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

Optimal. Leaf size=265 \[ -\frac {\left (a^4+12 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a^7 d}+\frac {b \left (3 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^6 d}-\frac {2 \left (a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^5 d}+\frac {3 b \cot ^4(c+d x)}{4 a^4 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}-\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (\tan (c+d x))}{a^8 d}+\frac {b \left (3 a^4+20 a^2 b^2+21 b^4\right ) \log (a+b \tan (c+d x))}{a^8 d}-\frac {b \left (a^2+b^2\right )^2}{2 a^6 d (a+b \tan (c+d x))^2}-\frac {2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))} \]

[Out]

-(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))

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Rubi [A]
time = 0.17, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3597, 908} \begin {gather*} \frac {3 b \cot ^4(c+d x)}{4 a^4 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}-\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))}-\frac {b \left (a^2+b^2\right )^2}{2 a^6 d (a+b \tan (c+d x))^2}+\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 {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 {\left (a^4+12 a^2 b^2+15 b^4\right ) \cot (c+d x)}{a^7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a^4 + 12*a^2*b^2 + 15*b^4)*Cot[c + d*x])/(a^7*d)) + (b*(3*a^2 + 5*b^2)*Cot[c + d*x]^2)/(a^6*d) - (2*(a^2 +
3*b^2)*Cot[c + d*x]^3)/(3*a^5*d) + (3*b*Cot[c + d*x]^4)/(4*a^4*d) - Cot[c + d*x]^5/(5*a^3*d) - (b*(3*a^4 + 20
*a^2*b^2 + 21*b^4)*Log[Tan[c + d*x]])/(a^8*d) + (b*(3*a^4 + 20*a^2*b^2 + 21*b^4)*Log[a + b*Tan[c + d*x]])/(a^8
*d) - (b*(a^2 + b^2)^2)/(2*a^6*d*(a + b*Tan[c + d*x])^2) - (2*b*(a^2 + b^2)*(a^2 + 3*b^2))/(a^7*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*} \int \frac {\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {b \text {Subst}\left (\int \frac {\left (b^2+x^2\right )^2}{x^6 (a+x)^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \text {Subst}\left (\int \left (\frac {b^4}{a^3 x^6}-\frac {3 b^4}{a^4 x^5}+\frac {2 b^2 \left (a^2+3 b^2\right )}{a^5 x^4}-\frac {2 \left (3 a^2 b^2+5 b^4\right )}{a^6 x^3}+\frac {a^4+12 a^2 b^2+15 b^4}{a^7 x^2}+\frac {-3 a^4-20 a^2 b^2-21 b^4}{a^8 x}+\frac {\left (a^2+b^2\right )^2}{a^6 (a+x)^3}+\frac {2 \left (a^4+4 a^2 b^2+3 b^4\right )}{a^7 (a+x)^2}+\frac {3 a^4+20 a^2 b^2+21 b^4}{a^8 (a+x)}\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\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))}\\ \end {align*}

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Mathematica [A]
time = 4.99, size = 494, normalized size = 1.86 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-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)] - (2
4*a^7 + 619*a^5*b^2 + 630*a^3*b^4 - 675*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)] -
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)]) + 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 + 210*b^6)*Tan[c + d*x])))/(a^8*d*(a + b*Tan[c + d*x])^2)

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Maple [A]
time = 0.45, size = 246, normalized size = 0.93

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(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))-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)))

