\(\int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx\) [450]

Optimal result

Integrand size = 21, antiderivative size = 97 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=-\left (\left (a^4-6 a^2 b^2+b^4\right ) x\right )-\frac {4 a b^3 \log (\cos (c+d x))}{d}+\frac {4 a^3 b \log (\sin (c+d x))}{d}+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d} \] Output:

-(a^4-6*a^2*b^2+b^4)*x-4*a*b^3*ln(cos(d*x+c))/d+4*a^3*b*ln(sin(d*x+c))/d+b 
^2*(a^2+b^2)*tan(d*x+c)/d-a^2*cot(d*x+c)*(a+b*tan(d*x+c))^2/d
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.18 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=-\frac {a^4 \cot (c+d x)+\frac {1}{2} i (a-i b)^4 \log (i-\cot (c+d x))-\frac {1}{2} i (a+i b)^4 \log (i+\cot (c+d x))-4 a b^3 \log (\tan (c+d x))-b^4 \tan (c+d x)}{d} \] Input:

Integrate[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]
 

Output:

-((a^4*Cot[c + d*x] + (I/2)*(a - I*b)^4*Log[I - Cot[c + d*x]] - (I/2)*(a + 
 I*b)^4*Log[I + Cot[c + d*x]] - 4*a*b^3*Log[Tan[c + d*x]] - b^4*Tan[c + d* 
x])/d)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4048, 3042, 4120, 25, 3042, 4107, 3042, 25, 3956}

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.

\(\displaystyle \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x))^4}{\tan (c+d x)^2}dx\)

\(\Big \downarrow \) 4048

\(\displaystyle \int \cot (c+d x) (a+b \tan (c+d x)) \left (4 b a^2-\left (a^2-3 b^2\right ) \tan (c+d x) a+b \left (a^2+b^2\right ) \tan ^2(c+d x)\right )dx-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (c+d x)) \left (4 b a^2-\left (a^2-3 b^2\right ) \tan (c+d x) a+b \left (a^2+b^2\right ) \tan (c+d x)^2\right )}{\tan (c+d x)}dx-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}\)

\(\Big \downarrow \) 4120

\(\displaystyle -\int -\cot (c+d x) \left (4 b a^3+4 b^3 \tan ^2(c+d x) a-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)\right )dx+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \int \cot (c+d x) \left (4 b a^3+4 b^3 \tan ^2(c+d x) a-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)\right )dx+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {4 b a^3+4 b^3 \tan (c+d x)^2 a-\left (a^4-6 b^2 a^2+b^4\right ) \tan (c+d x)}{\tan (c+d x)}dx+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}\)

\(\Big \downarrow \) 4107

\(\displaystyle 4 a^3 b \int \cot (c+d x)dx+4 a b^3 \int \tan (c+d x)dx+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-x \left (a^4-6 a^2 b^2+b^4\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 4 a^3 b \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx+4 a b^3 \int \tan (c+d x)dx+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-x \left (a^4-6 a^2 b^2+b^4\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -4 a^3 b \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx+4 a b^3 \int \tan (c+d x)dx+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-x \left (a^4-6 a^2 b^2+b^4\right )\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {4 a^3 b \log (-\sin (c+d x))}{d}+\frac {b^2 \left (a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \cot (c+d x) (a+b \tan (c+d x))^2}{d}-x \left (a^4-6 a^2 b^2+b^4\right )-\frac {4 a b^3 \log (\cos (c+d x))}{d}\)

Input:

Int[Cot[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]
 

Output:

-((a^4 - 6*a^2*b^2 + b^4)*x) - (4*a*b^3*Log[Cos[c + d*x]])/d + (4*a^3*b*Lo 
g[-Sin[c + d*x]])/d + (b^2*(a^2 + b^2)*Tan[c + d*x])/d - (a^2*Cot[c + d*x] 
*(a + b*Tan[c + d*x])^2)/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

