\(\int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [542]

Optimal result

Integrand size = 29, antiderivative size = 73 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^3(c+d x)}{3 a d}+\frac {\csc ^4(c+d x)}{4 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {\csc ^6(c+d x)}{6 a d} \]

[Out]

-1/3*csc(d*x+c)^3/a/d+1/4*csc(d*x+c)^4/a/d+1/5*csc(d*x+c)^5/a/d-1/6*csc(d*x+c)^6/a/d

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 76} \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^4(c+d x)}{4 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]

[In]

Int[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]

[Out]

-1/3*Csc[c + d*x]^3/(a*d) + Csc[c + d*x]^4/(4*a*d) + Csc[c + d*x]^5/(5*a*d) - Csc[c + d*x]^6/(6*a*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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

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]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.66 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\csc ^3(c+d x) \left (-20+15 \csc (c+d x)+12 \csc ^2(c+d x)-10 \csc ^3(c+d x)\right )}{60 a d} \]

[In]

Integrate[(Cot[c + d*x]^5*Csc[c + d*x]^2)/(a + a*Sin[c + d*x]),x]

[Out]

(Csc[c + d*x]^3*(-20 + 15*Csc[c + d*x] + 12*Csc[c + d*x]^2 - 10*Csc[c + d*x]^3))/(60*a*d)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68

method	result	size
derivativedivides	\(-\frac {\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}}{d a}\)	\(50\)
default	\(-\frac {\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\csc ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{3}}{d a}\)	\(50\)
risch	\(\frac {4 i \left (-15 i {\mathrm e}^{8 i \left (d x +c \right )}+10 \,{\mathrm e}^{9 i \left (d x +c \right )}-10 i {\mathrm e}^{6 i \left (d x +c \right )}-6 \,{\mathrm e}^{7 i \left (d x +c \right )}-15 i {\mathrm e}^{4 i \left (d x +c \right )}+6 \,{\mathrm e}^{5 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{15 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}\)	\(104\)
parallelrisch	\(\frac {-5 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5-20 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+45 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}\)	\(137\)
norman	\(\frac {-\frac {1}{384 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d a}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{96 d a}+\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}-\frac {5 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}-\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}+\frac {5 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{96 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}+\frac {7 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1920 d a}-\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\)	\(242\)

[In]

int(cos(d*x+c)^5*csc(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/d/a*(1/6*csc(d*x+c)^6-1/5*csc(d*x+c)^5-1/4*csc(d*x+c)^4+1/3*csc(d*x+c)^3)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {15 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (5 \, \cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) - 5}{60 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/60*(15*cos(d*x + c)^2 - 4*(5*cos(d*x + c)^2 - 2)*sin(d*x + c) - 5)/(a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^
4 + 3*a*d*cos(d*x + c)^2 - a*d)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**7/(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {20 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) + 10}{60 \, a d \sin \left (d x + c\right )^{6}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*(20*sin(d*x + c)^3 - 15*sin(d*x + c)^2 - 12*sin(d*x + c) + 10)/(a*d*sin(d*x + c)^6)

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {20 \, \sin \left (d x + c\right )^{3} - 15 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) + 10}{60 \, a d \sin \left (d x + c\right )^{6}} \]

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/60*(20*sin(d*x + c)^3 - 15*sin(d*x + c)^2 - 12*sin(d*x + c) + 10)/(a*d*sin(d*x + c)^6)

Mupad [B] (verification not implemented)

Time = 10.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63 \[ \int \frac {\cot ^5(c+d x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-20\,{\sin \left (c+d\,x\right )}^3+15\,{\sin \left (c+d\,x\right )}^2+12\,\sin \left (c+d\,x\right )-10}{60\,a\,d\,{\sin \left (c+d\,x\right )}^6} \]

[In]

int(cos(c + d*x)^5/(sin(c + d*x)^7*(a + a*sin(c + d*x))),x)

[Out]

(12*sin(c + d*x) + 15*sin(c + d*x)^2 - 20*sin(c + d*x)^3 - 10)/(60*a*d*sin(c + d*x)^6)