\(\int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx\) [669]

Optimal result

Integrand size = 25, antiderivative size = 119 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \log (\sin (c+d x))}{d} \]

[Out]

a*csc(d*x+c)/d-3/2*a*csc(d*x+c)^2/d-a*csc(d*x+c)^3/d+3/4*a*csc(d*x+c)^4/d+3/5*a*csc(d*x+c)^5/d-1/6*a*csc(d*x+c
)^6/d-1/7*a*csc(d*x+c)^7/d-a*ln(sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2915, 12, 90} \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc (c+d x)}{d}-\frac {a \log (\sin (c+d x))}{d} \]

[In]

Int[Cot[c + d*x]^7*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

(a*Csc[c + d*x])/d - (3*a*Csc[c + d*x]^2)/(2*d) - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^4)/(4*d) + (3*a*Csc
[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^6)/(6*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*Log[Sin[c + d*x]])/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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a^8 (a-x)^3 (a+x)^4}{x^8} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {a \text {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^8} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \text {Subst}\left (\int \left (\frac {a^7}{x^8}+\frac {a^6}{x^7}-\frac {3 a^5}{x^6}-\frac {3 a^4}{x^5}+\frac {3 a^3}{x^4}+\frac {3 a^2}{x^3}-\frac {a}{x^2}-\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \log (\sin (c+d x))}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.97 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \csc (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \left (6 \cot ^2(c+d x)-3 \cot ^4(c+d x)+2 \cot ^6(c+d x)+12 \log (\cos (c+d x))+12 \log (\tan (c+d x))\right )}{12 d} \]

[In]

Integrate[Cot[c + d*x]^7*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]

[Out]

(a*Csc[c + d*x])/d - (a*Csc[c + d*x]^3)/d + (3*a*Csc[c + d*x]^5)/(5*d) - (a*Csc[c + d*x]^7)/(7*d) - (a*(6*Cot[
c + d*x]^2 - 3*Cot[c + d*x]^4 + 2*Cot[c + d*x]^6 + 12*Log[Cos[c + d*x]] + 12*Log[Tan[c + d*x]]))/(12*d)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\csc ^{3}\left (d x +c \right )+\frac {3 \left (\csc ^{2}\left (d x +c \right )\right )}{2}-\csc \left (d x +c \right )-\ln \left (\csc \left (d x +c \right )\right )\right )}{d}\)	\(83\)
default	\(-\frac {a \left (\frac {\left (\csc ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\csc ^{6}\left (d x +c \right )\right )}{6}-\frac {3 \left (\csc ^{5}\left (d x +c \right )\right )}{5}-\frac {3 \left (\csc ^{4}\left (d x +c \right )\right )}{4}+\csc ^{3}\left (d x +c \right )+\frac {3 \left (\csc ^{2}\left (d x +c \right )\right )}{2}-\csc \left (d x +c \right )-\ln \left (\csc \left (d x +c \right )\right )\right )}{d}\)	\(83\)
risch	\(i a x +\frac {2 i a c}{d}+\frac {2 i a \left (105 \,{\mathrm e}^{13 i \left (d x +c \right )}-210 \,{\mathrm e}^{11 i \left (d x +c \right )}-315 i {\mathrm e}^{12 i \left (d x +c \right )}+903 \,{\mathrm e}^{9 i \left (d x +c \right )}+945 i {\mathrm e}^{10 i \left (d x +c \right )}-636 \,{\mathrm e}^{7 i \left (d x +c \right )}-1820 i {\mathrm e}^{8 i \left (d x +c \right )}+903 \,{\mathrm e}^{5 i \left (d x +c \right )}+1820 i {\mathrm e}^{6 i \left (d x +c \right )}-210 \,{\mathrm e}^{3 i \left (d x +c \right )}-945 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}+315 i {\mathrm e}^{2 i \left (d x +c \right )}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(203\)
parallelrisch	\(-\frac {\left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )+\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {7 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {7 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {49 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {49 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-28 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-28 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+49 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+49 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+203 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+203 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-245 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-245 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+896 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-896 \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right ) a}{896 d}\)	\(208\)
norman	\(\frac {-\frac {a}{896 d}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d}+\frac {11 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1120 d}+\frac {11 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {7 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}-\frac {25 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}+\frac {7 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}+\frac {35 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {7 a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {25 a \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}-\frac {7 a \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d}+\frac {11 a \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}+\frac {11 a \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1120 d}-\frac {a \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d}-\frac {a \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{896 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(307\)

