{11 i \left (d x +c \right )}-18182 i {\mathrm e}^{12 i \left (d x +c \right )}+350 \,{\mathrm e}^{9 i \left (d x +c \right )}+36514 i {\mathrm e}^{10 i \left (d x +c \right )}+6868 \,{\mathrm e}^{7 i \left (d x +c \right )}-690 i {\mathrm e}^{4 i \left (d x +c \right )}+804 \,{\mathrm e}^{5 i \left (d x +c \right )}-90 i {\mathrm e}^{2 i \left (d x +c \right )}+300 \,{\mathrm e}^{3 i \left (d x +c \right )}+45 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{1920 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{10} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{256 a d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{256 d a}\)	\(277\)
parallelrisch	\(\frac {\left (-45 \cos \left (10 d x +10 c \right )-9450 \cos \left (2 d x +2 c \right )-5400 \cos \left (4 d x +4 c \right )-2025 \cos \left (6 d x +6 c \right )-450 \cos \left (8 d x +8 c \right )-5670\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\left (45 \cos \left (10 d x +10 c \right )+9450 \cos \left (2 d x +2 c \right )+5400 \cos \left (4 d x +4 c \right )+2025 \cos \left (6 d x +6 c \right )+450 \cos \left (8 d x +8 c \right )+5670\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-60600 \sin \left (3 d x +3 c \right )+3768 \sin \left (5 d x +5 c \right )+870 \sin \left (7 d x +7 c \right )+90 \sin \left (9 d x +9 c \right )+64 \cos \left (10 d x +10 c \right )+95360 \cos \left (2 d x +2 c \right )-33280 \cos \left (4 d x +4 c \right )+2880 \cos \left (6 d x +6 c \right )+640 \cos \left (8 d x +8 c \right )+133020 \sin \left (d x +c \right )-65664}{3840 a d \left (\cos \left (10 d x +10 c \right )+10 \cos \left (8 d x +8 c \right )+45 \cos \left (6 d x +6 c \right )+120 \cos \left (4 d x +4 c \right )+210 \cos \left (2 d x +2 c \right )+126\right )}\)	\(315\)

[In]

int(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d/a*(1/256/(sin(d*x+c)-1)^4+1/192/(sin(d*x+c)-1)^3-3/512/(sin(d*x+c)-1)^2-3/512*ln(sin(d*x+c)-1)-1/160/(1+si
n(d*x+c))^5+3/256/(1+sin(d*x+c))^4+1/384/(1+sin(d*x+c))^3-3/512/(1+sin(d*x+c))^2-3/256/(1+sin(d*x+c))+3/512*ln
(1+sin(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.97 \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {90 \, \cos \left (d x + c\right )^{8} - 30 \, \cos \left (d x + c\right )^{6} - 12 \, \cos \left (d x + c\right )^{4} + 176 \, \cos \left (d x + c\right )^{2} - 45 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 45 \, {\left (\cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{8}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (45 \, \cos \left (d x + c\right )^{6} + 30 \, \cos \left (d x + c\right )^{4} - 616 \, \cos \left (d x + c\right )^{2} + 432\right )} \sin \left (d x + c\right ) - 96}{7680 \, {\left (a d \cos \left (d x + c\right )^{8} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{8}\right )}} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/7680*(90*cos(d*x + c)^8 - 30*cos(d*x + c)^6 - 12*cos(d*x + c)^4 + 176*cos(d*x + c)^2 - 45*(cos(d*x + c)^8*s
in(d*x + c) + cos(d*x + c)^8)*log(sin(d*x + c) + 1) + 45*(cos(d*x + c)^8*sin(d*x + c) + cos(d*x + c)^8)*log(-s
in(d*x + c) + 1) - 2*(45*cos(d*x + c)^6 + 30*cos(d*x + c)^4 - 616*cos(d*x + c)^2 + 432)*sin(d*x + c) - 96)/(a*
d*cos(d*x + c)^8*sin(d*x + c) + a*d*cos(d*x + c)^8)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**9*sin(d*x+c)**4/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.11 \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {2 \, {\left (45 \, \sin \left (d x + c\right )^{8} + 45 \, \sin \left (d x + c\right )^{7} - 165 \, \sin \left (d x + c\right )^{6} - 165 \, \sin \left (d x + c\right )^{5} + 219 \, \sin \left (d x + c\right )^{4} - 421 \, \sin \left (d x + c\right )^{3} - 211 \, \sin \left (d x + c\right )^{2} + 109 \, \sin \left (d x + c\right ) + 64\right )}}{a \sin \left (d x + c\right )^{9} + a \sin \left (d x + c\right )^{8} - 4 \, a \sin \left (d x + c\right )^{7} - 4 \, a \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} + 6 \, a \sin \left (d x + c\right )^{4} - 4 \, a \sin \left (d x + c\right )^{3} - 4 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac {45 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {45 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{7680 \, d} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/7680*(2*(45*sin(d*x + c)^8 + 45*sin(d*x + c)^7 - 165*sin(d*x + c)^6 - 165*sin(d*x + c)^5 + 219*sin(d*x + c)
^4 - 421*sin(d*x + c)^3 - 211*sin(d*x + c)^2 + 109*sin(d*x + c) + 64)/(a*sin(d*x + c)^9 + a*sin(d*x + c)^8 - 4
*a*sin(d*x + c)^7 - 4*a*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 + 6*a*sin(d*x + c)^4 - 4*a*sin(d*x + c)^3 - 4*a*si
n(d*x + c)^2 + a*sin(d*x + c) + a) - 45*log(sin(d*x + c) + 1)/a + 45*log(sin(d*x + c) - 1)/a)/d

