\(\int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [72]

Optimal result

Integrand size = 21, antiderivative size = 265 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (a^4+12 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a^7 d}+\frac {b \left (3 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^6 d}-\frac {2 \left (a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^5 d}+\frac {3 b \cot ^4(c+d x)}{4 a^4 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (\tan (c+d x))}{a^8 d}+\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (a+b \tan (c+d x))}{a^8 d}-\frac {b \left (a^2+b^2\right )^2}{2 a^6 d (a+b \tan (c+d x))^2}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))} \] Output:

-(a^4+12*a^2*b^2+15*b^4)*cot(d*x+c)/a^7/d+b*(3*a^2+5*b^2)*cot(d*x+c)^2/a^6 
/d-2/3*(a^2+3*b^2)*cot(d*x+c)^3/a^5/d+3/4*b*cot(d*x+c)^4/a^4/d-1/5*cot(d*x 
+c)^5/a^3/d-b*(3*a^4+20*a^2*b^2+21*b^4)*ln(tan(d*x+c))/a^8/d+b*(3*a^4+20*a 
^2*b^2+21*b^4)*ln(a+b*tan(d*x+c))/a^8/d-1/2*b*(a^2+b^2)^2/a^6/d/(a+b*tan(d 
*x+c))^2-2*b*(a^2+b^2)*(a^2+3*b^2)/a^7/d/(a+b*tan(d*x+c))
 

Mathematica [A] (verified)

Time = 4.29 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.86 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\csc ^5(c+d x) \left (\sec ^2(c+d x) \left (\left (8 a^7+567 a^5 b^2+630 a^3 b^4-1215 a b^6\right ) \cos (3 (c+d x))-\left (24 a^7+619 a^5 b^2+630 a^3 b^4-675 a b^6\right ) \cos (5 (c+d x))+8 a^7 \cos (7 (c+d x))+187 a^5 b^2 \cos (7 (c+d x))+210 a^3 b^4 \cos (7 (c+d x))-135 a b^6 \cos (7 (c+d x))-126 a^6 b \sin (3 (c+d x))+1665 a^4 b^3 \sin (3 (c+d x))+4635 a^2 b^5 \sin (3 (c+d x))+1890 b^7 \sin (3 (c+d x))+10 a^6 b \sin (5 (c+d x))-1215 a^4 b^3 \sin (5 (c+d x))-2565 a^2 b^5 \sin (5 (c+d x))-630 b^7 \sin (5 (c+d x))+16 a^6 b \sin (7 (c+d x))+345 a^4 b^3 \sin (7 (c+d x))+585 a^2 b^5 \sin (7 (c+d x))+90 b^7 \sin (7 (c+d x))\right )+960 b \left (3 a^4+20 a^2 b^2+21 b^4\right ) (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x))) \sin ^5(c+d x) (a+b \tan (c+d x))^2+5 \sec (c+d x) \left (40 a^7-27 a^5 b^2-42 a^3 b^4+135 a b^6-3 b \left (8 a^6+89 a^4 b^2+345 a^2 b^4+210 b^6\right ) \tan (c+d x)\right )\right )}{960 a^8 d (a+b \tan (c+d x))^2} \] Input:

Integrate[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]
 

Output:

