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

Optimal. Leaf size=76 \[ \frac{i e^{c (a+b x)}}{b c}-\frac{2 i e^{c (a+b x)} \text{Hypergeometric2F1}\left (1,-\frac{i b c}{2 e},1-\frac{i b c}{2 e},e^{2 i (d+e x)}\right )}{b c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.078103, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4443, 2194, 2251} \[ \frac{i e^{c (a+b x)}}{b c}-\frac{2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac{i b c}{2 e};1-\frac{i b c}{2 e};e^{2 i (d+e x)}\right )}{b c} \]

Antiderivative was successfully verified.

[In]

Int[E^(c*(a + b*x))*Cot[d + e*x],x]

[Out]

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

Rule 4443

Int[Cot[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Dist[(-I)^n, Int[ExpandInteg
rand[(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]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rubi steps

\begin{align*} \int e^{c (a+b x)} \cot (d+e x) \, dx &=-\left (i \int \left (-e^{c (a+b x)}-\frac{2 e^{c (a+b x)}}{-1+e^{2 i (d+e x)}}\right ) \, dx\right )\\ &=i \int e^{c (a+b x)} \, dx+2 i \int \frac{e^{c (a+b x)}}{-1+e^{2 i (d+e x)}} \, dx\\ &=\frac{i e^{c (a+b x)}}{b c}-\frac{2 i e^{c (a+b x)} \, _2F_1\left (1,-\frac{i b c}{2 e};1-\frac{i b c}{2 e};e^{2 i (d+e x)}\right )}{b c}\\ \end{align*}

Mathematica [B]  time = 1.25579, size = 163, normalized size = 2.14 \[ \frac{e^{c (a+b x)} \left (2 i b c e^{2 i (d+e x)} \text{Hypergeometric2F1}\left (1,1-\frac{i b c}{2 e},2-\frac{i b c}{2 e},e^{2 i (d+e x)}\right )+i (b c+2 i e) \left (-2 e^{2 i d} \text{Hypergeometric2F1}\left (1,-\frac{i b c}{2 e},1-\frac{i b c}{2 e},e^{2 i (d+e x)}\right )+e^{2 i d}+1\right )\right )}{b c \left (-1+e^{2 i d}\right ) (b c+2 i e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{c \left ( bx+a \right ) }}\cot \left ( ex+d \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot \left (e x + d\right ) e^{\left ({\left (b x + a\right )} c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cot \left (e x + d\right ) e^{\left (b c x + a c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a c} \int e^{b c x} \cot{\left (d + e x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cot \left (e x + d\right ) e^{\left ({\left (b x + a\right )} c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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