\(\int \frac {1}{(a+b \cot ^2(c+d x))^{7/2}} \, dx\) [37]

Optimal result

Integrand size = 16, antiderivative size = 190 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^{7/2} d}+\frac {b \cot (c+d x)}{5 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{5/2}}+\frac {(9 a-4 b) b \cot (c+d x)}{15 a^2 (a-b)^2 d \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 (a-b)^3 d \sqrt {a+b \cot ^2(c+d x)}} \]

[Out]

-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)*c
ot(d*x+c)/a^3/(a-b)^3/d/(a+b*cot(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3742, 425, 541, 12, 385, 209} \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\frac {b (9 a-4 b) \cot (c+d x)}{15 a^2 d (a-b)^2 \left (a+b \cot ^2(c+d x)\right )^{3/2}}+\frac {b \left (33 a^2-26 a b+8 b^2\right ) \cot (c+d x)}{15 a^3 d (a-b)^3 \sqrt {a+b \cot ^2(c+d x)}}-\frac {\arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d (a-b)^{7/2}}+\frac {b \cot (c+d x)}{5 a d (a-b) \left (a+b \cot ^2(c+d x)\right )^{5/2}} \]

[In]

Int[(a + b*Cot[c + d*x]^2)^(-7/2),x]

[Out]

-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(7/2)*d)) + (b*Cot[c + d*x])/(5*a*(a
- b)*d*(a + b*Cot[c + d*x]^2)^(5/2)) + ((9*a - 4*b)*b*Cot[c + d*x])/(15*a^2*(a - b)^2*d*(a + b*Cot[c + d*x]^2)
^(3/2)) + (b*(33*a^2 - 26*a*b + 8*b^2)*Cot[c + d*x])/(15*a^3*(a - b)^3*d*Sqrt[a + b*Cot[c + d*x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3742

Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x]
, x]}, Dist[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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b x^2\right )^{7/2}} \, dx,x,\cot (c+d x)\right )}{d} \\ & = \frac {b \cot (c+d x)}{5 a (a-b) d \left (a+b \cot ^2(c+d x)\right )^{5/2}}-\frac {\text {Subst}\left (\int \frac {5 a-4 b-4 b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\cot (c+d x)\right )}{5 a (a-b) 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 {\text {Subst}\left (\int \frac {15 a^2-18 a b+8 b^2-2 (9 a-4 b) b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\cot (c+d x)\right )}{15 a^2 (a-b)^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)}}-\frac {\text {Subst}\left (\int \frac {15 a^3}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{15 a^3 (a-b)^3 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)}}-\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{(a-b)^3 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)}}-\frac {\text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{(a-b)^3 d} \\ & = -\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)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Result too large to show} \]

[In]

Integrate[(a + b*Cot[c + d*x]^2)^(-7/2),x]

[Out]

-1/4725*(Cot[c + d*x]*(-33075*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]] + (99225*(a - b)*ArcSin[Sqrt[((a - b)*C
os[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]]*Co
s[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[Sqrt[((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*ArcS
in[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)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^4)/a^5 - (15120*b^3*ArcSin[Sqrt[((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] + 50715*(((a - b)*Cos[c + d*x
]^2)/a)^(5/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] - (154350*b*(((a - b)*Cos[c + d*x]^2)/a)^(3/2)*C
ot[c + d*x]^2*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (101430*b*(((a - b)*Cos[c + d*x]^2)/a)^(5/2
)*Cot[c + d*x]^2*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a - (123480*b^2*(((a - b)*Cos[c + d*x]^2)/a)
^(3/2)*Cot[c + d*x]^4*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (81144*b^2*(((a - b)*Cos[c + d*x]
^2)/a)^(5/2)*Cot[c + d*x]^4*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 - (35280*b^3*(((a - b)*Cos[c
+ d*x]^2)/a)^(3/2)*Cot[c + d*x]^6*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + (23184*b^3*(((a - b)*
Cos[c + d*x]^2)/a)^(5/2)*Cot[c + d*x]^6*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 1420*(((a - b)*
Cos[c + d*x]^2)/a)^(9/2)*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a
*Tan[c + d*x]^2))/a] + (3540*b*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^2*Hypergeometric2F1[2, 2, 11/2,
((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (3000*b^2*(((a - b)*Cos[c +
d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2
*(b + a*Tan[c + d*x]^2))/a])/a^2 + (880*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^6*Hypergeometric2F
1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 600*(((a - b)
*Cos[c + d*x]^2)/a)^(9/2)*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*
x]^2*(b + a*Tan[c + d*x]^2))/a] + (1680*b*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^2*HypergeometricPFQ[
{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (1560*b
^2*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c +
d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (480*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*
Cot[c + d*x]^6*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a
*Tan[c + d*x]^2))/a])/a^3 + 80*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}
, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + (240*b*(((a - b)*Cos[c + d*x]^
2)/a)^(9/2)*Cot[c + d*x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos
[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (240*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*Hyperge
ometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2)
)/a])/a^2 + (80*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^6*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 1
1/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 33075*Sqrt[((a - b)*C
os[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2] + (66150*b*Cot[c + d*x]^2*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c
+ d*x]^2))/a^2])/a + (52920*b^2*Cot[c + d*x]^4*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2])/a^2
+ (15120*b^3*Cot[c + d*x]^6*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2])/a^3 - (198450*(a - b)^
2*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]])/(a^3*(Tan[c + d*x] + Tan[c + d*x]^3)^2)))/(a^3*d*(((a - b)*Cos[c
+ d*x]^2)/a)^(7/2)*(1 + Cot[c + d*x]^2)*Sqrt[a + b*Cot[c + d*x]^2]*(1 + (b*Cot[c + d*x]^2)/a)^2*Sqrt[(Cos[c +
d*x]^2*(b + a*Tan[c + d*x]^2))/a])

