Integrand size = 16, antiderivative size = 190 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^{7/2} d}+\frac {b \cot (c+d x)}{5 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{5/2}}+\frac {(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a+b \cot ^2(c+d x)}} \]
-arctan(cot(d*x+c)*(a-b)^(1/2)/(a+b*cot(d*x+c)^2)^(1/2))/(a-b)^(7/2)/d+1/5 *b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)^(5/2)+1/15*(9*a-4*b)*b*cot(d*x+ c)/a^2/(a-b)^2/d/(a+b*cot(d*x+c)^2)^(3/2)+1/15*b*(33*a^2-26*a*b+8*b^2)*cot (d*x+c)/a^3/(a-b)^3/d/(a+b*cot(d*x+c)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 15.97 (sec) , antiderivative size = 2553, normalized size of antiderivative = 13.44 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Result too large to show} \]
-1/4725*(Cot[c + d*x]*(-33075*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]] + ( 99225*(a - b)*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2)/a - (99225*(a - b)^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4) /a^2 + (33075*(a - b)^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d *x]^6)/a^3 - (66150*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x ]^2)/a + (198450*(a - b)*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^2)/a^2 + (66150*(a - b)^3*b*ArcSin[Sqrt[((a - b)*Cos [c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^2)/a^4 - (52920*b^2*ArcSin[Sq rt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^4)/a^2 + (158760*(a - b)*b^2* ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^4)/a^ 3 - (158760*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4*Cot[c + d*x]^4)/a^4 + (52920*(a - b)^3*b^2*ArcSin[Sqrt[((a - b)*Co s[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^4)/a^5 - (15120*b^3*ArcSin[S qrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^6)/a^3 + (45360*(a - b)*b^3* ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^6)/a^ 4 - (45360*(a - b)^2*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4*Cot[c + d*x]^6)/a^5 + (15120*(a - b)^3*b^3*ArcSin[Sqrt[((a - b)*Cos [c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^6)/a^6 - 77175*(((a - b)*Cos[ c + d*x]^2)/a)^(3/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + 507 15*(((a - b)*Cos[c + d*x]^2)/a)^(5/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c...
Time = 0.38 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4144, 316, 402, 402, 27, 291, 216}
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.
\(\displaystyle \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a+b \tan \left (c+d x+\frac {\pi }{2}\right )^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4144 |
\(\displaystyle -\frac {\int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )^{7/2}}d\cot (c+d x)}{d}\) |
\(\Big \downarrow \) 316 |
\(\displaystyle -\frac {\frac {\int \frac {-4 b \cot ^2(c+d x)+5 a-4 b}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )^{5/2}}d\cot (c+d x)}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle -\frac {\frac {\frac {\int \frac {15 a^2-18 b a+8 b^2-2 (9 a-4 b) b \cot ^2(c+d x)}{\left (\cot ^2(c+d x)+1\right ) \left (b \cot ^2(c+d x)+a\right )^{3/2}}d\cot (c+d x)}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {15 a^3}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)}{a (a-b)}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\frac {\frac {15 a^2 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)}{a-b}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {\frac {\frac {\frac {15 a^2 \int \frac {1}{1-\frac {(b-a) \cot ^2(c+d x)}{b \cot ^2(c+d x)+a}}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a}}}{a-b}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {\frac {\frac {\frac {15 a^2 \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^{3/2}}-\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{a (a-b) \sqrt {a+b \cot ^2(c+d x)}}}{3 a (a-b)}-\frac {b (9 a-4 b) \cot (c+d x)}{3 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{3/2}}}{5 a (a-b)}-\frac {b \cot (c+d x)}{5 a (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}}}{d}\) |
-((-1/5*(b*Cot[c + d*x])/(a*(a - b)*(a + b*Cot[c + d*x]^2)^(5/2)) + (-1/3* ((9*a - 4*b)*b*Cot[c + d*x])/(a*(a - b)*(a + b*Cot[c + d*x]^2)^(3/2)) + (( 15*a^2*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(a - b)^(3/2) - (b*(33*a^2 - 26*a*b + 8*b^2)*Cot[c + d*x])/(a*(a - b)*Sqrt[a + b*Cot[c + d*x]^2]))/(3*a*(a - b)))/(5*a*(a - b)))/d)
3.1.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(a + b* (ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Time = 0.04 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {\frac {b \left (\frac {\cot \left (d x +c \right )}{5 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 \cot \left (d x +c \right )}{15 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {8 \cot \left (d x +c \right )}{15 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{a}\right )}{a -b}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{4} b^{2}}+\frac {b \left (\frac {\cot \left (d x +c \right )}{3 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \cot \left (d x +c \right )}{3 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{2}}+\frac {b \cot \left (d x +c \right )}{\left (a -b \right )^{3} a \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{d}\) | \(253\) |
default | \(\frac {\frac {b \left (\frac {\cot \left (d x +c \right )}{5 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 \cot \left (d x +c \right )}{15 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {8 \cot \left (d x +c \right )}{15 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{a}\right )}{a -b}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{4} b^{2}}+\frac {b \left (\frac {\cot \left (d x +c \right )}{3 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \cot \left (d x +c \right )}{3 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{2}}+\frac {b \cot \left (d x +c \right )}{\left (a -b \right )^{3} a \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{d}\) | \(253\) |
1/d*(1/(a-b)*b*(1/5*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(5/2)+4/5/a*(1/3*cot(d *x+c)/a/(a+b*cot(d*x+c)^2)^(3/2)+2/3/a^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/ 2)))-1/(a-b)^4*(b^4*(a-b))^(1/2)/b^2*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a +b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))+1/(a-b)^2*b*(1/3*cot(d*x+c)/a/(a+b*cot( d*x+c)^2)^(3/2)+2/3/a^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2))+b/(a-b)^3*cot (d*x+c)/a/(a+b*cot(d*x+c)^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 700 vs. \(2 (172) = 344\).
