\(\int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [77]

Optimal result

Integrand size = 21, antiderivative size = 300 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\left (a^4+20 a^2 b^2+35 b^4\right ) \cot (c+d x)}{a^8 d}+\frac {2 b \left (2 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^7 d}-\frac {2 \left (a^2+5 b^2\right ) \cot ^3(c+d x)}{3 a^6 d}+\frac {b \cot ^4(c+d x)}{a^5 d}-\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (\tan (c+d x))}{a^9 d}+\frac {4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (a+b \tan (c+d x))}{a^9 d}-\frac {b \left (a^2+b^2\right )^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac {b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right )}{a^8 d (a+b \tan (c+d x))} \]

[Out]

-(a^4+20*a^2*b^2+35*b^4)*cot(d*x+c)/a^8/d+2*b*(2*a^2+5*b^2)*cot(d*x+c)^2/a^7/d-2/3*(a^2+5*b^2)*cot(d*x+c)^3/a^
6/d+b*cot(d*x+c)^4/a^5/d-1/5*cot(d*x+c)^5/a^4/d-4*b*(a^4+10*a^2*b^2+14*b^4)*ln(tan(d*x+c))/a^9/d+4*b*(a^4+10*a
^2*b^2+14*b^4)*ln(a+b*tan(d*x+c))/a^9/d-1/3*b*(a^2+b^2)^2/a^6/d/(a+b*tan(d*x+c))^3-b*(a^2+b^2)*(a^2+3*b^2)/a^7
/d/(a+b*tan(d*x+c))^2-b*(3*a^4+20*a^2*b^2+21*b^4)/a^8/d/(a+b*tan(d*x+c))

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3597, 908} \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {b \cot ^4(c+d x)}{a^5 d}-\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))^2}+\frac {2 b \left (2 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^7 d}-\frac {b \left (a^2+b^2\right )^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac {2 \left (a^2+5 b^2\right ) \cot ^3(c+d x)}{3 a^6 d}-\frac {4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (\tan (c+d x))}{a^9 d}+\frac {4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (a+b \tan (c+d x))}{a^9 d}-\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right )}{a^8 d (a+b \tan (c+d x))}-\frac {\left (a^4+20 a^2 b^2+35 b^4\right ) \cot (c+d x)}{a^8 d} \]

[In]

Int[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^4,x]

[Out]

-(((a^4 + 20*a^2*b^2 + 35*b^4)*Cot[c + d*x])/(a^8*d)) + (2*b*(2*a^2 + 5*b^2)*Cot[c + d*x]^2)/(a^7*d) - (2*(a^2
+ 5*b^2)*Cot[c + d*x]^3)/(3*a^6*d) + (b*Cot[c + d*x]^4)/(a^5*d) - Cot[c + d*x]^5/(5*a^4*d) - (4*b*(a^4 + 10*a
^2*b^2 + 14*b^4)*Log[Tan[c + d*x]])/(a^9*d) + (4*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[a + b*Tan[c + d*x]])/(a^9*d
) - (b*(a^2 + b^2)^2)/(3*a^6*d*(a + b*Tan[c + d*x])^3) - (b*(a^2 + b^2)*(a^2 + 3*b^2))/(a^7*d*(a + b*Tan[c + d
*x])^2) - (b*(3*a^4 + 20*a^2*b^2 + 21*b^4))/(a^8*d*(a + b*Tan[c + d*x]))