Maple [A] (verified)

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	\(\frac {\frac {b \left (\frac {\cot \left (d x +c \right )}{5 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 \cot \left (d x +c \right )}{15 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {8 \cot \left (d x +c \right )}{15 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{a}\right )}{a -b}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{4} b^{2}}+\frac {b \left (\frac {\cot \left (d x +c \right )}{3 a \left (a +b \cot \left (d x +c \right )^{2}\right )^{\frac {3}{2}}}+\frac {2 \cot \left (d x +c \right )}{3 a^{2} \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{\left (a -b \right )^{2}}+\frac {b \cot \left (d x +c \right )}{\left (a -b \right )^{3} a \sqrt {a +b \cot \left (d x +c \right )^{2}}}}{d}\)	\(253\)

[In]

int(1/(a+b*cot(d*x+c)^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

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))

Fricas [B] (verification not implemented)

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 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cot(d*x+c)^2)^(7/2),x, algorithm="fricas")

[Out]

[-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))*
sqrt(-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)*sqr
t(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*co
s(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 + 21*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*cos(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)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x +
2*c)/((a - b)*cos(2*d*x + 2*c) - b)) - 2*(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 + 21*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*cos(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)]

Sympy [F]

\[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{\left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]

[In]

integrate(1/(a+b*cot(d*x+c)**2)**(7/2),x)

[Out]

Integral((a + b*cot(c + d*x)**2)**(-7/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(a+b*cot(d*x+c)^2)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

Giac [B] (verification not implemented)

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 \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*cot(d*x+c)^2)^(7/2),x, algorithm="giac")

[Out]

