3.1.16 \(\int \frac {\tan ^3(d+e x)}{(a+b \cot (d+e x)+c \cot ^2(d+e x))^{3/2}} \, dx\) [16]

3.1.16.1 Optimal result

Integrand size = 33, antiderivative size = 1008 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{a^{3/2} e}+\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{7/2} e}-\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right ) \cot (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {2 \left (b^2-2 a c+b c \cot (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}-\frac {b \left (15 b^2-52 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^3 \left (b^2-4 a c\right ) e}-\frac {2 \left (b^2-2 a c+b c \cot (d+e x)\right ) \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}+\frac {\left (5 b^2-12 a c\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right ) e} \]

output

-arctanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1 
/2))/a^(3/2)/e+3/8*(-4*a*c+5*b^2)*arctanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/( 
a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))/a^(7/2)/e+2*(b^2-2*a*c+b*c*cot(e*x+d 
))/a/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)-2*(a*(b^2-2*(a-c 
)*c)+b*c*(a+c)*cot(e*x+d))/(b^2+(a-c)^2)/(-4*a*c+b^2)/e/(a+b*cot(e*x+d)+c* 
cot(e*x+d)^2)^(1/2)+1/2*arctanh(1/2*(b^2-(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1 
/2))-b*cot(e*x+d)*(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))*2^(1/2)/(a+b*cot(e* 
x+d)+c*cot(e*x+d)^2)^(1/2)/(2*a-2*c+(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2- 
b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2))*(2*a-2*c+(a^2-2*a*c+ 
b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^ 
(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-1/2*arctanh(1/2*(b^2-b*cot(e*x+d 
)*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))-(a-c)*(a-c+(a^2-2*a*c+b^2+c^2)^(1/2) 
))*2^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)/(2*a-2*c-(a^2-2*a*c+b^2+c 
^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2+(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2) 
)*(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2+(a-c)*(a^2- 
2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-1/4*b*(-52 
*a*c+15*b^2)*(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2)*tan(e*x+d)/a^3/(-4*a*c+ 
b^2)/e-2*(b^2-2*a*c+b*c*cot(e*x+d))*tan(e*x+d)^2/a/(-4*a*c+b^2)/e/(a+b*cot 
(e*x+d)+c*cot(e*x+d)^2)^(1/2)+1/2*(-12*a*c+5*b^2)*(a+b*cot(e*x+d)+c*cot(e* 
x+d)^2)^(1/2)*tan(e*x+d)^2/a^2/(-4*a*c+b^2)/e
 

3.1.16.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.66 (sec) , antiderivative size = 1401, normalized size of antiderivative = 1.39 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\frac {2 \left (-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )+\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \arctan \left (\frac {i b+2 c-(-2 i a-b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{-4 a+4 i b+4 c}-\frac {4 \sqrt {a+i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )-\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \arctan \left (\frac {-i b+2 c-(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{-4 a-4 i b+4 c}\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}-\frac {2 \tan ^3(d+e x) \left (-b^2+2 a c-a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {2 \left (b^3+a b (a-3 c)+a \left (2 a^2+b^2-2 a c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}+\frac {4 \left (b^2-4 a c\right ) \left (\frac {a^2}{\left (b^2-4 a c\right ) \left (\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}\right )}\right )^{3/2} \left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right ) \left (-\frac {a \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}{b^2-4 a c}\right )^{3/2}}{a^2 \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )^{3/2} \sqrt {1-\frac {\left (-\frac {a b}{b^2-4 a c}-\frac {2 a^2 \tan (d+e x)}{b^2-4 a c}\right )^2}{\frac {a^2 b^2}{\left (b^2-4 a c\right )^2}-\frac {4 a^3 c}{\left (b^2-4 a c\right )^2}}}}-\frac {2 \left (b \tan ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {\frac {\left (-6 a^2 b^2 c+24 a^3 c^2\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{4 a^{5/2}}+\frac {\left (6 a^2 b c-12 a^3 c \tan (d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{2 a^2}}{3 a}\right )}{c \left (b^2-4 a c\right )}\right )}{e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}}+\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (-\frac {2 \tan ^5(d+e x) \left (-b^2+2 a c-a b \tan (d+e x)\right )}{c \left (b^2-4 a c\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}-\frac {2 \left (b \tan ^4(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {-10 a c \tan ^3(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {20 a b c \tan ^2(d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}+\frac {\frac {\left (240 a^3 b^2 c^2-45 a^2 b^2 c \left (5 b^2-12 a c\right )+60 a^3 c^2 \left (5 b^2-12 a c\right )\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )}{4 a^{5/2}}+\frac {\left (-240 a^3 b c^2+45 a^2 b c \left (5 b^2-12 a c\right )-30 a^3 c \left (5 b^2-12 a c\right ) \tan (d+e x)\right ) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}{2 a^2}}{3 a}}{4 a}}{5 a}\right )}{c \left (b^2-4 a c\right )}\right )}{e \sqrt {\cot ^2(d+e x) \left (c+b \tan (d+e x)+a \tan ^2(d+e x)\right )}} \]