Rule 908

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 3597

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {\left (b^2+x^2\right )^2}{x^6 (a+x)^4} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {b \text {Subst}\left (\int \left (\frac {b^4}{a^4 x^6}-\frac {4 b^4}{a^5 x^5}+\frac {2 b^2 \left (a^2+5 b^2\right )}{a^6 x^4}-\frac {4 \left (2 a^2 b^2+5 b^4\right )}{a^7 x^3}+\frac {a^4+20 a^2 b^2+35 b^4}{a^8 x^2}-\frac {4 \left (a^4+10 a^2 b^2+14 b^4\right )}{a^9 x}+\frac {\left (a^2+b^2\right )^2}{a^6 (a+x)^4}+\frac {2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 (a+x)^3}+\frac {3 a^4+20 a^2 b^2+21 b^4}{a^8 (a+x)^2}+\frac {4 \left (a^4+10 a^2 b^2+14 b^4\right )}{a^9 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\left (a^4+20 a^2 b^2+35 b^4\right ) \cot (c+d x)}{a^8 d}+\frac {2 b \left (2 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^7 d}-\frac {2 \left (a^2+5 b^2\right ) \cot ^3(c+d x)}{3 a^6 d}+\frac {b \cot ^4(c+d x)}{a^5 d}-\frac {\cot ^5(c+d x)}{5 a^4 d}-\frac {4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (\tan (c+d x))}{a^9 d}+\frac {4 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (a+b \tan (c+d x))}{a^9 d}-\frac {b \left (a^2+b^2\right )^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac {b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))^2}-\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right )}{a^8 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(673\) vs. \(2(300)=600\).

Time = 3.10 (sec) , antiderivative size = 673, normalized size of antiderivative = 2.24 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (-7680 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3+7680 b \left (a^4+10 a^2 b^2+14 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3+\csc ^5(c+d x) \left (-200 a^8+380 a^6 b^2+3070 a^4 b^4+11375 a^2 b^6+11025 b^8-4 \left (52 a^8+194 a^6 b^2+1510 a^4 b^4+5705 a^2 b^6+4410 b^8\right ) \cos (2 (c+d x))+4 \left (4 a^8-16 a^6 b^2+1010 a^4 b^4+4585 a^2 b^6+2205 b^8\right ) \cos (4 (c+d x))+16 a^8 \cos (6 (c+d x))+776 a^6 b^2 \cos (6 (c+d x))-1000 a^4 b^4 \cos (6 (c+d x))-8540 a^2 b^6 \cos (6 (c+d x))-2520 b^8 \cos (6 (c+d x))-8 a^8 \cos (8 (c+d x))-316 a^6 b^2 \cos (8 (c+d x))-70 a^4 b^4 \cos (8 (c+d x))+1645 a^2 b^6 \cos (8 (c+d x))+315 b^8 \cos (8 (c+d x))+264 a^7 b \sin (2 (c+d x))+372 a^5 b^3 \sin (2 (c+d x))+4830 a^3 b^5 \sin (2 (c+d x))+1470 a b^7 \sin (2 (c+d x))+144 a^7 b \sin (4 (c+d x))-2476 a^5 b^3 \sin (4 (c+d x))-9730 a^3 b^5 \sin (4 (c+d x))-1470 a b^7 \sin (4 (c+d x))-24 a^7 b \sin (6 (c+d x))+2756 a^5 b^3 \sin (6 (c+d x))+7670 a^3 b^5 \sin (6 (c+d x))+630 a b^7 \sin (6 (c+d x))-24 a^7 b \sin (8 (c+d x))-922 a^5 b^3 \sin (8 (c+d x))-2095 a^3 b^5 \sin (8 (c+d x))-105 a b^7 \sin (8 (c+d x))\right )\right )}{1920 a^9 d (a+b \tan (c+d x))^4} \]

[In]

Integrate[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^4,x]

[Out]

(Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-7680*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[Sin[c + d*x]]*(a*Co
s[c + d*x] + b*Sin[c + d*x])^3 + 7680*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*(a*Co
s[c + d*x] + b*Sin[c + d*x])^3 + Csc[c + d*x]^5*(-200*a^8 + 380*a^6*b^2 + 3070*a^4*b^4 + 11375*a^2*b^6 + 11025
*b^8 - 4*(52*a^8 + 194*a^6*b^2 + 1510*a^4*b^4 + 5705*a^2*b^6 + 4410*b^8)*Cos[2*(c + d*x)] + 4*(4*a^8 - 16*a^6*
b^2 + 1010*a^4*b^4 + 4585*a^2*b^6 + 2205*b^8)*Cos[4*(c + d*x)] + 16*a^8*Cos[6*(c + d*x)] + 776*a^6*b^2*Cos[6*(
c + d*x)] - 1000*a^4*b^4*Cos[6*(c + d*x)] - 8540*a^2*b^6*Cos[6*(c + d*x)] - 2520*b^8*Cos[6*(c + d*x)] - 8*a^8*
Cos[8*(c + d*x)] - 316*a^6*b^2*Cos[8*(c + d*x)] - 70*a^4*b^4*Cos[8*(c + d*x)] + 1645*a^2*b^6*Cos[8*(c + d*x)]
+ 315*b^8*Cos[8*(c + d*x)] + 264*a^7*b*Sin[2*(c + d*x)] + 372*a^5*b^3*Sin[2*(c + d*x)] + 4830*a^3*b^5*Sin[2*(c
+ d*x)] + 1470*a*b^7*Sin[2*(c + d*x)] + 144*a^7*b*Sin[4*(c + d*x)] - 2476*a^5*b^3*Sin[4*(c + d*x)] - 9730*a^3
*b^5*Sin[4*(c + d*x)] - 1470*a*b^7*Sin[4*(c + d*x)] - 24*a^7*b*Sin[6*(c + d*x)] + 2756*a^5*b^3*Sin[6*(c + d*x)
] + 7670*a^3*b^5*Sin[6*(c + d*x)] + 630*a*b^7*Sin[6*(c + d*x)] - 24*a^7*b*Sin[8*(c + d*x)] - 922*a^5*b^3*Sin[8
*(c + d*x)] - 2095*a^3*b^5*Sin[8*(c + d*x)] - 105*a*b^7*Sin[8*(c + d*x)])))/(1920*a^9*d*(a + b*Tan[c + d*x])^4
)

Maple [A] (verified)

Time = 7.96 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {b \left (3 a^{4}+20 a^{2} b^{2}+21 b^{4}\right )}{a^{8} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b}{3 a^{6} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (a^{4}+4 a^{2} b^{2}+3 b^{4}\right )}{a^{7} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b \left (a^{4}+10 a^{2} b^{2}+14 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{9}}-\frac {1}{5 a^{4} \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+10 b^{2}}{3 a^{6} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+20 a^{2} b^{2}+35 b^{4}}{a^{8} \tan \left (d x +c \right )}+\frac {b}{a^{5} \tan \left (d x +c \right )^{4}}+\frac {2 b \left (2 a^{2}+5 b^{2}\right )}{a^{7} \tan \left (d x +c \right )^{2}}-\frac {4 b \left (a^{4}+10 a^{2} b^{2}+14 b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{9}}}{d}\)	\(280\)
default	\(\frac {-\frac {b \left (3 a^{4}+20 a^{2} b^{2}+21 b^{4}\right )}{a^{8} \left (a +b \tan \left (d x +c \right )\right )}-\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b}{3 a^{6} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {b \left (a^{4}+4 a^{2} b^{2}+3 b^{4}\right )}{a^{7} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b \left (a^{4}+10 a^{2} b^{2}+14 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{9}}-\frac {1}{5 a^{4} \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+10 b^{2}}{3 a^{6} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+20 a^{2} b^{2}+35 b^{4}}{a^{8} \tan \left (d x +c \right )}+\frac {b}{a^{5} \tan \left (d x +c \right )^{4}}+\frac {2 b \left (2 a^{2}+5 b^{2}\right )}{a^{7} \tan \left (d x +c \right )^{2}}-\frac {4 b \left (a^{4}+10 a^{2} b^{2}+14 b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{9}}}{d}\)	\(280\)
risch	\(\text {Expression too large to display}\)	\(1317\)

