\(\int \cot (c+d x) \sin (a+b x) \, dx\) [237]

Optimal result

Integrand size = 13, antiderivative size = 139 \[ \int \cot (c+d x) \sin (a+b x) \, dx=-\frac {i e^{-i (a+b x)}}{2 b}-\frac {i e^{i (a+b x)}}{2 b}+\frac {i e^{-i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b}+\frac {i e^{i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b} \]

[Out]

-1/2*I/b/exp(I*(b*x+a))-1/2*I*exp(I*(b*x+a))/b+I*hypergeom([1, -1/2*b/d],[1-1/2*b/d],exp(2*I*(d*x+c)))/b/exp(I
*(b*x+a))+I*exp(I*(b*x+a))*hypergeom([1, 1/2*b/d],[1+1/2*b/d],exp(2*I*(d*x+c)))/b

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4655, 2225, 2283} \[ \int \cot (c+d x) \sin (a+b x) \, dx=\frac {i e^{-i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b}+\frac {i e^{i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},\frac {b}{2 d}+1,e^{2 i (c+d x)}\right )}{b}-\frac {i e^{-i (a+b x)}}{2 b}-\frac {i e^{i (a+b x)}}{2 b} \]

[In]

Int[Cot[c + d*x]*Sin[a + b*x],x]

[Out]

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

Rule 2225

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 2283

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

Rule 4655

Int[Cot[(c_.) + (d_.)*(x_)]*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Int[-E^((-I)*(a + b*x))/2 + E^(I*(a + b*x))/
2 + 1/(E^(I*(a + b*x))*(1 - E^(2*I*(c + d*x)))) - E^(I*(a + b*x))/(1 - E^(2*I*(c + d*x))), x] /; FreeQ[{a, b,
c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{2} e^{-i (a+b x)}+\frac {1}{2} e^{i (a+b x)}+\frac {e^{-i (a+b x)}}{1-e^{2 i (c+d x)}}-\frac {e^{i (a+b x)}}{1-e^{2 i (c+d x)}}\right ) \, dx \\ & = -\left (\frac {1}{2} \int e^{-i (a+b x)} \, dx\right )+\frac {1}{2} \int e^{i (a+b x)} \, dx+\int \frac {e^{-i (a+b x)}}{1-e^{2 i (c+d x)}} \, dx-\int \frac {e^{i (a+b x)}}{1-e^{2 i (c+d x)}} \, dx \\ & = -\frac {i e^{-i (a+b x)}}{2 b}-\frac {i e^{i (a+b x)}}{2 b}+\frac {i e^{-i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b}+\frac {i e^{i (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},e^{2 i (c+d x)}\right )}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.37 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.87 \[ \int \cot (c+d x) \sin (a+b x) \, dx=\frac {-\cos (a) \cos (b x) \cot (c)-\frac {i e^{-i (a-2 c+b x)} \left (b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {b}{2 d},2-\frac {b}{2 d},e^{2 i (c+d x)}\right )-(b-2 d) \operatorname {Hypergeometric2F1}\left (1,-\frac {b}{2 d},1-\frac {b}{2 d},e^{2 i (c+d x)}\right )\right )}{(b-2 d) \left (-1+e^{2 i c}\right )}-\frac {i e^{i (a+2 c+b x)} \left (b e^{2 i d x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {b}{2 d},2+\frac {b}{2 d},e^{2 i (c+d x)}\right )-(b+2 d) \operatorname {Hypergeometric2F1}\left (1,\frac {b}{2 d},1+\frac {b}{2 d},e^{2 i (c+d x)}\right )\right )}{(b+2 d) \left (-1+e^{2 i c}\right )}+\cot (c) \sin (a) \sin (b x)}{b} \]

[In]

Integrate[Cot[c + d*x]*Sin[a + b*x],x]

[Out]

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

Maple [F]

\[\int \cot \left (d x +c \right ) \sin \left (x b +a \right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \cot (c+d x) \sin (a+b x) \, dx=\int { \cot \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \cot (c+d x) \sin (a+b x) \, dx=\int \sin {\left (a + b x \right )} \cot {\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \cot (c+d x) \sin (a+b x) \, dx=\int { \cot \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \cot (c+d x) \sin (a+b x) \, dx=\int { \cot \left (d x + c\right ) \sin \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(cot(c + d*x)*sin(a + b*x),x)

[Out]

int(cot(c + d*x)*sin(a + b*x), x)