\(\int (a+b \cot ^2(c+d x))^{5/2} \, dx\) [31]

Optimal result

Integrand size = 16, antiderivative size = 171 \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=-\frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{8 d}-\frac {(7 a-4 b) b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{8 d}-\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d} \]

[Out]

-(a-b)^(5/2)*arctan(cot(d*x+c)*(a-b)^(1/2)/(a+b*cot(d*x+c)^2)^(1/2))/d-1/4*b*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(3/
2)/d-1/8*(15*a^2-20*a*b+8*b^2)*arctanh(cot(d*x+c)*b^(1/2)/(a+b*cot(d*x+c)^2)^(1/2))*b^(1/2)/d-1/8*(7*a-4*b)*b*
cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 542, 537, 223, 212, 385, 209} \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{8 d}-\frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d}-\frac {b (7 a-4 b) \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{8 d} \]

[In]

Int[(a + b*Cot[c + d*x]^2)^(5/2),x]

[Out]

-(((a - b)^(5/2)*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*(15*a^2 - 20*a*b
+ 8*b^2)*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(8*d) - ((7*a - 4*b)*b*Cot[c + d*x]*Sqrt
[a + b*Cot[c + d*x]^2])/(8*d) - (b*Cot[c + d*x]*(a + b*Cot[c + d*x]^2)^(3/2))/(4*d)

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 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 427

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c
+ d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 542

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(n*(p + q + 1) + 1))), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

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 {\left (a+b x^2\right )^{5/2}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d}-\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x^2} \left (a (4 a-b)+(7 a-4 b) b x^2\right )}{1+x^2} \, dx,x,\cot (c+d x)\right )}{4 d} \\ & = -\frac {(7 a-4 b) b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{8 d}-\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d}-\frac {\text {Subst}\left (\int \frac {a \left (8 a^2-9 a b+4 b^2\right )+b \left (15 a^2-20 a b+8 b^2\right ) x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{8 d} \\ & = -\frac {(7 a-4 b) b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{8 d}-\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d}-\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{8 d} \\ & = -\frac {(7 a-4 b) b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{8 d}-\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d}-\frac {(a-b)^3 \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 )}{d}-\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{8 d} \\ & = -\frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{8 d}-\frac {(7 a-4 b) b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{8 d}-\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\frac {8 (a-b)^{5/2} \arctan \left (\frac {\sqrt {b}+\sqrt {b} \cot ^2(c+d x)-\cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{\sqrt {a-b}}\right )-b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)} \left (9 a-4 b+2 b \cot ^2(c+d x)\right )+\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \log \left (-\sqrt {b} \cot (c+d x)+\sqrt {a+b \cot ^2(c+d x)}\right )}{8 d} \]

[In]

Integrate[(a + b*Cot[c + d*x]^2)^(5/2),x]

[Out]

(8*(a - b)^(5/2)*ArcTan[(Sqrt[b] + Sqrt[b]*Cot[c + d*x]^2 - Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2])/Sqrt[a -
b]] - b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2]*(9*a - 4*b + 2*b*Cot[c + d*x]^2) + Sqrt[b]*(15*a^2 - 20*a*b +
8*b^2)*Log[-(Sqrt[b]*Cot[c + d*x]) + Sqrt[a + b*Cot[c + d*x]^2]])/(8*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(149)=298\).