[In]

int(csc(d*x+c)^6/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-b*(3*a^4+20*a^2*b^2+21*b^4)/a^8/(a+b*tan(d*x+c))-1/3*(a^4+2*a^2*b^2+b^4)*b/a^6/(a+b*tan(d*x+c))^3-b*(a^4
+4*a^2*b^2+3*b^4)/a^7/(a+b*tan(d*x+c))^2+4*b*(a^4+10*a^2*b^2+14*b^4)/a^9*ln(a+b*tan(d*x+c))-1/5/a^4/tan(d*x+c)
^5-1/3*(2*a^2+10*b^2)/a^6/tan(d*x+c)^3-(a^4+20*a^2*b^2+35*b^4)/a^8/tan(d*x+c)+1/a^5*b/tan(d*x+c)^4+2*b*(2*a^2+
5*b^2)/a^7/tan(d*x+c)^2-4*b*(a^4+10*a^2*b^2+14*b^4)/a^9*ln(tan(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1536 vs. \(2 (294) = 588\).

Time = 0.37 (sec) , antiderivative size = 1536, normalized size of antiderivative = 5.12 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/15*(110*a^6*b^4 + 510*a^4*b^6 + 420*a^2*b^8 - 4*(2*a^10 + 81*a^8*b^2 + 29*a^6*b^4 - 660*a^4*b^6 - 630*a^2*b
^8)*cos(d*x + c)^8 + 2*(10*a^10 + 423*a^8*b^2 - 47*a^6*b^4 - 4320*a^4*b^6 - 3990*a^2*b^8)*cos(d*x + c)^6 - 15*
(a^10 + 47*a^8*b^2 - 44*a^6*b^4 - 658*a^4*b^6 - 588*a^2*b^8)*cos(d*x + c)^4 + 20*(9*a^8*b^2 - 28*a^6*b^4 - 219
*a^4*b^6 - 189*a^2*b^8)*cos(d*x + c)^2 + 30*(a^6*b^4 + 11*a^4*b^6 + 24*a^2*b^8 + 14*b^10 - (3*a^8*b^2 + 32*a^6
*b^4 + 61*a^4*b^6 + 18*a^2*b^8 - 14*b^10)*cos(d*x + c)^8 + (9*a^8*b^2 + 95*a^6*b^4 + 172*a^4*b^6 + 30*a^2*b^8
- 56*b^10)*cos(d*x + c)^6 - 3*(3*a^8*b^2 + 31*a^6*b^4 + 50*a^4*b^6 - 6*a^2*b^8 - 28*b^10)*cos(d*x + c)^4 + (3*
a^8*b^2 + 29*a^6*b^4 + 28*a^4*b^6 - 54*a^2*b^8 - 56*b^10)*cos(d*x + c)^2 + ((a^9*b + 8*a^7*b^3 - 9*a^5*b^5 - 5
8*a^3*b^7 - 42*a*b^9)*cos(d*x + c)^7 - (2*a^9*b + 13*a^7*b^3 - 51*a^5*b^5 - 188*a^3*b^7 - 126*a*b^9)*cos(d*x +
c)^5 + (a^9*b + 2*a^7*b^3 - 75*a^5*b^5 - 202*a^3*b^7 - 126*a*b^9)*cos(d*x + c)^3 + 3*(a^7*b^3 + 11*a^5*b^5 +
24*a^3*b^7 + 14*a*b^9)*cos(d*x + c))*sin(d*x + c))*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x +
c)^2 + b^2) - 30*(a^6*b^4 + 11*a^4*b^6 + 24*a^2*b^8 + 14*b^10 - (3*a^8*b^2 + 32*a^6*b^4 + 61*a^4*b^6 + 18*a^2
*b^8 - 14*b^10)*cos(d*x + c)^8 + (9*a^8*b^2 + 95*a^6*b^4 + 172*a^4*b^6 + 30*a^2*b^8 - 56*b^10)*cos(d*x + c)^6
- 3*(3*a^8*b^2 + 31*a^6*b^4 + 50*a^4*b^6 - 6*a^2*b^8 - 28*b^10)*cos(d*x + c)^4 + (3*a^8*b^2 + 29*a^6*b^4 + 28*
a^4*b^6 - 54*a^2*b^8 - 56*b^10)*cos(d*x + c)^2 + ((a^9*b + 8*a^7*b^3 - 9*a^5*b^5 - 58*a^3*b^7 - 42*a*b^9)*cos(
d*x + c)^7 - (2*a^9*b + 13*a^7*b^3 - 51*a^5*b^5 - 188*a^3*b^7 - 126*a*b^9)*cos(d*x + c)^5 + (a^9*b + 2*a^7*b^3
- 75*a^5*b^5 - 202*a^3*b^7 - 126*a*b^9)*cos(d*x + c)^3 + 3*(a^7*b^3 + 11*a^5*b^5 + 24*a^3*b^7 + 14*a*b^9)*cos
(d*x + c))*sin(d*x + c))*log(-1/4*cos(d*x + c)^2 + 1/4) - 2*(2*(6*a^9*b + 259*a^7*b^3 + 783*a^5*b^5 + 340*a^3*
b^7 - 210*a*b^9)*cos(d*x + c)^7 - (15*a^9*b + 1141*a^7*b^3 + 3546*a^5*b^5 + 1270*a^3*b^7 - 1260*a*b^9)*cos(d*x
+ c)^5 + 5*(151*a^7*b^3 + 483*a^5*b^5 + 100*a^3*b^7 - 252*a*b^9)*cos(d*x + c)^3 - 15*(9*a^7*b^3 + 29*a^5*b^5
- 6*a^3*b^7 - 28*a*b^9)*cos(d*x + c))*sin(d*x + c))/((3*a^13*b + 2*a^11*b^3 - a^9*b^5)*d*cos(d*x + c)^8 - (9*a
^13*b + 5*a^11*b^3 - 4*a^9*b^5)*d*cos(d*x + c)^6 + 3*(3*a^13*b + a^11*b^3 - 2*a^9*b^5)*d*cos(d*x + c)^4 - (3*a
^13*b - a^11*b^3 - 4*a^9*b^5)*d*cos(d*x + c)^2 - (a^11*b^3 + a^9*b^5)*d - ((a^14 - 2*a^12*b^2 - 3*a^10*b^4)*d*
cos(d*x + c)^7 - (2*a^14 - 7*a^12*b^2 - 9*a^10*b^4)*d*cos(d*x + c)^5 + (a^14 - 8*a^12*b^2 - 9*a^10*b^4)*d*cos(
d*x + c)^3 + 3*(a^12*b^2 + a^10*b^4)*d*cos(d*x + c))*sin(d*x + c))