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

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4048

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m 
 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 
/(d*(n + 1)*(c^2 + d^2))   Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + 
f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c 
*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) 
*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( 
n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ 
[n, -1] && IntegerQ[2*m]
 

rule 4107

Int[((A_) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2 
)/tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[B*x, x] + (Simp[A   Int[1/Tan[ 
e + f*x], x], x] + Simp[C   Int[Tan[e + f*x], x], x]) /; FreeQ[{e, f, A, B, 
 C}, x] && NeQ[A, C]
 

rule 4120

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) 
*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f 
_.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 
 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2))   Int[(c + d*Tan[e + f*x])^n*Si 
mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* 
d*(n + 2) - b*(c*C - B*d*(n + 2)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, 
b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[c^2 + d^2, 0] && 
  !LtQ[n, -1]
 

Maple [A] (verified)

Time = 1.41 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {b^{4} \left (\tan \left (d x +c \right )-d x -c \right )-4 a \,b^{3} \ln \left (\cos \left (d x +c \right )\right )+6 b^{2} a^{2} \left (d x +c \right )+4 a^{3} b \ln \left (\sin \left (d x +c \right )\right )+a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(83\)
default	\(\frac {b^{4} \left (\tan \left (d x +c \right )-d x -c \right )-4 a \,b^{3} \ln \left (\cos \left (d x +c \right )\right )+6 b^{2} a^{2} \left (d x +c \right )+4 a^{3} b \ln \left (\sin \left (d x +c \right )\right )+a^{4} \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(83\)
parallelrisch	\(-\frac {-2 a^{3} b \left (2 \ln \left (\tan \left (d x +c \right )\right )-\ln \left (\sec \left (d x +c \right )^{2}\right )\right )-2 \ln \left (\sec \left (d x +c \right )^{2}\right ) a \,b^{3}+a^{4} x d -6 a^{2} b^{2} d x +b^{4} d x +\cot \left (d x +c \right ) a^{4}-b^{4} \tan \left (d x +c \right )}{d}\)	\(92\)
norman	\(\frac {\frac {b^{4} \tan \left (d x +c \right )^{2}}{d}+\left (-a^{4}+6 b^{2} a^{2}-b^{4}\right ) x \tan \left (d x +c \right )-\frac {a^{4}}{d}}{\tan \left (d x +c \right )}+\frac {4 a^{3} b \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {2 a b \left (a^{2}-b^{2}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{d}\)	\(105\)
risch	\(-4 i a^{3} b x +4 i a \,b^{3} x -a^{4} x +6 a^{2} b^{2} x -b^{4} x +\frac {8 i a \,b^{3} c}{d}-\frac {8 i a^{3} b c}{d}-\frac {2 i \left (a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+a^{4}+b^{4}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(169\)

Input:

int(cot(d*x+c)^2*(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)
 

Output:

1/d*(b^4*(tan(d*x+c)-d*x-c)-4*a*b^3*ln(cos(d*x+c))+6*b^2*a^2*(d*x+c)+4*a^3 
*b*ln(sin(d*x+c))+a^4*(-cot(d*x+c)-d*x-c))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.18 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {2 \, a^{3} b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) - 2 \, a b^{3} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + b^{4} \tan \left (d x + c\right )^{2} - a^{4} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} d x \tan \left (d x + c\right )}{d \tan \left (d x + c\right )} \] Input:

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^4,x, algorithm="fricas")
 

Output:

(2*a^3*b*log(tan(d*x + c)^2/(tan(d*x + c)^2 + 1))*tan(d*x + c) - 2*a*b^3*l 
og(1/(tan(d*x + c)^2 + 1))*tan(d*x + c) + b^4*tan(d*x + c)^2 - a^4 - (a^4 
- 6*a^2*b^2 + b^4)*d*x*tan(d*x + c))/(d*tan(d*x + c))
 

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.35 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=\begin {cases} \tilde {\infty } a^{4} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\left (c \right )}\right )^{4} \cot ^{2}{\left (c \right )} & \text {for}\: d = 0 \\\tilde {\infty } a^{4} x & \text {for}\: c = - d x \\- a^{4} x - \frac {a^{4}}{d \tan {\left (c + d x \right )}} - \frac {2 a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {4 a^{3} b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 6 a^{2} b^{2} x + \frac {2 a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} - b^{4} x + \frac {b^{4} \tan {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases} \] Input:

integrate(cot(d*x+c)**2*(a+b*tan(d*x+c))**4,x)
 