Time = 0.42 (sec) , antiderivative size = 1452, normalized size of antiderivative = 7.64 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display} \]
[-1/60*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 - (a^6 - 3*a^5*b + 3*a^4*b ^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 + 3*(a^6 - a^5*b - a^4*b^2 + a^3*b^3)*cos (2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c))*sq rt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + 2*((a - b)*cos( 2*d*x + 2*c) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(co s(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2* d*x + 2*c)) + 4*(45*a^5*b - 15*a^4*b^2 - 47*a^3*b^3 + 11*a^2*b^4 + 14*a*b^ 5 - 8*b^6 + (45*a^5*b - 165*a^4*b^2 + 233*a^3*b^3 - 159*a^2*b^4 + 54*a*b^5 - 8*b^6)*cos(2*d*x + 2*c)^2 - 2*(45*a^5*b - 90*a^4*b^2 + 27*a^3*b^3 + 44* a^2*b^4 - 34*a*b^5 + 8*b^6)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2* c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^10 - 7*a^9*b + 2 1*a^8*b^2 - 35*a^7*b^3 + 35*a^6*b^4 - 21*a^5*b^5 + 7*a^4*b^6 - a^3*b^7)*d* cos(2*d*x + 2*c)^3 - 3*(a^10 - 5*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 - 5*a^6*b^4 + 9*a^5*b^5 - 5*a^4*b^6 + a^3*b^7)*d*cos(2*d*x + 2*c)^2 + 3*(a^10 - 3*a^9 *b + a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - a^5*b^5 + 3*a^4*b^6 - a^3*b^7)*d*co s(2*d*x + 2*c) - (a^10 - a^9*b - 3*a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 - 3*a^5 *b^5 - a^4*b^6 + a^3*b^7)*d), 1/30*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^ 3 - (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 + 3*(a^6 - a^ 5*b - a^4*b^2 + a^3*b^3)*cos(2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a ^3*b^3)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)...
\[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{\left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Exception generated. \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 3249 vs. \(2 (172) = 344\).
Time = 2.09 (sec) , antiderivative size = 3249, normalized size of antiderivative = 17.10 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display} \]
1/15*(30*arctan(-1/2*(sqrt(b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(b*tan(1/2*d*x + 1/2*c)^4 + 4*a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c)^2 + b) + sqrt(b))/sqrt(a - b))/((a^3*sgn(sin(d*x + c)) - 3*a^2*b*sgn(sin(d*x + c) ) + 3*a*b^2*sgn(sin(d*x + c)) - b^3*sgn(sin(d*x + c)))*sqrt(a - b)) - (((( ((33*a^20*b^3*sgn(sin(d*x + c)) - 620*a^19*b^4*sgn(sin(d*x + c)) + 5525*a^ 18*b^5*sgn(sin(d*x + c)) - 31050*a^17*b^6*sgn(sin(d*x + c)) + 123420*a^16* b^7*sgn(sin(d*x + c)) - 368832*a^15*b^8*sgn(sin(d*x + c)) + 859860*a^14*b^ 9*sgn(sin(d*x + c)) - 1601400*a^13*b^10*sgn(sin(d*x + c)) + 2419950*a^12*b ^11*sgn(sin(d*x + c)) - 2996760*a^11*b^12*sgn(sin(d*x + c)) + 3058198*a^10 *b^13*sgn(sin(d*x + c)) - 2576860*a^9*b^14*sgn(sin(d*x + c)) + 1790100*a^8 *b^15*sgn(sin(d*x + c)) - 1020000*a^7*b^16*sgn(sin(d*x + c)) + 472260*a^6* b^17*sgn(sin(d*x + c)) - 175032*a^5*b^18*sgn(sin(d*x + c)) + 50745*a^4*b^1 9*sgn(sin(d*x + c)) - 11100*a^3*b^20*sgn(sin(d*x + c)) + 1725*a^2*b^21*sgn (sin(d*x + c)) - 170*a*b^22*sgn(sin(d*x + c)) + 8*b^23*sgn(sin(d*x + c)))* tan(1/2*d*x + 1/2*c)^2/(a^24 - 21*a^23*b + 210*a^22*b^2 - 1330*a^21*b^3 + 5985*a^20*b^4 - 20349*a^19*b^5 + 54264*a^18*b^6 - 116280*a^17*b^7 + 203490 *a^16*b^8 - 293930*a^15*b^9 + 352716*a^14*b^10 - 352716*a^13*b^11 + 293930 *a^12*b^12 - 203490*a^11*b^13 + 116280*a^10*b^14 - 54264*a^9*b^15 + 20349* a^8*b^16 - 5985*a^7*b^17 + 1330*a^6*b^18 - 210*a^5*b^19 + 21*a^4*b^20 - a^ 3*b^21) + 5*(60*a^21*b^2*sgn(sin(d*x + c)) - 1165*a^20*b^3*sgn(sin(d*x ...
Timed out. \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{7/2}} \,d x \]