[In]

int(cos(d*x+c)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-a/d*(1/7*csc(d*x+c)^7+1/6*csc(d*x+c)^6-3/5*csc(d*x+c)^5-3/4*csc(d*x+c)^4+csc(d*x+c)^3+3/2*csc(d*x+c)^2-csc(d*
x+c)-ln(csc(d*x+c)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.45 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {420 \, a \cos \left (d x + c\right )^{6} - 840 \, a \cos \left (d x + c\right )^{4} + 672 \, a \cos \left (d x + c\right )^{2} - 420 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 35 \, {\left (18 \, a \cos \left (d x + c\right )^{4} - 27 \, a \cos \left (d x + c\right )^{2} + 11 \, a\right )} \sin \left (d x + c\right ) - 192 \, a}{420 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/420*(420*a*cos(d*x + c)^6 - 840*a*cos(d*x + c)^4 + 672*a*cos(d*x + c)^2 - 420*(a*cos(d*x + c)^6 - 3*a*cos(d*
x + c)^4 + 3*a*cos(d*x + c)^2 - a)*log(1/2*sin(d*x + c))*sin(d*x + c) + 35*(18*a*cos(d*x + c)^4 - 27*a*cos(d*x
+ c)^2 + 11*a)*sin(d*x + c) - 192*a)/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*
x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**7*csc(d*x+c)**8*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.79 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {420 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac {420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/420*(420*a*log(sin(d*x + c)) - (420*a*sin(d*x + c)^6 - 630*a*sin(d*x + c)^5 - 420*a*sin(d*x + c)^4 + 315*a*
sin(d*x + c)^3 + 252*a*sin(d*x + c)^2 - 70*a*sin(d*x + c) - 60*a)/sin(d*x + c)^7)/d

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.89 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {420 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {1089 \, a \sin \left (d x + c\right )^{7} + 420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \]

[In]

integrate(cos(d*x+c)^7*csc(d*x+c)^8*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/420*(420*a*log(abs(sin(d*x + c))) - (1089*a*sin(d*x + c)^7 + 420*a*sin(d*x + c)^6 - 630*a*sin(d*x + c)^5 -
420*a*sin(d*x + c)^4 + 315*a*sin(d*x + c)^3 + 252*a*sin(d*x + c)^2 - 70*a*sin(d*x + c) - 60*a)/sin(d*x + c)^7)
/d

Mupad [B] (verification not implemented)

Time = 10.34 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.27 \[ \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {35\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {35\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {29\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]

[In]

int((cos(c + d*x)^7*(a + a*sin(c + d*x)))/sin(c + d*x)^8,x)

[Out]

(35*a*cot(c/2 + (d*x)/2))/(128*d) + (35*a*tan(c/2 + (d*x)/2))/(128*d) + (a*log(tan(c/2 + (d*x)/2)^2 + 1))/d -
(29*a*cot(c/2 + (d*x)/2)^2)/(128*d) - (7*a*cot(c/2 + (d*x)/2)^3)/(128*d) + (a*cot(c/2 + (d*x)/2)^4)/(32*d) + (
7*a*cot(c/2 + (d*x)/2)^5)/(640*d) - (a*cot(c/2 + (d*x)/2)^6)/(384*d) - (a*cot(c/2 + (d*x)/2)^7)/(896*d) - (29*
a*tan(c/2 + (d*x)/2)^2)/(128*d) - (7*a*tan(c/2 + (d*x)/2)^3)/(128*d) + (a*tan(c/2 + (d*x)/2)^4)/(32*d) + (7*a*
tan(c/2 + (d*x)/2)^5)/(640*d) - (a*tan(c/2 + (d*x)/2)^6)/(384*d) - (a*tan(c/2 + (d*x)/2)^7)/(896*d) - (a*log(t
an(c/2 + (d*x)/2)))/d