\(\int (a+b \cot ^2(c+d x))^{3/2} \, dx\) [32]

Optimal result

Integrand size = 16, antiderivative size = 126 \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=-\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {(3 a-2 b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3742, 427, 537, 223, 212, 385, 209} \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=-\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {\sqrt {b} (3 a-2 b) \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d} \]

[In]

Int[(a + b*Cot[c + d*x]^2)^(3/2),x]

[Out]

-(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - ((3*a - 2*b)*Sqrt[b]*ArcT
anh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(2*d) - (b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2])/(2
*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 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 )^{3/2}}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d} \\ & = -\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}-\frac {\text {Subst}\left (\int \frac {a (2 a-b)+(3 a-2 b) b x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{2 d} \\ & = -\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}-\frac {(a-b)^2 \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 {((3 a-2 b) b) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\cot (c+d x)\right )}{2 d} \\ & = -\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d}-\frac {(a-b)^2 \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 {((3 a-2 b) b) \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 )}{2 d} \\ & = -\frac {(a-b)^{3/2} \arctan \left (\frac {\sqrt {a-b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{d}-\frac {(3 a-2 b) \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \cot (c+d x)}{\sqrt {a+b \cot ^2(c+d x)}}\right )}{2 d}-\frac {b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{2 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13 \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\frac {2 (a-b)^{3/2} \arctan \left (\frac {\sqrt {b}+\sqrt {b} \cot ^2(c+d x)-\cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}}{\sqrt {a-b}}\right )-b \cot (c+d x) \sqrt {a+b \cot ^2(c+d x)}+(3 a-2 b) \sqrt {b} \log \left (-\sqrt {b} \cot (c+d x)+\sqrt {a+b \cot ^2(c+d x)}\right )}{2 d} \]

[In]

Integrate[(a + b*Cot[c + d*x]^2)^(3/2),x]

[Out]

(2*(a - b)^(3/2)*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] + (3*a - 2*b)*Sqrt[b]*Log[-(Sqrt[b]*Cot[c + d*x]) + Sqrt[a + b
*Cot[c + d*x]^2]])/(2*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(108)=216\).

Time = 0.04 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.37

method	result	size
derivativedivides	\(\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{d}-\frac {b \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{2 d}-\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{2 d}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}-\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\)	\(298\)
default	\(\frac {b^{\frac {3}{2}} \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{d}-\frac {b \cot \left (d x +c \right ) \sqrt {a +b \cot \left (d x +c \right )^{2}}}{2 d}-\frac {3 \sqrt {b}\, a \ln \left (\sqrt {b}\, \cot \left (d x +c \right )+\sqrt {a +b \cot \left (d x +c \right )^{2}}\right )}{2 d}-\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {2 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}-\frac {a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \cot \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \cot \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\)	\(298\)

[In]

int((a+b*cot(d*x+c)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/d*b^(3/2)*ln(b^(1/2)*cot(d*x+c)+(a+b*cot(d*x+c)^2)^(1/2))-1/2*b*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)/d-3/2/d*
b^(1/2)*a*ln(b^(1/2)*cot(d*x+c)+(a+b*cot(d*x+c)^2)^(1/2))-1/d*(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))+2/d*a/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^2*(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. 258 vs. \(2 (108) = 216\).

Time = 0.32 (sec) , antiderivative size = 1071, normalized size of antiderivative = 8.50 \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*cot(d*x+c)^2)^(3/2),x, algorithm="fricas")

[Out]

[-1/4*(2*(a - b)*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) + (3*a - 2*b)*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*(b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a
- b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), 1/2*((3*a - 2*b)*sqrt(-b)*arctan(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)/(b*cos(2*d*x + 2*c) + b))*sin(2*d*x + 2*c) -
(a - b)*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)/(co
s(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + b)*sin(2*d*x + 2*c) - (b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x
+ 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), -1/4*(4*(a - b)^(3/2)*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) + (3*a - 2*b)*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*(b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*
c)), -1/2*(2*(a - b)^(3/2)*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) - (3*a - 2*b)*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) + (b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*s
in(2*d*x + 2*c))]

Sympy [F]

\[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\int \left (a + b \cot ^{2}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((a+b*cot(d*x+c)**2)**(3/2),x)

[Out]

Integral((a + b*cot(c + d*x)**2)**(3/2), x)

Maxima [F]

\[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\int { {\left (b \cot \left (d x + c\right )^{2} + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*cot(d*x+c)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((b*cot(d*x + c)^2 + a)^(3/2), x)

Giac [F(-2)]

Exception generated. \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*cot(d*x+c)^2)^(3/2),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. \[ \int \left (a+b \cot ^2(c+d x)\right )^{3/2} \, dx=\int {\left (b\,{\mathrm {cot}\left (c+d\,x\right )}^2+a\right )}^{3/2} \,d x \]

[In]

int((a + b*cot(c + d*x)^2)^(3/2),x)

[Out]

int((a + b*cot(c + d*x)^2)^(3/2), x)