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

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

[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]))

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Rubi [A]  time = 0.239401, antiderivative size = 265, normalized size of antiderivative = 1., 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 = {3516, 894} \[ -\frac{2 b \left (a^2+b^2\right ) \left (a^2+3 b^2\right )}{a^7 d (a+b \tan (c+d x))}-\frac{b \left (a^2+b^2\right )^2}{2 a^6 d (a+b \tan (c+d x))^2}-\frac{2 \left (a^2+3 b^2\right ) \cot ^3(c+d x)}{3 a^5 d}+\frac{b \left (3 a^2+5 b^2\right ) \cot ^2(c+d x)}{a^6 d}-\frac{\left (12 a^2 b^2+a^4+15 b^4\right ) \cot (c+d x)}{a^7 d}-\frac{b \left (20 a^2 b^2+3 a^4+21 b^4\right ) \log (\tan (c+d x))}{a^8 d}+\frac{b \left (20 a^2 b^2+3 a^4+21 b^4\right ) \log (a+b \tan (c+d x))}{a^8 d}+\frac{3 b \cot ^4(c+d x)}{4 a^4 d}-\frac{\cot ^5(c+d x)}{5 a^3 d} \]

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 3516

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]

Rule 894

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]))

Rubi steps

\begin{align*} \int \frac{\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac{b \operatorname{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 \operatorname{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*}

Mathematica [A]  time = 4.67453, size = 494, normalized size = 1.86 \[ -\frac{\csc ^5(c+d x) \left (5 \sec (c+d x) \left (-3 b \left (89 a^4 b^2+345 a^2 b^4+8 a^6+210 b^6\right ) \tan (c+d x)-27 a^5 b^2-42 a^3 b^4+40 a^7+135 a b^6\right )+\sec ^2(c+d x) \left (1665 a^4 b^3 \sin (3 (c+d x))-1215 a^4 b^3 \sin (5 (c+d x))+345 a^4 b^3 \sin (7 (c+d x))+4635 a^2 b^5 \sin (3 (c+d x))-2565 a^2 b^5 \sin (5 (c+d x))+585 a^2 b^5 \sin (7 (c+d x))+187 a^5 b^2 \cos (7 (c+d x))+210 a^3 b^4 \cos (7 (c+d x))+\left (567 a^5 b^2+630 a^3 b^4+8 a^7-1215 a b^6\right ) \cos (3 (c+d x))-\left (619 a^5 b^2+630 a^3 b^4+24 a^7-675 a b^6\right ) \cos (5 (c+d x))-126 a^6 b \sin (3 (c+d x))+10 a^6 b \sin (5 (c+d x))+16 a^6 b \sin (7 (c+d x))+8 a^7 \cos (7 (c+d x))-135 a b^6 \cos (7 (c+d x))+1890 b^7 \sin (3 (c+d x))-630 b^7 \sin (5 (c+d x))+90 b^7 \sin (7 (c+d x))\right )+960 b \left (20 a^2 b^2+3 a^4+21 b^4\right ) \sin ^5(c+d x) (a+b \tan (c+d x))^2 (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))\right )}{960 a^8 d (a+b \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 - 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*Si
n[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])))/(960*a^8*d*(a + b*Tan[c + d*x])^2)

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Maple [A]  time = 0.137, size = 410, normalized size = 1.6 \begin{align*} -{\frac{1}{5\,d{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2}{3\,d{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-2\,{\frac{{b}^{2}}{d{a}^{5} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{d{a}^{3}\tan \left ( dx+c \right ) }}-12\,{\frac{{b}^{2}}{d{a}^{5}\tan \left ( dx+c \right ) }}-15\,{\frac{{b}^{4}}{d{a}^{7}\tan \left ( dx+c \right ) }}+{\frac{3\,b}{4\,d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}}+3\,{\frac{b}{d{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}+5\,{\frac{{b}^{3}}{d{a}^{6} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}-3\,{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}-20\,{\frac{{b}^{3}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{6}}}-21\,{\frac{{b}^{5}\ln \left ( \tan \left ( dx+c \right ) \right ) }{d{a}^{8}}}+3\,{\frac{b\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{4}}}+20\,{\frac{{b}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{6}}}+21\,{\frac{{b}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{a}^{8}}}-{\frac{b}{2\,{a}^{2}d \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{3}}{d{a}^{4} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{b}^{5}}{2\,d{a}^{6} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{b}{d{a}^{3} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-8\,{\frac{{b}^{3}}{d{a}^{5} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-6\,{\frac{{b}^{5}}{d{a}^{7} \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}

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)

[Out]

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

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Maxima [A]  time = 1.21861, size = 379, normalized size = 1.43 \begin{align*} \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{align*}

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]  time = 3.01524, size = 2344, normalized size = 8.85 \begin{align*} \text{result too large to display} \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 1.23827, size = 516, normalized size = 1.95 \begin{align*} -\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{align*}

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