3.1.6 \(\int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx\) [6]

3.1.6.1 Optimal result

Integrand size = 23, antiderivative size = 75 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2} a \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{3/2}}+\frac {2 a}{d e \sqrt {e \cot (c+d x)}} \]

output

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

3.1.6.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.55 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\frac {a \left (6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )+3 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )-3 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )+24 \sqrt {\tan (c+d x)}+8 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)\right )}{12 d (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)} \]

input

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

output

(a*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 6*Sqrt[2]*ArcTan[1 
+ Sqrt[2]*Sqrt[Tan[c + d*x]]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x 
]] + Tan[c + d*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c 
+ d*x]] + 24*Sqrt[Tan[c + d*x]] + 8*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c 
+ d*x]^2]*Tan[c + d*x]^(3/2)))/(12*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^( 
3/2))
 

3.1.6.3 Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4012, 3042, 4015, 221}

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.

\(\displaystyle \int \frac {a \cot (c+d x)+a}{(e \cot (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a-a \tan \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4012

\(\displaystyle \frac {\int \frac {a e-a e \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{e^2}+\frac {2 a}{d e \sqrt {e \cot (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {a e+a \tan \left (c+d x+\frac {\pi }{2}\right ) e}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}+\frac {2 a}{d e \sqrt {e \cot (c+d x)}}\)

\(\Big \downarrow \) 4015

\(\displaystyle \frac {2 a}{d e \sqrt {e \cot (c+d x)}}-\frac {2 a^2 \int \frac {1}{2 a^2 e^2-(a e+a \cot (c+d x) e)^2 \tan (c+d x)}d\frac {a e+a \cot (c+d x) e}{\sqrt {e \cot (c+d x)}}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 a}{d e \sqrt {e \cot (c+d x)}}-\frac {\sqrt {2} a \text {arctanh}\left (\frac {a e \cot (c+d x)+a e}{\sqrt {2} a \sqrt {e} \sqrt {e \cot (c+d x)}}\right )}{d e^{3/2}}\)

input

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

output

-((Sqrt[2]*a*ArcTanh[(a*e + a*e*Cot[c + d*x])/(Sqrt[2]*a*Sqrt[e]*Sqrt[e*Co 
t[c + d*x]])])/(d*e^(3/2))) + (2*a)/(d*e*Sqrt[e*Cot[c + d*x]])
 

3.1.6.3.1 Defintions of rubi rules used

rule 221

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4012

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ 
(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2)   Int[(a + b*Tan[e + f*x] 
)^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a 
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 
]
 

rule 4015

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[-2*(d^2/f)   Subst[Int[1/(2*c*d + b*x^2), x], x, (c 
- d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && 
 EqQ[c^2 - d^2, 0]
 

3.1.6.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(62)=124\).

Time = 0.05 (sec) , antiderivative size = 294, normalized size of antiderivative = 3.92

method	result	size
derivativedivides	\(-\frac {a \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\)	\(294\)
default	\(-\frac {a \left (\frac {\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d}\)	\(294\)
parts	\(-\frac {2 a e \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d}-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \,e^{2}}\)	\(297\)

input

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

output

-1/d*a*(2/e*(1/8/e*(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*co 
t(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+ 
c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c) 
)^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))-1/8/(e^2 
)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2) 
+(e^2)^(1/2))/(e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2) 
^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^ 
(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-2/e/(e*cot(d*x+c))^(1/2))
 

3.1.6.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (62) = 124\).

Time = 0.27 (sec) , antiderivative size = 321, normalized size of antiderivative = 4.28 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\left [\frac {4 \, a \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + \frac {\sqrt {2} {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \log \left (\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )}}{\sqrt {e}} + 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}{\sqrt {e}}}{2 \, {\left (d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}\right )}}, \frac {\sqrt {2} {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {-\frac {1}{e}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right ) + 2 \, a \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right )}{d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}}\right ] \]

input

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

output

[1/2*(4*a*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c) 
 + sqrt(2)*(a*e*cos(2*d*x + 2*c) + a*e)*log(sqrt(2)*sqrt((e*cos(2*d*x + 2* 
c) + e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2*c) - sin(2*d*x + 2*c) - 1)/sqrt(e 
) + 2*sin(2*d*x + 2*c) + 1)/sqrt(e))/(d*e^2*cos(2*d*x + 2*c) + d*e^2), (sq 
rt(2)*(a*e*cos(2*d*x + 2*c) + a*e)*sqrt(-1/e)*arctan(1/2*sqrt(2)*sqrt((e*c 
os(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sqrt(-1/e)*(cos(2*d*x + 2*c) + sin( 
2*d*x + 2*c) + 1)/(cos(2*d*x + 2*c) + 1)) + 2*a*sqrt((e*cos(2*d*x + 2*c) + 
 e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c))/(d*e^2*cos(2*d*x + 2*c) + d*e^2)]
 

3.1.6.6 Sympy [F]

\[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=a \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]

input

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

output

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

3.1.6.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\text {Exception raised: ValueError} \]

input

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

output

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

3.1.6.8 Giac [F]

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

input

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

output

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

3.1.6.9 Mupad [B] (verification not implemented)

Time = 13.33 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.12 \[ \int \frac {a+a \cot (c+d x)}{(e \cot (c+d x))^{3/2}} \, dx=\frac {2\,a}{d\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1+1{}\mathrm {i}\right )}{d\,e^{3/2}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (-1+1{}\mathrm {i}\right )}{d\,e^{3/2}} \]

input

int((a + a*cot(c + d*x))/(e*cot(c + d*x))^(3/2),x)
 

output

(2*a)/(d*e*(e*cot(c + d*x))^(1/2)) + ((-1)^(1/4)*a*atan(((-1)^(1/4)*(e*cot 
(c + d*x))^(1/2))/e^(1/2))*(1 + 1i))/(d*e^(3/2)) - ((-1)^(1/4)*a*atanh(((- 
1)^(1/4)*(e*cot(c + d*x))^(1/2))/e^(1/2))*(1 - 1i))/(d*e^(3/2))