Sympy [F]

\[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\int \frac {\csc ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{4}}\, dx \]

[In]

integrate(csc(d*x+c)**6/(a+b*tan(d*x+c))**4,x)

[Out]

Integral(csc(c + d*x)**6/(a + b*tan(c + d*x))**4, x)

Maxima [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {6 \, a^{6} b \tan \left (d x + c\right ) - 60 \, {\left (a^{4} b^{3} + 10 \, a^{2} b^{5} + 14 \, b^{7}\right )} \tan \left (d x + c\right )^{7} - 3 \, a^{7} - 150 \, {\left (a^{5} b^{2} + 10 \, a^{3} b^{4} + 14 \, a b^{6}\right )} \tan \left (d x + c\right )^{6} - 110 \, {\left (a^{6} b + 10 \, a^{4} b^{3} + 14 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{5} - 15 \, {\left (a^{7} + 10 \, a^{5} b^{2} + 14 \, a^{3} b^{4}\right )} \tan \left (d x + c\right )^{4} + 6 \, {\left (5 \, a^{6} b + 7 \, a^{4} b^{3}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (5 \, a^{7} + 7 \, a^{5} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{8} b^{3} \tan \left (d x + c\right )^{8} + 3 \, a^{9} b^{2} \tan \left (d x + c\right )^{7} + 3 \, a^{10} b \tan \left (d x + c\right )^{6} + a^{11} \tan \left (d x + c\right )^{5}} + \frac {60 \, {\left (a^{4} b + 10 \, a^{2} b^{3} + 14 \, b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9}} - \frac {60 \, {\left (a^{4} b + 10 \, a^{2} b^{3} + 14 \, b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{9}}}{15 \, d} \]