-1/960*(Csc[c + d*x]^5*(Sec[c + d*x]^2*((8*a^7 + 567*a^5*b^2 + 630*a^3*b^4 
 - 1215*a*b^6)*Cos[3*(c + d*x)] - (24*a^7 + 619*a^5*b^2 + 630*a^3*b^4 - 67 
5*a*b^6)*Cos[5*(c + d*x)] + 8*a^7*Cos[7*(c + d*x)] + 187*a^5*b^2*Cos[7*(c 
+ d*x)] + 210*a^3*b^4*Cos[7*(c + d*x)] - 135*a*b^6*Cos[7*(c + d*x)] - 126* 
a^6*b*Sin[3*(c + d*x)] + 1665*a^4*b^3*Sin[3*(c + d*x)] + 4635*a^2*b^5*Sin[ 
3*(c + d*x)] + 1890*b^7*Sin[3*(c + d*x)] + 10*a^6*b*Sin[5*(c + d*x)] - 121 
5*a^4*b^3*Sin[5*(c + d*x)] - 2565*a^2*b^5*Sin[5*(c + d*x)] - 630*b^7*Sin[5 
*(c + d*x)] + 16*a^6*b*Sin[7*(c + d*x)] + 345*a^4*b^3*Sin[7*(c + d*x)] + 5 
85*a^2*b^5*Sin[7*(c + d*x)] + 90*b^7*Sin[7*(c + d*x)]) + 960*b*(3*a^4 + 20 
*a^2*b^2 + 21*b^4)*(Log[Sin[c + d*x]] - Log[a*Cos[c + d*x] + b*Sin[c + d*x 
]])*Sin[c + d*x]^5*(a + b*Tan[c + d*x])^2 + 5*Sec[c + d*x]*(40*a^7 - 27*a^ 
5*b^2 - 42*a^3*b^4 + 135*a*b^6 - 3*b*(8*a^6 + 89*a^4*b^2 + 345*a^2*b^4 + 2 
10*b^6)*Tan[c + d*x])))/(a^8*d*(a + b*Tan[c + d*x])^2)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3999, 522, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sin (c+d x)^6 (a+b \tan (c+d x))^3}dx\)

\(\Big \downarrow \) 3999

\(\displaystyle \frac {b \int \frac {\cot ^6(c+d x) \left (\tan ^2(c+d x) b^2+b^2\right )^2}{b^6 (a+b \tan (c+d x))^3}d(b \tan (c+d x))}{d}\)

\(\Big \downarrow \) 522

\(\displaystyle \frac {b \int \left (\frac {\cot ^6(c+d x)}{a^3 b^2}-\frac {3 \cot ^5(c+d x)}{a^4 b}+\frac {2 \left (a^2+3 b^2\right ) \cot ^4(c+d x)}{a^5 b^2}-\frac {2 \left (5 b^4+3 a^2 b^2\right ) \cot ^3(c+d x)}{a^6 b^3}+\frac {\left (a^4+12 b^2 a^2+15 b^4\right ) \cot ^2(c+d x)}{a^7 b^2}+\frac {\left (-3 a^4-20 b^2 a^2-21 b^4\right ) \cot (c+d x)}{a^8 b}+\frac {3 a^4+20 b^2 a^2+21 b^4}{a^8 (a+b \tan (c+d x))}+\frac {2 \left (a^4+4 b^2 a^2+3 b^4\right )}{a^7 (a+b \tan (c+d x))^2}+\frac {\left (a^2+b^2\right )^2}{a^6 (a+b \tan (c+d x))^3}\right )d(b \tan (c+d x))}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {3 \cot ^4(c+d x)}{4 a^4}-\frac {\cot ^5(c+d x)}{5 a^3 b}-\frac {2 \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 (a+b \tan (c+d x))}-\frac {\left (a^2+b^2\right )^2}{2 a^6 (a+b \tan (c+d x))^2}+\frac {\left (3 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^6}-\frac {2 \left (a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^5 b}-\frac {\left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (b \tan (c+d x))}{a^8}+\frac {\left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (a+b \tan (c+d x))}{a^8}-\frac {\left (a^4+12 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a^7 b}\right )}{d}\)

Input:

Int[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]
 

Output:

(b*(-(((a^4 + 12*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(a^7*b)) + ((3*a^2 + 5*b^ 
2)*Cot[c + d*x]^2)/a^6 - (2*(a^2 + 3*b^2)*Cot[c + d*x]^3)/(3*a^5*b) + (3*C 
ot[c + d*x]^4)/(4*a^4) - Cot[c + d*x]^5/(5*a^3*b) - ((3*a^4 + 20*a^2*b^2 + 
 21*b^4)*Log[b*Tan[c + d*x]])/a^8 + ((3*a^4 + 20*a^2*b^2 + 21*b^4)*Log[a + 
 b*Tan[c + d*x]])/a^8 - (a^2 + b^2)^2/(2*a^6*(a + b*Tan[c + d*x])^2) - (2* 
(a^2 + b^2)*(a^2 + 3*b^2))/(a^7*(a + b*Tan[c + d*x]))))/d
 

