\(\int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [94]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {4 \cos (c+d x)}{a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {4 \log (1+\cos (c+d x))}{a^3 d} \]

[Out]

-4*cos(d*x+c)/a^3/d+2*cos(d*x+c)^2/a^3/d-4/3*cos(d*x+c)^3/a^3/d+3/4*cos(d*x+c)^4/a^3/d-1/5*cos(d*x+c)^5/a^3/d+
4*ln(1+cos(d*x+c))/a^3/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 90} \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos (c+d x)}{a^3 d}+\frac {4 \log (\cos (c+d x)+1)}{a^3 d} \]

[In]

Int[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]

[Out]

(-4*Cos[c + d*x])/(a^3*d) + (2*Cos[c + d*x]^2)/(a^3*d) - (4*Cos[c + d*x]^3)/(3*a^3*d) + (3*Cos[c + d*x]^4)/(4*
a^3*d) - Cos[c + d*x]^5/(5*a^3*d) + (4*Log[1 + Cos[c + d*x]])/(a^3*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 2915

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

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^5(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^2 x^3}{a^3 (-a+x)} \, dx,x,-a \cos (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \frac {(-a-x)^2 x^3}{-a+x} \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^4-\frac {4 a^5}{a-x}+4 a^3 x+4 a^2 x^2+3 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^8 d} \\ & = -\frac {4 \cos (c+d x)}{a^3 d}+\frac {2 \cos ^2(c+d x)}{a^3 d}-\frac {4 \cos ^3(c+d x)}{3 a^3 d}+\frac {3 \cos ^4(c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x)}{5 a^3 d}+\frac {4 \log (1+\cos (c+d x))}{a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3857-4920 \cos (c+d x)+1320 \cos (2 (c+d x))-380 \cos (3 (c+d x))+90 \cos (4 (c+d x))-12 \cos (5 (c+d x))+7680 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{960 a^3 d} \]

[In]

Integrate[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]

[Out]

(3857 - 4920*Cos[c + d*x] + 1320*Cos[2*(c + d*x)] - 380*Cos[3*(c + d*x)] + 90*Cos[4*(c + d*x)] - 12*Cos[5*(c +
d*x)] + 7680*Log[Cos[(c + d*x)/2]])/(960*a^3*d)

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {-\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}-\frac {4 \cos \left (d x +c \right )^{3}}{3}+2 \cos \left (d x +c \right )^{2}-4 \cos \left (d x +c \right )+4 \ln \left (\cos \left (d x +c \right )+1\right )}{d \,a^{3}}\)	\(68\)
default	\(\frac {-\frac {\cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}-\frac {4 \cos \left (d x +c \right )^{3}}{3}+2 \cos \left (d x +c \right )^{2}-4 \cos \left (d x +c \right )+4 \ln \left (\cos \left (d x +c \right )+1\right )}{d \,a^{3}}\)	\(68\)
parallelrisch	\(\frac {-1920 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-3361-6 \cos \left (5 d x +5 c \right )+45 \cos \left (4 d x +4 c \right )-190 \cos \left (3 d x +3 c \right )+660 \cos \left (2 d x +2 c \right )-2460 \cos \left (d x +c \right )}{480 a^{3} d}\)	\(77\)
norman	\(\frac {-\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d a}-\frac {166}{15 a d}-\frac {78 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d a}-\frac {154 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d a}-\frac {278 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} a^{2}}-\frac {4 \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{3} d}\)	\(128\)
risch	\(-\frac {4 i x}{a^{3}}-\frac {41 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a^{3} d}-\frac {41 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a^{3} d}-\frac {8 i c}{a^{3} d}+\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{3}}+\frac {3 \cos \left (4 d x +4 c \right )}{32 d \,a^{3}}-\frac {19 \cos \left (3 d x +3 c \right )}{48 d \,a^{3}}+\frac {11 \cos \left (2 d x +2 c \right )}{8 d \,a^{3}}\)	\(141\)

[In]

int(sin(d*x+c)^5/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d/a^3*(-1/5*cos(d*x+c)^5+3/4*cos(d*x+c)^4-4/3*cos(d*x+c)^3+2*cos(d*x+c)^2-4*cos(d*x+c)+4*ln(cos(d*x+c)+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right ) - 240 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{60 \, a^{3} d} \]

[In]

integrate(sin(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/60*(12*cos(d*x + c)^5 - 45*cos(d*x + c)^4 + 80*cos(d*x + c)^3 - 120*cos(d*x + c)^2 + 240*cos(d*x + c) - 240
*log(1/2*cos(d*x + c) + 1/2))/(a^3*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate(sin(d*x+c)**5/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {12 \, \cos \left (d x + c\right )^{5} - 45 \, \cos \left (d x + c\right )^{4} + 80 \, \cos \left (d x + c\right )^{3} - 120 \, \cos \left (d x + c\right )^{2} + 240 \, \cos \left (d x + c\right )}{a^{3}} - \frac {240 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{60 \, d} \]

[In]

integrate(sin(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*((12*cos(d*x + c)^5 - 45*cos(d*x + c)^4 + 80*cos(d*x + c)^3 - 120*cos(d*x + c)^2 + 240*cos(d*x + c))/a^3
- 240*log(cos(d*x + c) + 1)/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.69 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {60 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {\frac {85 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {20 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {200 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {205 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 29}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{15 \, d} \]

[In]

integrate(sin(d*x+c)^5/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/15*(60*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1))/a^3 + (85*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)
- 20*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 200*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 205*(cos(d*x
+ c) - 1)^4/(cos(d*x + c) + 1)^4 - 137*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 - 29)/(a^3*((cos(d*x + c) -
1)/(cos(d*x + c) + 1) - 1)^5))/d

Mupad [B] (verification not implemented)

Time = 13.36 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {4\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^3}-\frac {4\,\cos \left (c+d\,x\right )}{a^3}+\frac {2\,{\cos \left (c+d\,x\right )}^2}{a^3}-\frac {4\,{\cos \left (c+d\,x\right )}^3}{3\,a^3}+\frac {3\,{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^3}}{d} \]

[In]

int(sin(c + d*x)^5/(a + a/cos(c + d*x))^3,x)

[Out]

((4*log(cos(c + d*x) + 1))/a^3 - (4*cos(c + d*x))/a^3 + (2*cos(c + d*x)^2)/a^3 - (4*cos(c + d*x)^3)/(3*a^3) +
(3*cos(c + d*x)^4)/(4*a^3) - cos(c + d*x)^5/(5*a^3))/d