[In]

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/15*((6*a^6*b*tan(d*x + c) - 60*(a^4*b^3 + 10*a^2*b^5 + 14*b^7)*tan(d*x + c)^7 - 3*a^7 - 150*(a^5*b^2 + 10*a^
3*b^4 + 14*a*b^6)*tan(d*x + c)^6 - 110*(a^6*b + 10*a^4*b^3 + 14*a^2*b^5)*tan(d*x + c)^5 - 15*(a^7 + 10*a^5*b^2
+ 14*a^3*b^4)*tan(d*x + c)^4 + 6*(5*a^6*b + 7*a^4*b^3)*tan(d*x + c)^3 - 2*(5*a^7 + 7*a^5*b^2)*tan(d*x + c)^2)
/(a^8*b^3*tan(d*x + c)^8 + 3*a^9*b^2*tan(d*x + c)^7 + 3*a^10*b*tan(d*x + c)^6 + a^11*tan(d*x + c)^5) + 60*(a^4
*b + 10*a^2*b^3 + 14*b^5)*log(b*tan(d*x + c) + a)/a^9 - 60*(a^4*b + 10*a^2*b^3 + 14*b^5)*log(tan(d*x + c))/a^9
)/d

Giac [A] (verification not implemented)

none

Time = 0.76 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.43 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {\frac {60 \, {\left (a^{4} b + 10 \, a^{2} b^{3} + 14 \, b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{9}} - \frac {60 \, {\left (a^{4} b^{2} + 10 \, a^{2} b^{4} + 14 \, b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b} + \frac {5 \, {\left (22 \, a^{4} b^{4} \tan \left (d x + c\right )^{3} + 220 \, a^{2} b^{6} \tan \left (d x + c\right )^{3} + 308 \, b^{8} \tan \left (d x + c\right )^{3} + 75 \, a^{5} b^{3} \tan \left (d x + c\right )^{2} + 720 \, a^{3} b^{5} \tan \left (d x + c\right )^{2} + 987 \, a b^{7} \tan \left (d x + c\right )^{2} + 87 \, a^{6} b^{2} \tan \left (d x + c\right ) + 792 \, a^{4} b^{4} \tan \left (d x + c\right ) + 1059 \, a^{2} b^{6} \tan \left (d x + c\right ) + 35 \, a^{7} b + 294 \, a^{5} b^{3} + 381 \, a^{3} b^{5}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{3} a^{9}} - \frac {137 \, a^{4} b \tan \left (d x + c\right )^{5} + 1370 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 1918 \, b^{5} \tan \left (d x + c\right )^{5} - 15 \, a^{5} \tan \left (d x + c\right )^{4} - 300 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 525 \, a b^{4} \tan \left (d x + c\right )^{4} + 60 \, a^{4} b \tan \left (d x + c\right )^{3} + 150 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 10 \, a^{5} \tan \left (d x + c\right )^{2} - 50 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 15 \, a^{4} b \tan \left (d x + c\right ) - 3 \, a^{5}}{a^{9} \tan \left (d x + c\right )^{5}}}{15 \, d} \]

[In]