Defintions of rubi rules used

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3999

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ 
), x_Symbol] :> Simp[b/f   Subst[Int[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), 
 x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]
 

Maple [A] (verified)

Time = 7.07 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {1}{5 a^{3} \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+6 b^{2}}{3 a^{5} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+12 b^{2} a^{2}+15 b^{4}}{a^{7} \tan \left (d x +c \right )}+\frac {3 b}{4 a^{4} \tan \left (d x +c \right )^{4}}+\frac {b \left (3 a^{2}+5 b^{2}\right )}{a^{6} \tan \left (d x +c \right )^{2}}-\frac {b \left (3 a^{4}+20 b^{2} a^{2}+21 b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{8}}+\frac {b \left (3 a^{4}+20 b^{2} a^{2}+21 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{8}}-\frac {\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b}{2 a^{6} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {2 b \left (a^{4}+4 b^{2} a^{2}+3 b^{4}\right )}{a^{7} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(246\)
default	\(\frac {-\frac {1}{5 a^{3} \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+6 b^{2}}{3 a^{5} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+12 b^{2} a^{2}+15 b^{4}}{a^{7} \tan \left (d x +c \right )}+\frac {3 b}{4 a^{4} \tan \left (d x +c \right )^{4}}+\frac {b \left (3 a^{2}+5 b^{2}\right )}{a^{6} \tan \left (d x +c \right )^{2}}-\frac {b \left (3 a^{4}+20 b^{2} a^{2}+21 b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{8}}+\frac {b \left (3 a^{4}+20 b^{2} a^{2}+21 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{8}}-\frac {\left (a^{4}+2 b^{2} a^{2}+b^{4}\right ) b}{2 a^{6} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {2 b \left (a^{4}+4 b^{2} a^{2}+3 b^{4}\right )}{a^{7} \left (a +b \tan \left (d x +c \right )\right )}}{d}\)	\(246\)
risch	\(-\frac {2 i \left (-315 b^{6}+187 a^{4} b^{2}+120 a^{2} b^{4}+8 a^{6}+120 a^{6} {\mathrm e}^{6 i \left (d x +c \right )}+8 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}+6300 b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-315 b^{6} {\mathrm e}^{12 i \left (d x +c \right )}-4725 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+1890 b^{6} {\mathrm e}^{10 i \left (d x +c \right )}+80 a^{6} {\mathrm e}^{8 i \left (d x +c \right )}-4725 b^{6} {\mathrm e}^{8 i \left (d x +c \right )}-24 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}+1890 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-1650 i a^{3} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-2835 i a \,b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-100 i a^{5} b \,{\mathrm e}^{6 i \left (d x +c \right )}-1530 i a^{3} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+9 i a^{5} b \,{\mathrm e}^{4 i \left (d x +c \right )}+2610 i a^{3} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-315 i a \,b^{5} {\mathrm e}^{12 i \left (d x +c \right )}+20 i a^{5} b \,{\mathrm e}^{8 i \left (d x +c \right )}+135 i a^{5} b \,{\mathrm e}^{10 i \left (d x +c \right )}-300 i a^{3} b^{3} {\mathrm e}^{12 i \left (d x +c \right )}+945 i a \,b^{5} {\mathrm e}^{10 i \left (d x +c \right )}-35 i a^{5} b \,{\mathrm e}^{2 i \left (d x +c \right )}+4725 i a \,b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-45 i a^{5} b \,{\mathrm e}^{12 i \left (d x +c \right )}-3150 i a \,b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+900 i a^{3} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}-420 i a^{3} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+297 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-45 a^{4} b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+4320 a^{2} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-574 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-300 a^{2} b^{4} {\mathrm e}^{12 i \left (d x +c \right )}+1800 a^{2} b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-1980 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+270 a^{4} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-4080 a^{2} b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-695 a^{4} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+560 a^{4} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+120 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+630 i a \,b^{5}+16 i a^{5} b +390 i a^{3} b^{3}\right )}{15 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} a^{7} d}-\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{4} d}-\frac {20 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{6} d}-\frac {21 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{8} d}+\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{4} d}+\frac {20 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{6} d}+\frac {21 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{8} d}\)	\(919\)