Time = 0.18 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.70

method	result	size
derivativedivides	\(-\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{d}-\frac {b^{2} \cot \left (d x +c \right )^{3} \sqrt {a +b \cot \left (d x +c \right )^{2}}}{4 d}-\frac {9 b a \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{8 d}-\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{8 d}+\frac {b^{2} \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{2 d}+\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{2 d}+\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}-\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}-\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\)	\(462\)
default	\(-\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{d}-\frac {b^{2} \cot \left (d x +c \right )^{3} \sqrt {a +b \cot \left (d x +c \right )^{2}}}{4 d}-\frac {9 b a \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{8 d}-\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{8 d}+\frac {b^{2} \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{2 d}+\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{2 d}+\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}-\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}-\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\)	\(462\)

[In]

int((a+b*cot(d*x+c)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/d*b^(5/2)*ln(b^(1/2)*cot(d*x+c)+(a+b*cot(d*x+c)^2)^(1/2))-1/4/d*b^2*cot(d*x+c)^3*(a+b*cot(d*x+c)^2)^(1/2)-9
/8/d*b*a*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)-15/8/d*b^(1/2)*a^2*ln(b^(1/2)*cot(d*x+c)+(a+b*cot(d*x+c)^2)^(1/2)
)+1/2/d*b^2*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)+5/2/d*b^(3/2)*a*ln(b^(1/2)*cot(d*x+c)+(a+b*cot(d*x+c)^2)^(1/2)
)+1/d*b*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))-3/d*a*
(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))+3/d*a^2/b*(b^4
*(a-b))^(1/2)/(a-b)*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/d*a^3*(b^4*(a-b)
)^(1/2)/b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (149) = 298\).

Time = 0.32 (sec) , antiderivative size = 1520, normalized size of antiderivative = 8.89 \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*cot(d*x+c)^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/16*(8*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c
) + 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) + b)*sin(2*d
*x + 2*c) + (15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos(2*d*x + 2*c))*sqrt(b)*log(((a - 2*b)*cos(
2*d*x + 2*c) + 2*sqrt(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
- 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - 2*(4*b^2*cos(2*d*x + 2*c) - 3*(3*a*b - 2*b^2)*cos(2*d*x + 2*
c)^2 + 9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/((d*cos(2*d*x + 2*c) -
d)*sin(2*d*x + 2*c)), 1/16*(16*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*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) + a - b))*sin(2*d*x + 2*c) - (15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos(2*d*x + 2*c))*s
qrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqrt(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 - 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + 2*(4*b^2*cos(2*d*x + 2*c) - 3*(3*a
*b - 2*b^2)*cos(2*d*x + 2*c)^2 + 9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)
))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c)), -1/8*((15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos
(2*d*x + 2*c))*sqrt(-b)*arctan(sqrt(-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)/(b*cos(2*d*x + 2*c) + b))*sin(2*d*x + 2*c) + 4*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*cos(2*d*x +
2*c))*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) + 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) + b)*sin(2*d*x + 2*c) - (4*b^2*cos(2*d*x + 2*c) - 3*(3*a*b - 2*b^2)*cos(2*
d*x + 2*c)^2 + 9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/((d*cos(2*d*x +
2*c) - d)*sin(2*d*x + 2*c)), 1/8*(8*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(a - b)*ar
ctan(-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)*co
s(2*d*x + 2*c) + a - b))*sin(2*d*x + 2*c) - (15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos(2*d*x + 2
*c))*sqrt(-b)*arctan(sqrt(-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)
/(b*cos(2*d*x + 2*c) + b))*sin(2*d*x + 2*c) + (4*b^2*cos(2*d*x + 2*c) - 3*(3*a*b - 2*b^2)*cos(2*d*x + 2*c)^2 +
9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/((d*cos(2*d*x + 2*c) - d)*sin
(2*d*x + 2*c))]

Sympy [F]

\[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\int \left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]

[In]

integrate((a+b*cot(d*x+c)**2)**(5/2),x)

[Out]

Integral((a + b*cot(c + d*x)**2)**(5/2), x)

Maxima [F]

\[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\int { {\left (b \cot \left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*cot(d*x+c)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((b*cot(d*x + c)^2 + a)^(5/2), x)

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*cot(d*x+c)^2)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\int {\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x \]

[In]

int((a + b*cot(c + d*x)^2)^(5/2),x)

[Out]

int((a + b*cot(c + d*x)^2)^(5/2), x)