Integrand size = 16, antiderivative size = 171 \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=-\frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{8 d}-\frac {(7 a-4 b) b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{8 d}-\frac {b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{4 d} \] Output:
-(a-b)^(5/2)*arctan((a-b)^(1/2)*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2))/d-1/8 *b^(1/2)*(15*a^2-20*a*b+8*b^2)*arctanh(b^(1/2)*cot(d*x+c)/(a+b*cot(d*x+c)^ 2)^(1/2))/d-1/8*(7*a-4*b)*b*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)/d-1/4*b*co t(d*x+c)*(a+b*cot(d*x+c)^2)^(3/2)/d
Time = 0.67 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\frac {8 (a-b)^{5/2} \arctan \left (\frac {\sqrt {b}+\sqrt {b} \cot ^2(c+d x)-\cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{\sqrt {a-b}}\right )-b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)} \left (9 a-4 b+2 b \cot ^2(c+d x)\right )+\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \log \left (-\sqrt {b} \cot (c+d x)+\sqrt {a+b \cot ^2(c+d x)}\right )}{8 d} \] Input:
Integrate[(a + b*Cot[c + d*x]^2)^(5/2),x]
Output:
(8*(a - b)^(5/2)*ArcTan[(Sqrt[b] + Sqrt[b]*Cot[c + d*x]^2 - Cot[c + d*x]*S qrt[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)
Time = 0.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 4144, 318, 403, 398, 224, 219, 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.
\(\displaystyle \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a+b \tan \left (c+d x+\frac {\pi }{2}\right )^2\right )^{5/2}dx\) |
\(\Big \downarrow \) 4144 |
\(\displaystyle -\frac {\int \frac {\left (b \cot ^2(c+d x)+a\right )^{5/2}}{\cot ^2(c+d x)+1}d\cot (c+d x)}{d}\) |
\(\Big \downarrow \) 318 |
\(\displaystyle -\frac {\frac {1}{4} \int \frac {\sqrt {b \cot ^2(c+d x)+a} \left ((7 a-4 b) b \cot ^2(c+d x)+a (4 a-b)\right )}{\cot ^2(c+d x)+1}d\cot (c+d x)+\frac {1}{4} b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{d}\) |
\(\Big \downarrow \) 403 |
\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \int \frac {b \left (15 a^2-20 b a+8 b^2\right ) \cot ^2(c+d x)+a \left (8 a^2-9 b a+4 b^2\right )}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)+\frac {1}{2} b (7 a-4 b) \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}\right )+\frac {1}{4} b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{d}\) |
\(\Big \downarrow \) 398 |
\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (b \left (15 a^2-20 a b+8 b^2\right ) \int \frac {1}{\sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)+8 (a-b)^3 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)\right )+\frac {1}{2} b (7 a-4 b) \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}\right )+\frac {1}{4} b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (b \left (15 a^2-20 a b+8 b^2\right ) \int \frac {1}{1-\frac {b \cot ^2(c+d x)}{b \cot ^2(c+d x)+a}}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a}}+8 (a-b)^3 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)\right )+\frac {1}{2} b (7 a-4 b) \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}\right )+\frac {1}{4} b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^3 \int \frac {1}{\left (\cot ^2(c+d x)+1\right ) \sqrt {b \cot ^2(c+d x)+a}}d\cot (c+d x)+\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )\right )+\frac {1}{2} b (7 a-4 b) \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}\right )+\frac {1}{4} b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{d}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^3 \int \frac {1}{1-\frac {(b-a) \cot ^2(c+d x)}{b \cot ^2(c+d x)+a}}d\frac {\cot (c+d x)}{\sqrt {b \cot ^2(c+d x)+a}}+\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )\right )+\frac {1}{2} b (7 a-4 b) \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}\right )+\frac {1}{4} b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )+8 (a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )\right )+\frac {1}{2} b (7 a-4 b) \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}\right )+\frac {1}{4} b \cot (c+d x) \left (a+b \cot ^2(c+d x)\right )^{3/2}}{d}\) |
Input:
Int[(a + b*Cot[c + d*x]^2)^(5/2),x]
Output:
-(((b*Cot[c + d*x]*(a + b*Cot[c + d*x]^2)^(3/2))/4 + ((8*(a - b)^(5/2)*Arc Tan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]] + 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]])/2 + ((7*a - 4*b)*b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2])/2)/4)/d )
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[((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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
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[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(461\) vs. \(2(149)=298\).
Time = 0.34 (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 | \(-\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{d}-\frac {b^{2} \cot \left (d x +c \right )^{3} \sqrt {a +b \cot \left (d x +c \right )^{2}}}{4 d}-\frac {9 b a \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{8 d}-\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{8 d}+\frac {b^{2} \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{2 d}+\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{2 d}+\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}-\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}-\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\) | \(462\) |
Input:
int((a+b*cot(d*x+c)^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-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)*c ot(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)*co t(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (149) = 298\).
Time = 0.13 (sec) , antiderivative size = 1544, normalized size of antiderivative = 9.03 \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((a+b*cot(d*x+c)^2)^(5/2),x, algorithm="fricas")
Output:
[-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)*c os(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))*sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqr t(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((...
\[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\int \left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((a+b*cot(d*x+c)**2)**(5/2),x)
Output:
Integral((a + b*cot(c + d*x)**2)**(5/2), x)
\[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\int { {\left (b \cot \left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*cot(d*x+c)^2)^(5/2),x, algorithm="maxima")
Output:
integrate((b*cot(d*x + c)^2 + a)^(5/2), x)
Exception generated. \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*cot(d*x+c)^2)^(5/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\int {\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x \] Input:
int((a + b*cot(c + d*x)^2)^(5/2),x)
Output:
int((a + b*cot(c + d*x)^2)^(5/2), x)
\[ \int \left (a+b \cot ^2(c+d x)\right )^{5/2} \, dx=\left (\int \sqrt {\cot \left (d x +c \right )^{2} b +a}d x \right ) a^{2}+\left (\int \sqrt {\cot \left (d x +c \right )^{2} b +a}\, \cot \left (d x +c \right )^{4}d x \right ) b^{2}+2 \left (\int \sqrt {\cot \left (d x +c \right )^{2} b +a}\, \cot \left (d x +c \right )^{2}d x \right ) a b \] Input:
int((a+b*cot(d*x+c)^2)^(5/2),x)
Output:
int(sqrt(cot(c + d*x)**2*b + a),x)*a**2 + int(sqrt(cot(c + d*x)**2*b + a)* cot(c + d*x)**4,x)*b**2 + 2*int(sqrt(cot(c + d*x)**2*b + a)*cot(c + d*x)** 2,x)*a*b