\(\int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [66]

Optimal result

Integrand size = 21, antiderivative size = 219 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\left (a^2+b^2\right ) \left (a^2+5 b^2\right ) \cot (c+d x)}{a^6 d}+\frac {2 b \left (a^2+b^2\right ) \cot ^2(c+d x)}{a^5 d}-\frac {\left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^4 d}+\frac {b \cot ^4(c+d x)}{2 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}-\frac {b \left (a^2+b^2\right )^2}{a^6 d (a+b \tan (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 219, 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} \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {b \cot ^4(c+d x)}{2 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}-\frac {b \left (a^2+b^2\right )^2}{a^6 d (a+b \tan (c+d x))}-\frac {\left (a^2+b^2\right ) \left (a^2+5 b^2\right ) \cot (c+d x)}{a^6 d}+\frac {2 b \left (a^2+b^2\right ) \cot ^2(c+d x)}{a^5 d}-\frac {\left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^4 d} \]

[In]

Int[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]

[Out]

-(((a^2 + b^2)*(a^2 + 5*b^2)*Cot[c + d*x])/(a^6*d)) + (2*b*(a^2 + b^2)*Cot[c + d*x]^2)/(a^5*d) - ((2*a^2 + 3*b
^2)*Cot[c + d*x]^3)/(3*a^4*d) + (b*Cot[c + d*x]^4)/(2*a^3*d) - Cot[c + d*x]^5/(5*a^2*d) - (2*b*(a^2 + b^2)*(a^
2 + 3*b^2)*Log[Tan[c + d*x]])/(a^7*d) + (2*b*(a^2 + b^2)*(a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])/(a^7*d) - (b*(
a^2 + b^2)^2)/(a^6*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)^2} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {b \text {Subst}\left (\int \left (\frac {b^4}{a^2 x^6}-\frac {2 b^4}{a^3 x^5}+\frac {2 a^2 b^2+3 b^4}{a^4 x^4}-\frac {4 b^2 \left (a^2+b^2\right )}{a^5 x^3}+\frac {a^4+6 a^2 b^2+5 b^4}{a^6 x^2}-\frac {2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 x}+\frac {\left (a^2+b^2\right )^2}{a^6 (a+x)^2}+\frac {2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\left (a^2+b^2\right ) \left (a^2+5 b^2\right ) \cot (c+d x)}{a^6 d}+\frac {2 b \left (a^2+b^2\right ) \cot ^2(c+d x)}{a^5 d}-\frac {\left (2 a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^4 d}+\frac {b \cot ^4(c+d x)}{2 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^2 d}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (\tan (c+d x))}{a^7 d}+\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right ) \log (a+b \tan (c+d x))}{a^7 d}-\frac {b \left (a^2+b^2\right )^2}{a^6 d (a+b \tan (c+d x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(589\) vs. \(2(219)=438\).

Time = 7.71 (sec) , antiderivative size = 589, normalized size of antiderivative = 2.69 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\csc ^5(c+d x) \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{5 a^2 d (a+b \tan (c+d x))^2}+\frac {\left (-8 a^4 \cos (c+d x)-75 a^2 b^2 \cos (c+d x)-75 b^4 \cos (c+d x)\right ) \csc (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{15 a^6 d (a+b \tan (c+d x))^2}+\frac {b \left (a^2+2 b^2\right ) \csc ^2(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{a^5 d (a+b \tan (c+d x))^2}+\frac {\left (-4 a^2 \cos (c+d x)-15 b^2 \cos (c+d x)\right ) \csc ^3(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{15 a^4 d (a+b \tan (c+d x))^2}+\frac {b \csc ^4(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{2 a^3 d (a+b \tan (c+d x))^2}-\frac {2 \left (a^4 b+4 a^2 b^3+3 b^5\right ) \log (\sin (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{a^7 d (a+b \tan (c+d x))^2}+\frac {2 \left (a^4 b+4 a^2 b^3+3 b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{a^7 d (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left (a^4 b^2 \sin (c+d x)+2 a^2 b^4 \sin (c+d x)+b^6 \sin (c+d x)\right )}{a^7 d (a+b \tan (c+d x))^2} \]

[In]

Integrate[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]

[Out]

