\(\int e^{c (a+b x)} \cot ^3(d+e x) \, dx\) [24]

Optimal result

Integrand size = 18, antiderivative size = 188 \[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=-\frac {i e^{c (a+b x)}}{b c}+\frac {6 i e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}-\frac {12 i e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}+\frac {8 i e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (3,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c} \] Output:

-I*exp(c*(b*x+a))/b/c+6*I*exp(c*(b*x+a))*hypergeom([1, -1/2*I*b*c/e],[1-1/ 
2*I*b*c/e],exp(2*I*(e*x+d)))/b/c-12*I*exp(c*(b*x+a))*hypergeom([2, -1/2*I* 
b*c/e],[1-1/2*I*b*c/e],exp(2*I*(e*x+d)))/b/c+8*I*exp(c*(b*x+a))*hypergeom( 
[3, -1/2*I*b*c/e],[1-1/2*I*b*c/e],exp(2*I*(e*x+d)))/b/c
 

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.12 \[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=\frac {1}{2} e^{c (a+b x)} \left (-\frac {2 \cot (d)}{b c}-\frac {\csc ^2(d+e x)}{e}+\frac {2 \left (b^2 c^2-2 e^2\right ) e^{2 i d} \left (i b c e^{2 i e x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i b c}{2 e},2-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )+(-i b c+2 e) \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )\right )}{b c (b c+2 i e) e^2 \left (-1+e^{2 i d}\right )}+\frac {b c \csc (d) \csc (d+e x) \sin (e x)}{e^2}\right ) \] Input:

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

Output:

(E^(c*(a + b*x))*((-2*Cot[d])/(b*c) - Csc[d + e*x]^2/e + (2*(b^2*c^2 - 2*e 
^2)*E^((2*I)*d)*(I*b*c*E^((2*I)*e*x)*Hypergeometric2F1[1, 1 - ((I/2)*b*c)/ 
e, 2 - ((I/2)*b*c)/e, E^((2*I)*(d + e*x))] + ((-I)*b*c + 2*e)*Hypergeometr 
ic2F1[1, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b*c)/e, E^((2*I)*(d + e*x))]))/(b*c* 
(b*c + (2*I)*e)*e^2*(-1 + E^((2*I)*d))) + (b*c*Csc[d]*Csc[d + e*x]*Sin[e*x 
])/e^2))/2
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4943, 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.

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

\(\Big \downarrow \) 4943

\(\displaystyle i \int \left (-e^{c (a+b x)}+\frac {6 e^{c (a+b x)}}{1-e^{2 i (d+e x)}}-\frac {12 e^{c (a+b x)}}{\left (1-e^{2 i (d+e x)}\right )^2}+\frac {8 e^{c (a+b x)}}{\left (1-e^{2 i (d+e x)}\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle i \left (\frac {6 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}-\frac {12 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}+\frac {8 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (3,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}-\frac {e^{c (a+b x)}}{b c}\right )\)

Input:

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

Output:

I*(-(E^(c*(a + b*x))/(b*c)) + (6*E^(c*(a + b*x))*Hypergeometric2F1[1, ((-1 
/2*I)*b*c)/e, 1 - ((I/2)*b*c)/e, E^((2*I)*(d + e*x))])/(b*c) - (12*E^(c*(a 
 + b*x))*Hypergeometric2F1[2, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b*c)/e, E^((2*I 
)*(d + e*x))])/(b*c) + (8*E^(c*(a + b*x))*Hypergeometric2F1[3, ((-1/2*I)*b 
*c)/e, 1 - ((I/2)*b*c)/e, E^((2*I)*(d + e*x))])/(b*c))
 

Defintions of rubi rules used

rule 2009

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

rule 4943

Int[Cot[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symb 
ol] :> Simp[(-I)^n   Int[ExpandIntegrand[F^(c*(a + b*x))*((1 + E^(2*I*(d + 
e*x)))^n/(1 - E^(2*I*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e} 
, x] && IntegerQ[n]
 

Maple [F]

\[\int {\mathrm e}^{c \left (b x +a \right )} \cot \left (e x +d \right )^{3}d x\]

Input:

int(exp(c*(b*x+a))*cot(e*x+d)^3,x)
 