Input:

int(csc(d*x+c)^6/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/5/a^3/tan(d*x+c)^5-1/3*(2*a^2+6*b^2)/a^5/tan(d*x+c)^3-(a^4+12*a^2* 
b^2+15*b^4)/a^7/tan(d*x+c)+3/4/a^4*b/tan(d*x+c)^4+b*(3*a^2+5*b^2)/a^6/tan( 
d*x+c)^2-b*(3*a^4+20*a^2*b^2+21*b^4)/a^8*ln(tan(d*x+c))+b*(3*a^4+20*a^2*b^ 
2+21*b^4)/a^8*ln(a+b*tan(d*x+c))-1/2*(a^4+2*a^2*b^2+b^4)*b/a^6/(a+b*tan(d* 
x+c))^2-2*b*(a^4+4*a^2*b^2+3*b^4)/a^7/(a+b*tan(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1018 vs. \(2 (257) = 514\).

Time = 0.15 (sec) , antiderivative size = 1018, normalized size of antiderivative = 3.84 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
 

Output:

1/60*(4*(8*a^7 + 187*a^5*b^2 + 120*a^3*b^4 - 315*a*b^6)*cos(d*x + c)^7 - 4 
*(20*a^7 + 482*a^5*b^2 + 255*a^3*b^4 - 945*a*b^6)*cos(d*x + c)^5 + 10*(6*a 
^7 + 157*a^5*b^2 + 60*a^3*b^4 - 378*a*b^6)*cos(d*x + c)^3 - 30*(13*a^5*b^2 
 + 2*a^3*b^4 - 42*a*b^6)*cos(d*x + c) + 30*(2*(3*a^5*b^2 + 20*a^3*b^4 + 21 
*a*b^6)*cos(d*x + c)^7 - 6*(3*a^5*b^2 + 20*a^3*b^4 + 21*a*b^6)*cos(d*x + c 
)^5 + 6*(3*a^5*b^2 + 20*a^3*b^4 + 21*a*b^6)*cos(d*x + c)^3 - 2*(3*a^5*b^2 
+ 20*a^3*b^4 + 21*a*b^6)*cos(d*x + c) - (3*a^4*b^3 + 20*a^2*b^5 + 21*b^7 + 
 (3*a^6*b + 17*a^4*b^3 + a^2*b^5 - 21*b^7)*cos(d*x + c)^6 - (6*a^6*b + 31* 
a^4*b^3 - 18*a^2*b^5 - 63*b^7)*cos(d*x + c)^4 + (3*a^6*b + 11*a^4*b^3 - 39 
*a^2*b^5 - 63*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(2*a*b*cos(d*x + c)*si 
n(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) - 30*(2*(3*a^5*b^2 + 20*a^3 
*b^4 + 21*a*b^6)*cos(d*x + c)^7 - 6*(3*a^5*b^2 + 20*a^3*b^4 + 21*a*b^6)*co 
s(d*x + c)^5 + 6*(3*a^5*b^2 + 20*a^3*b^4 + 21*a*b^6)*cos(d*x + c)^3 - 2*(3 
*a^5*b^2 + 20*a^3*b^4 + 21*a*b^6)*cos(d*x + c) - (3*a^4*b^3 + 20*a^2*b^5 + 
 21*b^7 + (3*a^6*b + 17*a^4*b^3 + a^2*b^5 - 21*b^7)*cos(d*x + c)^6 - (6*a^ 
6*b + 31*a^4*b^3 - 18*a^2*b^5 - 63*b^7)*cos(d*x + c)^4 + (3*a^6*b + 11*a^4 
*b^3 - 39*a^2*b^5 - 63*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/4*cos(d*x 
 + c)^2 + 1/4) - (285*a^4*b^3 + 630*a^2*b^5 - 8*(8*a^6*b + 195*a^4*b^3 + 3 
15*a^2*b^5)*cos(d*x + c)^6 + 10*(7*a^6*b + 330*a^4*b^3 + 567*a^2*b^5)*cos( 
d*x + c)^4 + 15*(a^6*b - 135*a^4*b^3 - 252*a^2*b^5)*cos(d*x + c)^2)*sin...
 