-1/5*(Csc[c + d*x]^5*Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^2*d*(a + b*Tan[c + d*x])^2) + ((-8*a
^4*Cos[c + d*x] - 75*a^2*b^2*Cos[c + d*x] - 75*b^4*Cos[c + d*x])*Csc[c + d*x]*Sec[c + d*x]^2*(a*Cos[c + d*x] +
b*Sin[c + d*x])^2)/(15*a^6*d*(a + b*Tan[c + d*x])^2) + (b*(a^2 + 2*b^2)*Csc[c + d*x]^2*Sec[c + d*x]^2*(a*Cos[
c + d*x] + b*Sin[c + d*x])^2)/(a^5*d*(a + b*Tan[c + d*x])^2) + ((-4*a^2*Cos[c + d*x] - 15*b^2*Cos[c + d*x])*Cs
c[c + d*x]^3*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(15*a^4*d*(a + b*Tan[c + d*x])^2) + (b*Csc[c
+ d*x]^4*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*a^3*d*(a + b*Tan[c + d*x])^2) - (2*(a^4*b + 4*
a^2*b^3 + 3*b^5)*Log[Sin[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^7*d*(a + b*Tan[c + d
*x])^2) + (2*(a^4*b + 4*a^2*b^3 + 3*b^5)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] +
b*Sin[c + d*x])^2)/(a^7*d*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(a^4*b^
2*Sin[c + d*x] + 2*a^2*b^4*Sin[c + d*x] + b^6*Sin[c + d*x]))/(a^7*d*(a + b*Tan[c + d*x])^2)

Maple [A] (verified)

Time = 2.48 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {-\frac {1}{5 a^{2} \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+3 b^{2}}{3 a^{4} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+6 a^{2} b^{2}+5 b^{4}}{a^{6} \tan \left (d x +c \right )}+\frac {b}{2 a^{3} \tan \left (d x +c \right )^{4}}+\frac {2 b \left (a^{2}+b^{2}\right )}{a^{5} \tan \left (d x +c \right )^{2}}-\frac {2 b \left (a^{4}+4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{7}}-\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b}{a^{6} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \left (a^{4}+4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{7}}}{d}\)	\(205\)
default	\(\frac {-\frac {1}{5 a^{2} \tan \left (d x +c \right )^{5}}-\frac {2 a^{2}+3 b^{2}}{3 a^{4} \tan \left (d x +c \right )^{3}}-\frac {a^{4}+6 a^{2} b^{2}+5 b^{4}}{a^{6} \tan \left (d x +c \right )}+\frac {b}{2 a^{3} \tan \left (d x +c \right )^{4}}+\frac {2 b \left (a^{2}+b^{2}\right )}{a^{5} \tan \left (d x +c \right )^{2}}-\frac {2 b \left (a^{4}+4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (\tan \left (d x +c \right )\right )}{a^{7}}-\frac {\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b}{a^{6} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \left (a^{4}+4 a^{2} b^{2}+3 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{a^{7}}}{d}\)	\(205\)
risch	\(-\frac {4 i \left (-45 a^{2} b^{3}+4 i a^{5}-45 b^{5}-4 a^{4} b +45 i a \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-120 i a^{3} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-180 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-180 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-180 i a \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+240 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+15 i a^{3} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+270 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+60 a^{2} b^{3} {\mathrm e}^{10 i \left (d x +c \right )}+40 a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}-255 a^{2} b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+210 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+45 i a^{3} b^{2}+45 i a \,b^{4}+20 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}-60 a^{4} b \,{\mathrm e}^{8 i \left (d x +c \right )}+40 i a^{5} {\mathrm e}^{6 i \left (d x +c \right )}-16 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+450 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-420 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-225 b^{5} {\mathrm e}^{8 i \left (d x +c \right )}+45 b^{5} {\mathrm e}^{10 i \left (d x +c \right )}+450 b^{5} {\mathrm e}^{6 i \left (d x +c \right )}+225 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}-450 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+9 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+15 a^{4} b \,{\mathrm e}^{10 i \left (d x +c \right )}\right )}{15 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right ) a^{6} d}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{3} d}+\frac {8 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{5} d}+\frac {6 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{a^{7} d}-\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}-\frac {8 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{5} d}-\frac {6 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{7} d}\)	\(681\)

[In]

int(csc(d*x+c)^6/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/5/a^2/tan(d*x+c)^5-1/3*(2*a^2+3*b^2)/a^4/tan(d*x+c)^3-(a^4+6*a^2*b^2+5*b^4)/a^6/tan(d*x+c)+1/2/a^3*b/t
an(d*x+c)^4+2*b*(a^2+b^2)/a^5/tan(d*x+c)^2-2*b*(a^4+4*a^2*b^2+3*b^4)/a^7*ln(tan(d*x+c))-(a^4+2*a^2*b^2+b^4)*b/
a^6/(a+b*tan(d*x+c))+2*b*(a^4+4*a^2*b^2+3*b^4)/a^7*ln(a+b*tan(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 787 vs. \(2 (213) = 426\).