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Maxima [A]
time = 0.35, size = 281, normalized size = 1.06 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*tan(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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1018 vs. \(2 (257) = 514\).
time = 0.41, size = 1018, normalized size = 3.84 \begin {gather*} \frac {4 \, {\left (8 \, a^{7} + 187 \, a^{5} b^{2} + 120 \, a^{3} b^{4} - 315 \, a b^{6}\right )} \cos \left (d x + c\right )^{7} - 4 \, {\left (20 \, a^{7} + 482 \, a^{5} b^{2} + 255 \, a^{3} b^{4} - 945 \, a b^{6}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (6 \, a^{7} + 157 \, a^{5} b^{2} + 60 \, a^{3} b^{4} - 378 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 30 \, {\left (13 \, a^{5} b^{2} + 2 \, a^{3} b^{4} - 42 \, a b^{6}\right )} \cos \left (d x + c\right ) + 30 \, {\left (2 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right )^{7} - 6 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{4} b^{3} + 20 \, a^{2} b^{5} + 21 \, b^{7} + {\left (3 \, a^{6} b + 17 \, a^{4} b^{3} + a^{2} b^{5} - 21 \, b^{7}\right )} \cos \left (d x + c\right )^{6} - {\left (6 \, a^{6} b + 31 \, a^{4} b^{3} - 18 \, a^{2} b^{5} - 63 \, b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{6} b + 11 \, a^{4} b^{3} - 39 \, a^{2} b^{5} - 63 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\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 (2 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right )^{7} - 6 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right )^{5} + 6 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (3 \, a^{5} b^{2} + 20 \, a^{3} b^{4} + 21 \, a b^{6}\right )} \cos \left (d x + c\right ) - {\left (3 \, a^{4} b^{3} + 20 \, a^{2} b^{5} + 21 \, b^{7} + {\left (3 \, a^{6} b + 17 \, a^{4} b^{3} + a^{2} b^{5} - 21 \, b^{7}\right )} \cos \left (d x + c\right )^{6} - {\left (6 \, a^{6} b + 31 \, a^{4} b^{3} - 18 \, a^{2} b^{5} - 63 \, b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{6} b + 11 \, a^{4} b^{3} - 39 \, a^{2} b^{5} - 63 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\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 (285 \, a^{4} b^{3} + 630 \, a^{2} b^{5} - 8 \, {\left (8 \, a^{6} b + 195 \, a^{4} b^{3} + 315 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{6} + 10 \, {\left (7 \, a^{6} b + 330 \, a^{4} b^{3} + 567 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (a^{6} b - 135 \, a^{4} b^{3} - 252 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, {\left (2 \, a^{9} b d \cos \left (d x + c\right )^{7} - 6 \, a^{9} b d \cos \left (d x + c\right )^{5} + 6 \, a^{9} b d \cos \left (d x + c\right )^{3} - 2 \, a^{9} b d \cos \left (d x + c\right ) - {\left (a^{8} b^{2} d + {\left (a^{10} - a^{8} b^{2}\right )} d \cos \left (d x + c\right )^{6} - {\left (2 \, a^{10} - 3 \, a^{8} b^{2}\right )} d \cos \left (d x + c\right )^{4} + {\left (a^{10} - 3 \, a^{8} b^{2}\right )} d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*sin(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)*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(-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 + 315*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(d*x + c))/(2*a^9*b*d*cos(d*x + c)^7 - 6*a^9*b*d*cos(d*x + c)^5
+ 6*a^9*b*d*cos(d*x + c)^3 - 2*a^9*b*d*cos(d*x + c) - (a^8*b^2*d + (a^10 - a^8*b^2)*d*cos(d*x + c)^6 - (2*a^1
0 - 3*a^8*b^2)*d*cos(d*x + c)^4 + (a^10 - 3*a^8*b^2)*d*cos(d*x + c)^2)*sin(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{6}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.63, size = 382, normalized size = 1.44 \begin {gather*} -\frac {\frac {60 \, {\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}} - \frac {60 \, {\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} + \frac {30 \, {\left (9 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} + 60 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 63 \, b^{7} \tan \left (d x + c\right )^{2} + 22 \, a^{5} b^{2} \tan \left (d x + c\right ) + 136 \, a^{3} b^{4} \tan \left (d x + c\right ) + 138 \, a b^{6} \tan \left (d x + c\right ) + 14 \, a^{6} b + 78 \, a^{4} b^{3} + 76 \, a^{2} b^{5}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{8}} - \frac {411 \, a^{4} b \tan \left (d x + c\right )^{5} + 2740 \, a^{2} b^{3} \tan \left (d x + c\right )^{5} + 2877 \, b^{5} \tan \left (d x + c\right )^{5} - 60 \, a^{5} \tan \left (d x + c\right )^{4} - 720 \, a^{3} b^{2} \tan \left (d x + c\right )^{4} - 900 \, a b^{4} \tan \left (d x + c\right )^{4} + 180 \, a^{4} b \tan \left (d x + c\right )^{3} + 300 \, a^{2} b^{3} \tan \left (d x + c\right )^{3} - 40 \, a^{5} \tan \left (d x + c\right )^{2} - 120 \, a^{3} b^{2} \tan \left (d x + c\right )^{2} + 45 \, a^{4} b \tan \left (d x + c\right ) - 12 \, a^{5}}{a^{8} \tan \left (d x + c\right )^{5}}}{60 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(60*(3*a^4*b + 20*a^2*b^3 + 21*b^5)*log(abs(tan(d*x + c)))/a^8 - 60*(3*a^4*b^2 + 20*a^2*b^4 + 21*b^6)*lo
g(abs(b*tan(d*x + c) + a))/(a^8*b) + 30*(9*a^4*b^3*tan(d*x + c)^2 + 60*a^2*b^5*tan(d*x + c)^2 + 63*b^7*tan(d*x
+ c)^2 + 22*a^5*b^2*tan(d*x + c) + 136*a^3*b^4*tan(d*x + c) + 138*a*b^6*tan(d*x + c) + 14*a^6*b + 78*a^4*b^3
+ 76*a^2*b^5)/((b*tan(d*x + c) + a)^2*a^8) - (411*a^4*b*tan(d*x + c)^5 + 2740*a^2*b^3*tan(d*x + c)^5 + 2877*b^
5*tan(d*x + c)^5 - 60*a^5*tan(d*x + c)^4 - 720*a^3*b^2*tan(d*x + c)^4 - 900*a*b^4*tan(d*x + c)^4 + 180*a^4*b*t
an(d*x + c)^3 + 300*a^2*b^3*tan(d*x + c)^3 - 40*a^5*tan(d*x + c)^2 - 120*a^3*b^2*tan(d*x + c)^2 + 45*a^4*b*tan
(d*x + c) - 12*a^5)/(a^8*tan(d*x + c)^5))/d

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Mupad [B]
time = 5.32, size = 297, normalized size = 1.12 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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*tan(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))

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