Sympy [F]

\[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\csc ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(csc(d*x+c)**6/(a+b*tan(d*x+c))**3,x)
 

Output:

Integral(csc(c + d*x)**6/(a + b*tan(c + d*x))**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.06 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {21 \, a^{5} b \tan \left (d x + c\right ) - 60 \, {\left (3 \, a^{4} b^{2} + 20 \, a^{2} b^{4} + 21 \, b^{6}\right )} \tan \left (d x + c\right )^{6} - 12 \, a^{6} - 90 \, {\left (3 \, a^{5} b + 20 \, a^{3} b^{3} + 21 \, a b^{5}\right )} \tan \left (d x + c\right )^{5} - 20 \, {\left (3 \, a^{6} + 20 \, a^{4} b^{2} + 21 \, a^{2} b^{4}\right )} \tan \left (d x + c\right )^{4} + 5 \, {\left (20 \, a^{5} b + 21 \, a^{3} b^{3}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (20 \, a^{6} + 21 \, a^{4} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{7} b^{2} \tan \left (d x + c\right )^{7} + 2 \, a^{8} b \tan \left (d x + c\right )^{6} + a^{9} \tan \left (d x + c\right )^{5}} + \frac {60 \, {\left (3 \, a^{4} b + 20 \, a^{2} b^{3} + 21 \, b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8}} - \frac {60 \, {\left (3 \, a^{4} b + 20 \, a^{2} b^{3} + 21 \, b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{8}}}{60 \, d} \] Input:

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
 

Output:

1/60*((21*a^5*b*tan(d*x + c) - 60*(3*a^4*b^2 + 20*a^2*b^4 + 21*b^6)*tan(d* 
x + c)^6 - 12*a^6 - 90*(3*a^5*b + 20*a^3*b^3 + 21*a*b^5)*tan(d*x + c)^5 - 
20*(3*a^6 + 20*a^4*b^2 + 21*a^2*b^4)*tan(d*x + c)^4 + 5*(20*a^5*b + 21*a^3 
*b^3)*tan(d*x + c)^3 - 2*(20*a^6 + 21*a^4*b^2)*tan(d*x + c)^2)/(a^7*b^2*ta 
n(d*x + c)^7 + 2*a^8*b*tan(d*x + c)^6 + a^9*tan(d*x + c)^5) + 60*(3*a^4*b 
+ 20*a^2*b^3 + 21*b^5)*log(b*tan(d*x + c) + a)/a^8 - 60*(3*a^4*b + 20*a^2* 
b^3 + 21*b^5)*log(tan(d*x + c))/a^8)/d
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {{\left (3 \, a^{4} b + 20 \, a^{2} b^{3} + 21 \, b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{8} d} + \frac {{\left (3 \, a^{4} b^{2} + 20 \, a^{2} b^{4} + 21 \, b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b d} + \frac {21 \, a^{6} b \tan \left (d x + c\right ) - 12 \, a^{7} - 60 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \tan \left (d x + c\right )^{6} - 90 \, {\left (3 \, a^{6} b + 20 \, a^{4} b^{3} + 21 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{5} - 20 \, {\left (3 \, a^{7} + 20 \, a^{5} b^{2} + 21 \, a^{3} b^{4}\right )} \tan \left (d x + c\right )^{4} + 5 \, {\left (20 \, a^{6} b + 21 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (20 \, a^{7} + 21 \, a^{5} b^{2}\right )} \tan \left (d x + c\right )^{2}}{60 \, {\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{8} d \tan \left (d x + c\right )^{5}} \] Input:

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^3,x, algorithm="giac")
 

Output:

-(3*a^4*b + 20*a^2*b^3 + 21*b^5)*log(abs(tan(d*x + c)))/(a^8*d) + (3*a^4*b 
^2 + 20*a^2*b^4 + 21*b^6)*log(abs(b*tan(d*x + c) + a))/(a^8*b*d) + 1/60*(2 
1*a^6*b*tan(d*x + c) - 12*a^7 - 60*(3*a^5*b^2 + 20*a^3*b^4 + 21*a*b^6)*tan 
(d*x + c)^6 - 90*(3*a^6*b + 20*a^4*b^3 + 21*a^2*b^5)*tan(d*x + c)^5 - 20*( 
3*a^7 + 20*a^5*b^2 + 21*a^3*b^4)*tan(d*x + c)^4 + 5*(20*a^6*b + 21*a^4*b^3 
)*tan(d*x + c)^3 - 2*(20*a^7 + 21*a^5*b^2)*tan(d*x + c)^2)/((b*tan(d*x + c 
) + a)^2*a^8*d*tan(d*x + c)^5)
 