Time = 0.32 (sec) , antiderivative size = 787, normalized size of antiderivative = 3.59 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4 \, {\left (4 \, a^{6} + 45 \, a^{4} b^{2} + 45 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{6} - 75 \, a^{4} b^{2} - 90 \, a^{2} b^{4} - 10 \, {\left (4 \, a^{6} + 45 \, a^{4} b^{2} + 45 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (2 \, a^{6} + 23 \, a^{4} b^{2} + 24 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left ({\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} - 3 \, b^{6} - 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 2 \, {\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 30 \, {\left ({\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} - 3 \, b^{6} - 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 2 \, {\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{5} b + 4 \, a^{3} b^{3} + 3 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) + {\left (4 \, {\left (4 \, a^{5} b + 45 \, a^{3} b^{3} + 45 \, a b^{5}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (a^{5} b + 33 \, a^{3} b^{3} + 36 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (a^{5} b - 10 \, a^{3} b^{3} - 12 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, {\left (a^{7} b d \cos \left (d x + c\right )^{6} - 3 \, a^{7} b d \cos \left (d x + c\right )^{4} + 3 \, a^{7} b d \cos \left (d x + c\right )^{2} - a^{7} b d - {\left (a^{8} d \cos \left (d x + c\right )^{5} - 2 \, a^{8} d \cos \left (d x + c\right )^{3} + a^{8} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]

[In]

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/30*(4*(4*a^6 + 45*a^4*b^2 + 45*a^2*b^4)*cos(d*x + c)^6 - 75*a^4*b^2 - 90*a^2*b^4 - 10*(4*a^6 + 45*a^4*b^2 +
45*a^2*b^4)*cos(d*x + c)^4 + 15*(2*a^6 + 23*a^4*b^2 + 24*a^2*b^4)*cos(d*x + c)^2 + 30*((a^4*b^2 + 4*a^2*b^4 +
3*b^6)*cos(d*x + c)^6 - a^4*b^2 - 4*a^2*b^4 - 3*b^6 - 3*(a^4*b^2 + 4*a^2*b^4 + 3*b^6)*cos(d*x + c)^4 + 3*(a^4*
b^2 + 4*a^2*b^4 + 3*b^6)*cos(d*x + c)^2 - ((a^5*b + 4*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^5 - 2*(a^5*b + 4*a^3*b^3
+ 3*a*b^5)*cos(d*x + c)^3 + (a^5*b + 4*a^3*b^3 + 3*a*b^5)*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^4*b^2 + 4*a^2*b^4 + 3*b^6)*cos(d*x + c)^6 - a^4*b^2
- 4*a^2*b^4 - 3*b^6 - 3*(a^4*b^2 + 4*a^2*b^4 + 3*b^6)*cos(d*x + c)^4 + 3*(a^4*b^2 + 4*a^2*b^4 + 3*b^6)*cos(d*x
+ c)^2 - ((a^5*b + 4*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^5 - 2*(a^5*b + 4*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3 + (a^
5*b + 4*a^3*b^3 + 3*a*b^5)*cos(d*x + c))*sin(d*x + c))*log(-1/4*cos(d*x + c)^2 + 1/4) + (4*(4*a^5*b + 45*a^3*b
^3 + 45*a*b^5)*cos(d*x + c)^5 - 10*(a^5*b + 33*a^3*b^3 + 36*a*b^5)*cos(d*x + c)^3 - 15*(a^5*b - 10*a^3*b^3 - 1
2*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^7*b*d*cos(d*x + c)^6 - 3*a^7*b*d*cos(d*x + c)^4 + 3*a^7*b*d*cos(d*x +
c)^2 - a^7*b*d - (a^8*d*cos(d*x + c)^5 - 2*a^8*d*cos(d*x + c)^3 + a^8*d*cos(d*x + c))*sin(d*x + c))

Sympy [F]

\[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\int \frac {\csc ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

integrate(csc(d*x+c)**6/(a+b*tan(d*x+c))**2,x)

[Out]

Integral(csc(c + d*x)**6/(a + b*tan(c + d*x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.03 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {9 \, a^{4} b \tan \left (d x + c\right ) - 60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \tan \left (d x + c\right )^{5} - 6 \, a^{5} - 30 \, {\left (a^{5} + 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \tan \left (d x + c\right )^{4} + 10 \, {\left (4 \, a^{4} b + 3 \, a^{2} b^{3}\right )} \tan \left (d x + c\right )^{3} - 5 \, {\left (4 \, a^{5} + 3 \, a^{3} b^{2}\right )} \tan \left (d x + c\right )^{2}}{a^{6} b \tan \left (d x + c\right )^{6} + a^{7} \tan \left (d x + c\right )^{5}} + \frac {60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{7}}}{30 \, d} \]

[In]

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/30*((9*a^4*b*tan(d*x + c) - 60*(a^4*b + 4*a^2*b^3 + 3*b^5)*tan(d*x + c)^5 - 6*a^5 - 30*(a^5 + 4*a^3*b^2 + 3*
a*b^4)*tan(d*x + c)^4 + 10*(4*a^4*b + 3*a^2*b^3)*tan(d*x + c)^3 - 5*(4*a^5 + 3*a^3*b^2)*tan(d*x + c)^2)/(a^6*b
*tan(d*x + c)^6 + a^7*tan(d*x + c)^5) + 60*(a^4*b + 4*a^2*b^3 + 3*b^5)*log(b*tan(d*x + c) + a)/a^7 - 60*(a^4*b
+ 4*a^2*b^3 + 3*b^5)*log(tan(d*x + c))/a^7)/d