Output:

int(exp(c*(b*x+a))*cot(e*x+d)^3,x)
 

Fricas [F]

\[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=\int { \cot \left (e x + d\right )^{3} e^{\left ({\left (b x + a\right )} c\right )} \,d x } \] Input:

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

Output:

integral(cot(e*x + d)^3*e^(b*c*x + a*c), x)
 

Sympy [F]

\[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=e^{a c} \int e^{b c x} \cot ^{3}{\left (d + e x \right )}\, dx \] Input:

integrate(exp(c*(b*x+a))*cot(e*x+d)**3,x)
 

Output:

exp(a*c)*Integral(exp(b*c*x)*cot(d + e*x)**3, x)
 

Maxima [F]

\[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=\int { \cot \left (e x + d\right )^{3} e^{\left ({\left (b x + a\right )} c\right )} \,d x } \] Input:

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

Output:

-(4*e*cos(2*e*x + 2*d)^2*e^(b*c*x + a*c) + b*c*e^(b*c*x + a*c)*sin(2*e*x + 
 2*d) + 4*e*e^(b*c*x + a*c)*sin(2*e*x + 2*d)^2 - 2*e*cos(2*e*x + 2*d)*e^(b 
*c*x + a*c) - (b*c*e^(b*c*x + a*c)*sin(2*e*x + 2*d) + 2*e*cos(2*e*x + 2*d) 
*e^(b*c*x + a*c))*cos(4*e*x + 4*d) + 2*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c 
) + (b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c))*cos(4*e*x + 4*d)^2 + 4*(b^2*c^2* 
e^4*e^(a*c) - 2*e^6*e^(a*c))*cos(2*e*x + 2*d)^2 + (b^2*c^2*e^4*e^(a*c) - 2 
*e^6*e^(a*c))*sin(4*e*x + 4*d)^2 - 4*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c)) 
*sin(4*e*x + 4*d)*sin(2*e*x + 2*d) + 4*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c 
))*sin(2*e*x + 2*d)^2 + 2*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c) - 2*(b^2*c^ 
2*e^4*e^(a*c) - 2*e^6*e^(a*c))*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 4*(b^2 
*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c))*cos(2*e*x + 2*d))*integrate(1/4*e^(b*c*x 
)*sin(e*x + d)/(e^4*cos(e*x + d)^2 + e^4*sin(e*x + d)^2 + 2*e^4*cos(e*x + 
d) + e^4), x) - 2*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c) + (b^2*c^2*e^4*e^(a 
*c) - 2*e^6*e^(a*c))*cos(4*e*x + 4*d)^2 + 4*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e 
^(a*c))*cos(2*e*x + 2*d)^2 + (b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c))*sin(4*e 
*x + 4*d)^2 - 4*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c))*sin(4*e*x + 4*d)*sin 
(2*e*x + 2*d) + 4*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c))*sin(2*e*x + 2*d)^2 
 + 2*(b^2*c^2*e^4*e^(a*c) - 2*e^6*e^(a*c) - 2*(b^2*c^2*e^4*e^(a*c) - 2*e^6 
*e^(a*c))*cos(2*e*x + 2*d))*cos(4*e*x + 4*d) - 4*(b^2*c^2*e^4*e^(a*c) - 2* 
e^6*e^(a*c))*cos(2*e*x + 2*d))*integrate(1/4*e^(b*c*x)*sin(e*x + d)/(e^...
 

Giac [F]

\[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=\int { \cot \left (e x + d\right )^{3} e^{\left ({\left (b x + a\right )} c\right )} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^3\,{\mathrm {e}}^{c\,\left (a+b\,x\right )} \,d x \] Input:

int(cot(d + e*x)^3*exp(c*(a + b*x)),x)
 

Output:

int(cot(d + e*x)^3*exp(c*(a + b*x)), x)
 

Reduce [F]

\[ \int e^{c (a+b x)} \cot ^3(d+e x) \, dx=e^{a c} \left (\int e^{b c x} \cot \left (e x +d \right )^{3}d x \right ) \] Input:

int(exp(c*(b*x+a))*cot(e*x+d)^3,x)
 

Output:

e**(a*c)*int(e**(b*c*x)*cot(d + e*x)**3,x)