Giac [A] (verification not implemented)

none

Time = 0.57 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.81 \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {180 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {180 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {5 \, {\left (75 \, \sin \left (d x + c\right )^{4} - 300 \, \sin \left (d x + c\right )^{3} + 414 \, \sin \left (d x + c\right )^{2} - 196 \, \sin \left (d x + c\right ) + 31\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{4}} - \frac {411 \, \sin \left (d x + c\right )^{5} + 2415 \, \sin \left (d x + c\right )^{4} + 5730 \, \sin \left (d x + c\right )^{3} + 6730 \, \sin \left (d x + c\right )^{2} + 3515 \, \sin \left (d x + c\right ) + 703}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{5}}}{30720 \, d} \]

[In]

integrate(sec(d*x+c)^9*sin(d*x+c)^4/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/30720*(180*log(abs(sin(d*x + c) + 1))/a - 180*log(abs(sin(d*x + c) - 1))/a + 5*(75*sin(d*x + c)^4 - 300*sin(
d*x + c)^3 + 414*sin(d*x + c)^2 - 196*sin(d*x + c) + 31)/(a*(sin(d*x + c) - 1)^4) - (411*sin(d*x + c)^5 + 2415
*sin(d*x + c)^4 + 5730*sin(d*x + c)^3 + 6730*sin(d*x + c)^2 + 3515*sin(d*x + c) + 703)/(a*(sin(d*x + c) + 1)^5
))/d

Mupad [B] (verification not implemented)

Time = 17.88 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.58 \[ \int \frac {\sec ^5(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,a\,d}+\frac {-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{64}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{32}+\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{64}+\frac {957\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{160}+\frac {899\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{960}+\frac {5813\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{480}+\frac {1873\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{960}+\frac {4061\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {1873\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{960}+\frac {5813\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{480}+\frac {899\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{960}+\frac {957\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}+\frac {23\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64}-\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+14\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-28\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )} \]

[In]

int(sin(c + d*x)^4/(cos(c + d*x)^9*(a + a*sin(c + d*x))),x)

[Out]

(3*atanh(tan(c/2 + (d*x)/2)))/(128*a*d) + ((5*tan(c/2 + (d*x)/2)^3)/32 - (3*tan(c/2 + (d*x)/2)^2)/64 - (3*tan(
c/2 + (d*x)/2))/128 + (23*tan(c/2 + (d*x)/2)^4)/64 + (957*tan(c/2 + (d*x)/2)^5)/160 + (899*tan(c/2 + (d*x)/2)^
6)/960 + (5813*tan(c/2 + (d*x)/2)^7)/480 + (1873*tan(c/2 + (d*x)/2)^8)/960 + (4061*tan(c/2 + (d*x)/2)^9)/192 +
(1873*tan(c/2 + (d*x)/2)^10)/960 + (5813*tan(c/2 + (d*x)/2)^11)/480 + (899*tan(c/2 + (d*x)/2)^12)/960 + (957*
tan(c/2 + (d*x)/2)^13)/160 + (23*tan(c/2 + (d*x)/2)^14)/64 + (5*tan(c/2 + (d*x)/2)^15)/32 - (3*tan(c/2 + (d*x)
/2)^16)/64 - (3*tan(c/2 + (d*x)/2)^17)/128)/(d*(a + 2*a*tan(c/2 + (d*x)/2) - 7*a*tan(c/2 + (d*x)/2)^2 - 16*a*t
an(c/2 + (d*x)/2)^3 + 20*a*tan(c/2 + (d*x)/2)^4 + 56*a*tan(c/2 + (d*x)/2)^5 - 28*a*tan(c/2 + (d*x)/2)^6 - 112*
a*tan(c/2 + (d*x)/2)^7 + 14*a*tan(c/2 + (d*x)/2)^8 + 140*a*tan(c/2 + (d*x)/2)^9 + 14*a*tan(c/2 + (d*x)/2)^10 -
112*a*tan(c/2 + (d*x)/2)^11 - 28*a*tan(c/2 + (d*x)/2)^12 + 56*a*tan(c/2 + (d*x)/2)^13 + 20*a*tan(c/2 + (d*x)/
2)^14 - 16*a*tan(c/2 + (d*x)/2)^15 - 7*a*tan(c/2 + (d*x)/2)^16 + 2*a*tan(c/2 + (d*x)/2)^17 + a*tan(c/2 + (d*x)
/2)^18))