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.52 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {60 \, {\left (a^{4} b + 4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{7}} - \frac {60 \, {\left (a^{4} b^{2} + 4 \, a^{2} b^{4} + 3 \, b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{7} b} + \frac {30 \, {\left (2 \, a^{4} b^{2} \tan \left (d x + c\right ) + 8 \, a^{2} b^{4} \tan \left (d x + c\right ) + 6 \, b^{6} \tan \left (d x + c\right ) + 3 \, a^{5} b + 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )} a^{7}} - \frac {137 \, a^{4} b \tan \left (d x + c\right )^{5} + 548 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 411 \, b^{5} \tan \left (d x + c\right )^{5} - 30 \, a^{5} \tan \left (d x + c\right )^{4} - 180 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 150 \, a b^{4} \tan \left (d x + c\right )^{4} + 60 \, a^{4} b \tan \left (d x + c\right )^{3} + 60 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 20 \, a^{5} \tan \left (d x + c\right )^{2} - 30 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 15 \, a^{4} b \tan \left (d x + c\right ) - 6 \, a^{5}}{a^{7} \tan \left (d x + c\right )^{5}}}{30 \, d} \]

[In]

integrate(csc(d*x+c)^6/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

-1/30*(60*(a^4*b + 4*a^2*b^3 + 3*b^5)*log(abs(tan(d*x + c)))/a^7 - 60*(a^4*b^2 + 4*a^2*b^4 + 3*b^6)*log(abs(b*
tan(d*x + c) + a))/(a^7*b) + 30*(2*a^4*b^2*tan(d*x + c) + 8*a^2*b^4*tan(d*x + c) + 6*b^6*tan(d*x + c) + 3*a^5*
b + 10*a^3*b^3 + 7*a*b^5)/((b*tan(d*x + c) + a)*a^7) - (137*a^4*b*tan(d*x + c)^5 + 548*a^2*b^3*tan(d*x + c)^5
+ 411*b^5*tan(d*x + c)^5 - 30*a^5*tan(d*x + c)^4 - 180*a^3*b^2*tan(d*x + c)^4 - 150*a*b^4*tan(d*x + c)^4 + 60*
a^4*b*tan(d*x + c)^3 + 60*a^2*b^3*tan(d*x + c)^3 - 20*a^5*tan(d*x + c)^2 - 30*a^3*b^2*tan(d*x + c)^2 + 15*a^4*
b*tan(d*x + c) - 6*a^5)/(a^7*tan(d*x + c)^5))/d

Mupad [B] (verification not implemented)

Time = 5.99 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.08 \[ \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4\,b\,\mathrm {atanh}\left (\frac {2\,b\,\left (a^2+3\,b^2\right )\,\left (a^2+b^2\right )\,\left (a+2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{a\,\left (2\,a^4\,b+8\,a^2\,b^3+6\,b^5\right )}\right )\,\left (a^2+3\,b^2\right )\,\left (a^2+b^2\right )}{a^7\,d}-\frac {\frac {1}{5\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}{a^5}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (4\,a^2+3\,b^2\right )}{6\,a^3}-\frac {3\,b\,\mathrm {tan}\left (c+d\,x\right )}{10\,a^2}+\frac {2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^4+4\,a^2\,b^2+3\,b^4\right )}{a^6}-\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (4\,a^2+3\,b^2\right )}{3\,a^4}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^6+a\,{\mathrm {tan}\left (c+d\,x\right )}^5\right )} \]

[In]

int(1/(sin(c + d*x)^6*(a + b*tan(c + d*x))^2),x)

[Out]

(4*b*atanh((2*b*(a^2 + 3*b^2)*(a^2 + b^2)*(a + 2*b*tan(c + d*x)))/(a*(2*a^4*b + 6*b^5 + 8*a^2*b^3)))*(a^2 + 3*
b^2)*(a^2 + b^2))/(a^7*d) - (1/(5*a) + (tan(c + d*x)^4*(a^4 + 3*b^4 + 4*a^2*b^2))/a^5 + (tan(c + d*x)^2*(4*a^2
+ 3*b^2))/(6*a^3) - (3*b*tan(c + d*x))/(10*a^2) + (2*b*tan(c + d*x)^5*(a^4 + 3*b^4 + 4*a^2*b^2))/a^6 - (b*tan
(c + d*x)^3*(4*a^2 + 3*b^2))/(3*a^4))/(d*(a*tan(c + d*x)^5 + b*tan(c + d*x)^6))