Mupad [B] (verification not implemented)

Time = 2.61 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{a\,\left (3\,a^4\,b+20\,a^2\,b^3+21\,b^5\right )}\right )\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{a^8\,d}-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{3\,a^5}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (20\,a^2+21\,b^2\right )}{30\,a^3}-\frac {7\,b\,\mathrm {tan}\left (c+d\,x\right )}{20\,a^2}+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{a^7}+\frac {3\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (3\,a^4+20\,a^2\,b^2+21\,b^4\right )}{2\,a^6}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (20\,a^2+21\,b^2\right )}{12\,a^4}}{d\,\left (a^2\,{\mathrm {tan}\left (c+d\,x\right )}^5+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^6+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^7\right )} \] Input:

int(1/(sin(c + d*x)^6*(a + b*tan(c + d*x))^3),x)
 

Output:

(2*b*atanh((b*(a + 2*b*tan(c + d*x))*(3*a^4 + 21*b^4 + 20*a^2*b^2))/(a*(3* 
a^4*b + 21*b^5 + 20*a^2*b^3)))*(3*a^4 + 21*b^4 + 20*a^2*b^2))/(a^8*d) - (1 
/(5*a) + (tan(c + d*x)^4*(3*a^4 + 21*b^4 + 20*a^2*b^2))/(3*a^5) + (tan(c + 
 d*x)^2*(20*a^2 + 21*b^2))/(30*a^3) - (7*b*tan(c + d*x))/(20*a^2) + (b^2*t 
an(c + d*x)^6*(3*a^4 + 21*b^4 + 20*a^2*b^2))/a^7 + (3*b*tan(c + d*x)^5*(3* 
a^4 + 21*b^4 + 20*a^2*b^2))/(2*a^6) - (b*tan(c + d*x)^3*(20*a^2 + 21*b^2)) 
/(12*a^4))/(d*(a^2*tan(c + d*x)^5 + b^2*tan(c + d*x)^7 + 2*a*b*tan(c + d*x 
)^6))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1176, normalized size of antiderivative = 4.44 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:

int(csc(d*x+c)^6/(a+b*tan(d*x+c))^3,x)
 

Output:

(11520*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)* 
sin(c + d*x)**6*a**5*b**3 + 76800*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 
 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**6*a**3*b**5 + 80640*cos(c + d*x)* 
log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**6*a*b* 
*7 - 11520*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**5*b**3 - 
76800*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**6*a**3*b**5 - 80640 
*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**6*a*b**7 + 174*cos(c + d 
*x)*sin(c + d*x)**6*a**7*b + 2696*cos(c + d*x)*sin(c + d*x)**6*a**5*b**3 + 
 3360*cos(c + d*x)*sin(c + d*x)**6*a**3*b**5 - 512*cos(c + d*x)*sin(c + d* 
x)**4*a**7*b - 10112*cos(c + d*x)*sin(c + d*x)**4*a**5*b**3 - 13440*cos(c 
+ d*x)*sin(c + d*x)**4*a**3*b**5 - 128*cos(c + d*x)*sin(c + d*x)**2*a**7*b 
 - 1344*cos(c + d*x)*sin(c + d*x)**2*a**5*b**3 - 384*cos(c + d*x)*a**7*b - 
 5760*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)** 
7*a**6*b**2 - 32640*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)* 
sin(c + d*x)**7*a**4*b**4 - 1920*log(tan((c + d*x)/2)**2*a - 2*tan((c + d* 
x)/2)*b - a)*sin(c + d*x)**7*a**2*b**6 + 40320*log(tan((c + d*x)/2)**2*a - 
 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**7*b**8 + 5760*log(tan((c + d*x)/2 
)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**5*a**6*b**2 + 38400*log(t 
an((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**5*a**4*b**4 
 + 40320*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + ...