\(\int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx\) [431]

Optimal result

Integrand size = 29, antiderivative size = 138 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {11 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {4 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d} \]

[Out]

-11/16*arctanh(cos(d*x+c))/a^2/d+2*cot(d*x+c)/a^2/d+4/3*cot(d*x+c)^3/a^2/d+2/5*cot(d*x+c)^5/a^2/d-11/16*cot(d*
x+c)*csc(d*x+c)/a^2/d-11/24*cot(d*x+c)*csc(d*x+c)^3/a^2/d-1/6*cot(d*x+c)*csc(d*x+c)^5/a^2/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2948, 2836, 3853, 3855, 3852} \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {11 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}+\frac {4 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{16 a^2 d} \]

[In]

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

[Out]

(-11*ArcTanh[Cos[c + d*x]])/(16*a^2*d) + (2*Cot[c + d*x])/(a^2*d) + (4*Cot[c + d*x]^3)/(3*a^2*d) + (2*Cot[c +
d*x]^5)/(5*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x])/(16*a^2*d) - (11*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^2*d) -
(Cot[c + d*x]*Csc[c + d*x]^5)/(6*a^2*d)

Rule 2836

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int[Expan
dTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] &
& IGtQ[m, 0] && RationalQ[n]

Rule 2948

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

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

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 \csc ^7(c+d x) (a-a \sin (c+d x))^2 \, dx}{a^4} \\ & = \frac {\int \left (a^2 \csc ^5(c+d x)-2 a^2 \csc ^6(c+d x)+a^2 \csc ^7(c+d x)\right ) \, dx}{a^4} \\ & = \frac {\int \csc ^5(c+d x) \, dx}{a^2}+\frac {\int \csc ^7(c+d x) \, dx}{a^2}-\frac {2 \int \csc ^6(c+d x) \, dx}{a^2} \\ & = -\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a^2}+\frac {5 \int \csc ^5(c+d x) \, dx}{6 a^2}+\frac {2 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{a^2 d} \\ & = \frac {2 \cot (c+d x)}{a^2 d}+\frac {4 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a^2 d}-\frac {11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {3 \int \csc (c+d x) \, dx}{8 a^2}+\frac {5 \int \csc ^3(c+d x) \, dx}{8 a^2} \\ & = -\frac {3 \text {arctanh}(\cos (c+d x))}{8 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {4 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}+\frac {5 \int \csc (c+d x) \, dx}{16 a^2} \\ & = -\frac {11 \text {arctanh}(\cos (c+d x))}{16 a^2 d}+\frac {2 \cot (c+d x)}{a^2 d}+\frac {4 \cot ^3(c+d x)}{3 a^2 d}+\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {11 \cot (c+d x) \csc (c+d x)}{16 a^2 d}-\frac {11 \cot (c+d x) \csc ^3(c+d x)}{24 a^2 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.51 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.66 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\csc ^6(c+d x) \left (-2820 \cos (c+d x)+1870 \cos (3 (c+d x))-330 \cos (5 (c+d x))-1650 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2475 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-990 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+165 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1650 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2475 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+990 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-165 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+3840 \sin (2 (c+d x))-1536 \sin (4 (c+d x))+256 \sin (6 (c+d x))\right )}{7680 a^2 d} \]

[In]

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

[Out]

(Csc[c + d*x]^6*(-2820*Cos[c + d*x] + 1870*Cos[3*(c + d*x)] - 330*Cos[5*(c + d*x)] - 1650*Log[Cos[(c + d*x)/2]
] + 2475*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 990*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 165*Cos[6*(c +
d*x)]*Log[Cos[(c + d*x)/2]] + 1650*Log[Sin[(c + d*x)/2]] - 2475*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 990*C
os[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 165*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 3840*Sin[2*(c + d*x)] - 1
536*Sin[4*(c + d*x)] + 256*Sin[6*(c + d*x)]))/(7680*a^2*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.53 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.22

method	result	size
risch	\(\frac {165 \,{\mathrm e}^{11 i \left (d x +c \right )}-935 \,{\mathrm e}^{9 i \left (d x +c \right )}+2560 i {\mathrm e}^{6 i \left (d x +c \right )}+1410 \,{\mathrm e}^{7 i \left (d x +c \right )}-3840 i {\mathrm e}^{4 i \left (d x +c \right )}+1410 \,{\mathrm e}^{5 i \left (d x +c \right )}+1536 i {\mathrm e}^{2 i \left (d x +c \right )}-935 \,{\mathrm e}^{3 i \left (d x +c \right )}-256 i+165 \,{\mathrm e}^{i \left (d x +c \right )}}{120 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{2}}+\frac {11 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{2}}\)	\(168\)
parallelrisch	\(\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+75 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-75 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-200 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+200 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+465 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-465 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1200 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1320 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1200 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d \,a^{2}}\)	\(174\)
derivativedivides	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {31 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {40}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {20}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {31}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{64 d \,a^{2}}\)	\(176\)
default	\(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {31 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+44 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {5}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {40}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {20}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {31}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {4}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{64 d \,a^{2}}\)	\(176\)
norman	\(\frac {-\frac {1}{384 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{640 d a}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}-\frac {11 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}+\frac {11 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}-\frac {11 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}-\frac {7 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}+\frac {3 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{320 d a}-\frac {3 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{640 d a}+\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}+\frac {11 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a}+\frac {305 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}+\frac {481 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {11 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{2}}\)	\(321\)

[In]

int(cos(d*x+c)^4*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/120*(165*exp(11*I*(d*x+c))-935*exp(9*I*(d*x+c))+2560*I*exp(6*I*(d*x+c))+1410*exp(7*I*(d*x+c))-3840*I*exp(4*I
*(d*x+c))+1410*exp(5*I*(d*x+c))+1536*I*exp(2*I*(d*x+c))-935*exp(3*I*(d*x+c))-256*I+165*exp(I*(d*x+c)))/a^2/d/(
exp(2*I*(d*x+c))-1)^6-11/16/d/a^2*ln(exp(I*(d*x+c))+1)+11/16/d/a^2*ln(exp(I*(d*x+c))-1)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.48 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {330 \, \cos \left (d x + c\right )^{5} - 880 \, \cos \left (d x + c\right )^{3} - 165 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 165 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 630 \, \cos \left (d x + c\right )}{480 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d\right )}} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/480*(330*cos(d*x + c)^5 - 880*cos(d*x + c)^3 - 165*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1
)*log(1/2*cos(d*x + c) + 1/2) + 165*(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*
x + c) + 1/2) - 64*(8*cos(d*x + c)^5 - 20*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c) + 630*cos(d*x + c))/(
a^2*d*cos(d*x + c)^6 - 3*a^2*d*cos(d*x + c)^4 + 3*a^2*d*cos(d*x + c)^2 - a^2*d)

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**4*csc(d*x+c)**7/(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (126) = 252\).

Time = 0.22 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {1200 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {465 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {75 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {24 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{2}} - \frac {1320 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {{\left (\frac {24 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {75 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {200 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {465 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1200 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{2} \sin \left (d x + c\right )^{6}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/1920*((1200*sin(d*x + c)/(cos(d*x + c) + 1) - 465*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 200*sin(d*x + c)^3/
(cos(d*x + c) + 1)^3 - 75*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 24*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5*sin
(d*x + c)^6/(cos(d*x + c) + 1)^6)/a^2 - 1320*log(sin(d*x + c)/(cos(d*x + c) + 1))/a^2 - (24*sin(d*x + c)/(cos(
d*x + c) + 1) - 75*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 200*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 465*sin(d*x
+ c)^4/(cos(d*x + c) + 1)^4 + 1200*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos(d*x + c) + 1)^6/(a^2*sin(d*x
+ c)^6))/d

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.56 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {1320 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac {3234 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 465 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} + \frac {5 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 24 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 75 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 200 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 465 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1200 \, a^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{12}}}{1920 \, d} \]

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^7/(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/1920*(1320*log(abs(tan(1/2*d*x + 1/2*c)))/a^2 - (3234*tan(1/2*d*x + 1/2*c)^6 - 1200*tan(1/2*d*x + 1/2*c)^5 +
465*tan(1/2*d*x + 1/2*c)^4 - 200*tan(1/2*d*x + 1/2*c)^3 + 75*tan(1/2*d*x + 1/2*c)^2 - 24*tan(1/2*d*x + 1/2*c)
+ 5)/(a^2*tan(1/2*d*x + 1/2*c)^6) + (5*a^10*tan(1/2*d*x + 1/2*c)^6 - 24*a^10*tan(1/2*d*x + 1/2*c)^5 + 75*a^10
*tan(1/2*d*x + 1/2*c)^4 - 200*a^10*tan(1/2*d*x + 1/2*c)^3 + 465*a^10*tan(1/2*d*x + 1/2*c)^2 - 1200*a^10*tan(1/
2*d*x + 1/2*c))/a^12)/d

Mupad [B] (verification not implemented)

Time = 10.91 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.46 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-24\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+75\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-1200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-465\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+200\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-75\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1320\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]

[In]

int(cos(c + d*x)^4/(sin(c + d*x)^7*(a + a*sin(c + d*x))^2),x)

[Out]

(5*sin(c/2 + (d*x)/2)^12 - 5*cos(c/2 + (d*x)/2)^12 - 24*cos(c/2 + (d*x)/2)*sin(c/2 + (d*x)/2)^11 + 24*cos(c/2
+ (d*x)/2)^11*sin(c/2 + (d*x)/2) + 75*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 200*cos(c/2 + (d*x)/2)^3*si
n(c/2 + (d*x)/2)^9 + 465*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 1200*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)
/2)^7 + 1200*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 465*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 200*c
os(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 - 75*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 1320*log(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6)/(1920*a^2*d*cos(c/2 + (d*x)/2)^6*sin(c
/2 + (d*x)/2)^6)