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)) + 305819
8*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)) - 1020
000*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^19*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 + c)) + 10752*a^19*b^4*sgn(sin(d*x + c)) - 62729*a^18*b^5*sgn(s
in(d*x + c)) + 259530*a^17*b^6*sgn(sin(d*x + c)) - 809676*a^16*b^7*sgn(sin(d*x + c)) + 1977168*a^15*b^8*sgn(si
n(d*x + c)) - 3871716*a^14*b^9*sgn(sin(d*x + c)) + 6178752*a^13*b^10*sgn(sin(d*x + c)) - 8121750*a^12*b^11*sgn
(sin(d*x + c)) + 8850608*a^11*b^12*sgn(sin(d*x + c)) - 8020974*a^10*b^13*sgn(sin(d*x + c)) + 6045676*a^9*b^14*
sgn(sin(d*x + c)) - 3778692*a^8*b^15*sgn(sin(d*x + c)) + 1946160*a^7*b^16*sgn(sin(d*x + c)) - 817428*a^6*b^17*
sgn(sin(d*x + c)) + 275604*a^5*b^18*sgn(sin(d*x + c)) - 72837*a^4*b^19*sgn(sin(d*x + c)) + 14544*a^3*b^20*sgn(
sin(d*x + c)) - 2065*a^2*b^21*sgn(sin(d*x + c)) + 186*a*b^22*sgn(sin(d*x + c)) - 8*b^23*sgn(sin(d*x + c)))/(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))*tan(1/2*d*x + 1/2*c)^2 + 10*(72*a^22*b*sgn(sin(d*x + c)) - 1458*a^21*b^2*sgn(sin(d*x +
c)) + 14067*a^20*b^3*sgn(sin(d*x + c)) - 86018*a^19*b^4*sgn(sin(d*x + c)) + 374075*a^18*b^5*sgn(sin(d*x + c))
- 1230570*a^17*b^6*sgn(sin(d*x + c)) + 3179748*a^16*b^7*sgn(sin(d*x + c)) - 6614904*a^15*b^8*sgn(sin(d*x + c))
+ 11265084*a^14*b^9*sgn(sin(d*x + c)) - 15882420*a^13*b^10*sgn(sin(d*x + c)) + 18674058*a^12*b^11*sgn(sin(d*x
+ c)) - 18386316*a^11*b^12*sgn(sin(d*x + c)) + 15180490*a^10*b^13*sgn(sin(d*x + c)) - 10497364*a^9*b^14*sgn(s
in(d*x + c)) + 6055740*a^8*b^15*sgn(sin(d*x + c)) - 2893944*a^7*b^16*sgn(sin(d*x + c)) + 1133220*a^6*b^17*sgn(
sin(d*x + c)) - 357786*a^5*b^18*sgn(sin(d*x + c)) + 88923*a^4*b^19*sgn(sin(d*x + c)) - 16770*a^3*b^20*sgn(sin(
d*x + c)) + 2259*a^2*b^21*sgn(sin(d*x + c)) - 194*a*b^22*sgn(sin(d*x + c)) + 8*b^23*sgn(sin(d*x + c)))/(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^1
3 + 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))*tan(1/2*d*x + 1/2*c)^2 - 10*(72*a^22*b*sgn(sin(d*x + c)) - 1458*a^21*b^2*sgn(sin(d*x + c))
+ 14067*a^20*b^3*sgn(sin(d*x + c)) - 86018*a^19*b^4*sgn(sin(d*x + c)) + 374075*a^18*b^5*sgn(sin(d*x + c)) - 12
30570*a^17*b^6*sgn(sin(d*x + c)) + 3179748*a^16*b^7*sgn(sin(d*x + c)) - 6614904*a^15*b^8*sgn(sin(d*x + c)) + 1
1265084*a^14*b^9*sgn(sin(d*x + c)) - 15882420*a^13*b^10*sgn(sin(d*x + c)) + 18674058*a^12*b^11*sgn(sin(d*x + c
)) - 18386316*a^11*b^12*sgn(sin(d*x + c)) + 15180490*a^10*b^13*sgn(sin(d*x + c)) - 10497364*a^9*b^14*sgn(sin(d
*x + c)) + 6055740*a^8*b^15*sgn(sin(d*x + c)) - 2893944*a^7*b^16*sgn(sin(d*x + c)) + 1133220*a^6*b^17*sgn(sin(
d*x + c)) - 357786*a^5*b^18*sgn(sin(d*x + c)) + 88923*a^4*b^19*sgn(sin(d*x + c)) - 16770*a^3*b^20*sgn(sin(d*x
+ c)) + 2259*a^2*b^21*sgn(sin(d*x + c)) - 194*a*b^22*sgn(sin(d*x + c)) + 8*b^23*sgn(sin(d*x + c)))/(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 + 20
3490*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^2
0 - a^3*b^21))*tan(1/2*d*x + 1/2*c)^2 - 5*(60*a^21*b^2*sgn(sin(d*x + c)) - 1165*a^20*b^3*sgn(sin(d*x + c)) + 1
0752*a^19*b^4*sgn(sin(d*x + c)) - 62729*a^18*b^5*sgn(sin(d*x + c)) + 259530*a^17*b^6*sgn(sin(d*x + c)) - 80967
6*a^16*b^7*sgn(sin(d*x + c)) + 1977168*a^15*b^8*sgn(sin(d*x + c)) - 3871716*a^14*b^9*sgn(sin(d*x + c)) + 61787
52*a^13*b^10*sgn(sin(d*x + c)) - 8121750*a^12*b^11*sgn(sin(d*x + c)) + 8850608*a^11*b^12*sgn(sin(d*x + c)) - 8
020974*a^10*b^13*sgn(sin(d*x + c)) + 6045676*a^9*b^14*sgn(sin(d*x + c)) - 3778692*a^8*b^15*sgn(sin(d*x + c)) +
1946160*a^7*b^16*sgn(sin(d*x + c)) - 817428*a^6*b^17*sgn(sin(d*x + c)) + 275604*a^5*b^18*sgn(sin(d*x + c)) -
72837*a^4*b^19*sgn(sin(d*x + c)) + 14544*a^3*b^20*sgn(sin(d*x + c)) - 2065*a^2*b^21*sgn(sin(d*x + c)) + 186*a*
b^22*sgn(sin(d*x + c)) - 8*b^23*sgn(sin(d*x + c)))/(a^24 - 21*a^23*b + 210*a^22*b^2 - 1330*a^21*b^3 + 5985*a^2
0*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))*tan(1/2*d*x + 1/2*c)^2 - (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(s
in(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*s
gn(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^19*sgn(sin(d*x + c)) - 11100*a^3*b^20*sgn(sin(d*x + c)) + 1725*a^2*b^21*sgn(si
n(d*x + c)) - 170*a*b^22*sgn(sin(d*x + c)) + 8*b^23*sgn(sin(d*x + c)))/(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))/(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)^(5/2))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \cot ^2(c+d x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{7/2}} \,d x \]

[In]

int(1/(a + b*cot(c + d*x)^2)^(7/2),x)

[Out]

int(1/(a + b*cot(c + d*x)^2)^(7/2), x)