Output:

Piecewise((zoo*a**4*x, Eq(c, 0) & Eq(d, 0)), (x*(a + b*tan(c))**4*cot(c)** 
2, Eq(d, 0)), (zoo*a**4*x, Eq(c, -d*x)), (-a**4*x - a**4/(d*tan(c + d*x)) 
- 2*a**3*b*log(tan(c + d*x)**2 + 1)/d + 4*a**3*b*log(tan(c + d*x))/d + 6*a 
**2*b**2*x + 2*a*b**3*log(tan(c + d*x)**2 + 1)/d - b**4*x + b**4*tan(c + d 
*x)/d, True))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.91 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {4 \, a^{3} b \log \left (\tan \left (d x + c\right )\right ) + b^{4} \tan \left (d x + c\right ) - \frac {a^{4}}{\tan \left (d x + c\right )} - {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )} - 2 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{d} \] Input:

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^4,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

(4*a^3*b*log(tan(d*x + c)) + b^4*tan(d*x + c) - a^4/tan(d*x + c) - (a^4 - 
6*a^2*b^2 + b^4)*(d*x + c) - 2*(a^3*b - a*b^3)*log(tan(d*x + c)^2 + 1))/d
 

Giac [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.03 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {4 \, a^{3} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{d} + \frac {b^{4} \tan \left (d x + c\right )}{d} - \frac {a^{4}}{d \tan \left (d x + c\right )} - \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} {\left (d x + c\right )}}{d} - \frac {2 \, {\left (a^{3} b - a b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{d} \] Input:

integrate(cot(d*x+c)^2*(a+b*tan(d*x+c))^4,x, algorithm="giac")
 

Output:

4*a^3*b*log(abs(tan(d*x + c)))/d + b^4*tan(d*x + c)/d - a^4/(d*tan(d*x + c 
)) - (a^4 - 6*a^2*b^2 + b^4)*(d*x + c)/d - 2*(a^3*b - a*b^3)*log(tan(d*x + 
 c)^2 + 1)/d
 

Mupad [B] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {b^4\,\mathrm {tan}\left (c+d\,x\right )}{d}-\frac {a^4\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {4\,a^3\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2\,d} \] Input:

int(cot(c + d*x)^2*(a + b*tan(c + d*x))^4,x)
 

Output:

(b^4*tan(c + d*x))/d - (a^4*cot(c + d*x))/d - (log(tan(c + d*x) + 1i)*(a - 
 b*1i)^4*1i)/(2*d) + (log(tan(c + d*x) - 1i)*(a*1i - b)^4*1i)/(2*d) + (4*a 
^3*b*log(tan(c + d*x)))/d
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.70 \[ \int \cot ^2(c+d x) (a+b \tan (c+d x))^4 \, dx=\frac {-4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) a^{3} b +4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) a \,b^{3}-4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right ) a \,b^{3}-4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right ) a \,b^{3}+4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right ) a^{3} b -\cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} d x +6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{2} d x -\cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{4} d x +\sin \left (d x +c \right )^{2} a^{4}+\sin \left (d x +c \right )^{2} b^{4}-a^{4}}{\cos \left (d x +c \right ) \sin \left (d x +c \right ) d} \] Input:

int(cot(d*x+c)^2*(a+b*tan(d*x+c))^4,x)
 

Output:

( - 4*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**3*b + 4*co 
s(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a*b**3 - 4*cos(c + d* 
x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)*a*b**3 - 4*cos(c + d*x)*log(tan( 
(c + d*x)/2) + 1)*sin(c + d*x)*a*b**3 + 4*cos(c + d*x)*log(tan((c + d*x)/2 
))*sin(c + d*x)*a**3*b - cos(c + d*x)*sin(c + d*x)*a**4*d*x + 6*cos(c + d* 
x)*sin(c + d*x)*a**2*b**2*d*x - cos(c + d*x)*sin(c + d*x)*b**4*d*x + sin(c 
 + d*x)**2*a**4 + sin(c + d*x)**2*b**4 - a**4)/(cos(c + d*x)*sin(c + d*x)* 
d)