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/15*(60*(a^4*b + 10*a^2*b^3 + 14*b^5)*log(abs(tan(d*x + c)))/a^9 - 60*(a^4*b^2 + 10*a^2*b^4 + 14*b^6)*log(ab
s(b*tan(d*x + c) + a))/(a^9*b) + 5*(22*a^4*b^4*tan(d*x + c)^3 + 220*a^2*b^6*tan(d*x + c)^3 + 308*b^8*tan(d*x +
c)^3 + 75*a^5*b^3*tan(d*x + c)^2 + 720*a^3*b^5*tan(d*x + c)^2 + 987*a*b^7*tan(d*x + c)^2 + 87*a^6*b^2*tan(d*x
+ c) + 792*a^4*b^4*tan(d*x + c) + 1059*a^2*b^6*tan(d*x + c) + 35*a^7*b + 294*a^5*b^3 + 381*a^3*b^5)/((b*tan(d
*x + c) + a)^3*a^9) - (137*a^4*b*tan(d*x + c)^5 + 1370*a^2*b^3*tan(d*x + c)^5 + 1918*b^5*tan(d*x + c)^5 - 15*a
^5*tan(d*x + c)^4 - 300*a^3*b^2*tan(d*x + c)^4 - 525*a*b^4*tan(d*x + c)^4 + 60*a^4*b*tan(d*x + c)^3 + 150*a^2*
b^3*tan(d*x + c)^3 - 10*a^5*tan(d*x + c)^2 - 50*a^3*b^2*tan(d*x + c)^2 + 15*a^4*b*tan(d*x + c) - 3*a^5)/(a^9*t
an(d*x + c)^5))/d

Mupad [B] (verification not implemented)

Time = 6.83 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.12 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {8\,b\,\mathrm {atanh}\left (\frac {4\,b\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^4+10\,a^2\,b^2+14\,b^4\right )}{a\,\left (4\,a^4\,b+40\,a^2\,b^3+56\,b^5\right )}\right )\,\left (a^4+10\,a^2\,b^2+14\,b^4\right )}{a^9\,d}-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^4+10\,a^2\,b^2+14\,b^4\right )}{a^5}+\frac {2\,{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (5\,a^2+7\,b^2\right )}{15\,a^3}-\frac {2\,b\,\mathrm {tan}\left (c+d\,x\right )}{5\,a^2}+\frac {22\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^4+10\,a^2\,b^2+14\,b^4\right )}{3\,a^6}+\frac {10\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (a^4+10\,a^2\,b^2+14\,b^4\right )}{a^7}+\frac {4\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (a^4+10\,a^2\,b^2+14\,b^4\right )}{a^8}-\frac {2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (5\,a^2+7\,b^2\right )}{5\,a^4}}{d\,\left (a^3\,{\mathrm {tan}\left (c+d\,x\right )}^5+3\,a^2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^6+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^7+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^8\right )} \]

[In]

int(1/(sin(c + d*x)^6*(a + b*tan(c + d*x))^4),x)

[Out]

(8*b*atanh((4*b*(a + 2*b*tan(c + d*x))*(a^4 + 14*b^4 + 10*a^2*b^2))/(a*(4*a^4*b + 56*b^5 + 40*a^2*b^3)))*(a^4
+ 14*b^4 + 10*a^2*b^2))/(a^9*d) - (1/(5*a) + (tan(c + d*x)^4*(a^4 + 14*b^4 + 10*a^2*b^2))/a^5 + (2*tan(c + d*x
)^2*(5*a^2 + 7*b^2))/(15*a^3) - (2*b*tan(c + d*x))/(5*a^2) + (22*b*tan(c + d*x)^5*(a^4 + 14*b^4 + 10*a^2*b^2))
/(3*a^6) + (10*b^2*tan(c + d*x)^6*(a^4 + 14*b^4 + 10*a^2*b^2))/a^7 + (4*b^3*tan(c + d*x)^7*(a^4 + 14*b^4 + 10*
a^2*b^2))/a^8 - (2*b*tan(c + d*x)^3*(5*a^2 + 7*b^2))/(5*a^4))/(d*(a^3*tan(c + d*x)^5 + b^3*tan(c + d*x)^8 + 3*
a^2*b*tan(c + d*x)^6 + 3*a*b^2*tan(c + d*x)^7))