input

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

output

-((Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*((2*((-4*Sqrt[ 
a - I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) + (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTan 
[(I*b + 2*c - ((-2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a - I*b - c]*Sqrt[c + b 
*Tan[d + e*x] + a*Tan[d + e*x]^2])])/(-4*a + (4*I)*b + 4*c) - (4*Sqrt[a + 
I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) - (I/4)*(a - c)*(b^2 - 4*a*c))*ArcTan[((- 
I)*b + 2*c - ((2*I)*a - b)*Tan[d + e*x])/(2*Sqrt[a + I*b - c]*Sqrt[c + b*T 
an[d + e*x] + a*Tan[d + e*x]^2])])/(-4*a - (4*I)*b + 4*c)))/((b^2 + (a - c 
)^2)*(b^2 - 4*a*c)) - (2*Tan[d + e*x]^3*(-b^2 + 2*a*c - a*b*Tan[d + e*x])) 
/(c*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) - (2*(b^3 + 
 a*b*(a - 3*c) + a*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/((b^2 + (a - c)^2) 
*(b^2 - 4*a*c)*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]) + (4*(b^2 - 4* 
a*c)*(a^2/((b^2 - 4*a*c)*((a^2*b^2)/(b^2 - 4*a*c)^2 - (4*a^3*c)/(b^2 - 4*a 
*c)^2)))^(3/2)*(-((a*b)/(b^2 - 4*a*c)) - (2*a^2*Tan[d + e*x])/(b^2 - 4*a*c 
))*(-((a*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2))/(b^2 - 4*a*c)))^(3/2))/( 
a^2*(c + b*Tan[d + e*x] + a*Tan[d + e*x]^2)^(3/2)*Sqrt[1 - (-((a*b)/(b^2 - 
 4*a*c)) - (2*a^2*Tan[d + e*x])/(b^2 - 4*a*c))^2/((a^2*b^2)/(b^2 - 4*a*c)^ 
2 - (4*a^3*c)/(b^2 - 4*a*c)^2)]) - (2*(b*Tan[d + e*x]^2*Sqrt[c + b*Tan[d + 
 e*x] + a*Tan[d + e*x]^2] + (((-6*a^2*b^2*c + 24*a^3*c^2)*ArcTanh[(b + 2*a 
*Tan[d + e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])])/( 
4*a^(5/2)) + ((6*a^2*b*c - 12*a^3*c*Tan[d + e*x])*Sqrt[c + b*Tan[d + e*...
 

3.1.16.3 Rubi [A] (verified)

Time = 4.64 (sec) , antiderivative size = 985, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 4184, 7276, 2009}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot (d+e x)^3 \left (a+b \cot (d+e x)+c \cot (d+e x)^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\tan ^3(d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {\int \left (\frac {\tan ^3(d+e x)}{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}-\frac {\tan (d+e x)}{\left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}+\frac {\cot (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \left (c \cot ^2(d+e x)+b \cot (d+e x)+a\right )^{3/2}}\right )d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {\left (5 b^2-12 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \tan ^2(d+e x)}{2 a^2 \left (b^2-4 a c\right )}+\frac {2 \left (b^2+c \cot (d+e x) b-2 a c\right ) \tan ^2(d+e x)}{a \left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a} \tan (d+e x)}{4 a^3 \left (b^2-4 a c\right )}-\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{8 a^{7/2}}+\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{a^{3/2}}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \text {arctanh}\left (\frac {b^2-\left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) \cot (d+e x) b-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}-\frac {2 \left (b^2+c \cot (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}+\frac {2 \left (a \left (b^2-2 (a-c) c\right )+b c (a+c) \cot (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c \cot ^2(d+e x)+b \cot (d+e x)+a}}}{e}\)

input

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

output

-((ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*C 
ot[d + e*x]^2])]/a^(3/2) - (3*(5*b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x 
])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(7/2)) + 
 (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + 
 c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b^2 - (a - c)*(a - 
c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c 
 + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + 
c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2] 
]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a 
*c + c^2)^(3/2)) - (Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a 
^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(b 
^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) - b*(2*a - 2*c + Sqrt 
[a^2 + b^2 - 2*a*c + c^2])*Cot[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^ 
2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + 
b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(Sqrt[2 
]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) - (2*(b^2 - 2*a*c + b*c*Cot[d + e*x]))/ 
(a*(b^2 - 4*a*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (2*(a*(b^2 
 - 2*(a - c)*c) + b*c*(a + c)*Cot[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a 
*c)*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]) + (b*(15*b^2 - 52*a*c)*Sq 
rt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(4*a^3*(b^2 - 4...
 

3.1.16.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.1.16.4 Maple [F(-1)]

Timed out.

hanged

input

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

output

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

3.1.16.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20316 vs. \(2 (923) = 1846\).

Time = 7.93 (sec) , antiderivative size = 40633, normalized size of antiderivative = 40.31 \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.1.16.6 Sympy [F]

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

input

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

output

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

3.1.16.7 Maxima [F(-1)]

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

input

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

output

Timed out
 

3.1.16.8 Giac [F(-2)]

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

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument 
Value
 

3.1.16.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\tan ^3(d+e x)}{\left (a+b \cot (d+e x)+c \cot ^2(d+e x)\right )^{3/2}} \, dx=\text {Hanged} \]

input

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

output

\text{Hanged}