3.228 \(\int \cos (a+b x) \csc ^2(c+b x) \, dx\)

Optimal. Leaf size=35 \[ \frac{\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac{\cos (a-c) \csc (b x+c)}{b} \]

[Out]

-((Cos[a - c]*Csc[c + b*x])/b) + (ArcTanh[Cos[c + b*x]]*Sin[a - c])/b

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Rubi [A]  time = 0.028113, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4581, 2606, 8, 3770} \[ \frac{\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac{\cos (a-c) \csc (b x+c)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]*Csc[c + b*x]^2,x]

[Out]

-((Cos[a - c]*Csc[c + b*x])/b) + (ArcTanh[Cos[c + b*x]]*Sin[a - c])/b

Rule 4581

Int[Cos[v_]*Csc[w_]^(n_.), x_Symbol] :> Dist[Cos[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] - Dist[Sin[v - w],
Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3770

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

Rubi steps

\begin{align*} \int \cos (a+b x) \csc ^2(c+b x) \, dx &=\cos (a-c) \int \cot (c+b x) \csc (c+b x) \, dx-\sin (a-c) \int \csc (c+b x) \, dx\\ &=\frac{\tanh ^{-1}(\cos (c+b x)) \sin (a-c)}{b}-\frac{\cos (a-c) \operatorname{Subst}(\int 1 \, dx,x,\csc (c+b x))}{b}\\ &=-\frac{\cos (a-c) \csc (c+b x)}{b}+\frac{\tanh ^{-1}(\cos (c+b x)) \sin (a-c)}{b}\\ \end{align*}

Mathematica [C]  time = 0.0955334, size = 90, normalized size = 2.57 \[ -\frac{\cos (a-c) \csc (b x+c)}{b}+\frac{2 i \sin (a-c) \tan ^{-1}\left (\frac{(\cos (c)-i \sin (c)) \left (\cos (c) \cos \left (\frac{b x}{2}\right )-\sin (c) \sin \left (\frac{b x}{2}\right )\right )}{\sin (c) \cos \left (\frac{b x}{2}\right )+i \cos (c) \cos \left (\frac{b x}{2}\right )}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]*Csc[c + b*x]^2,x]

[Out]

-((Cos[a - c]*Csc[c + b*x])/b) + ((2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]
))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b

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Maple [B]  time = 0.452, size = 1062, normalized size = 30.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)/sin(b*x+c)^2,x)

[Out]

-1/b/(-1/2*cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+1/2*cos(c)*sin(a)*tan(1/2*b*x+1/2*a)^2+tan(1/2*b*x+1/2*a)*cos(a)
*cos(c)+tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+1/2*cos(a)*sin(c)-1/2*sin(a)*cos(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(
c)^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)/(cos(a)*sin(c)-sin(a)*cos(c))*tan(1/2*b*x+1/2*a)*cos(a)^2*cos(c)^2-2
/b/(-1/2*cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+1/2*cos(c)*sin(a)*tan(1/2*b*x+1/2*a)^2+tan(1/2*b*x+1/2*a)*cos(a)*c
os(c)+tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+1/2*cos(a)*sin(c)-1/2*sin(a)*cos(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)
^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)/(cos(a)*sin(c)-sin(a)*cos(c))*tan(1/2*b*x+1/2*a)*cos(a)*cos(c)*sin(a)*
sin(c)-1/b/(-1/2*cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+1/2*cos(c)*sin(a)*tan(1/2*b*x+1/2*a)^2+tan(1/2*b*x+1/2*a)*
cos(a)*cos(c)+tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+1/2*cos(a)*sin(c)-1/2*sin(a)*cos(c))/(cos(a)^2*cos(c)^2+cos(a)^
2*sin(c)^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)/(cos(a)*sin(c)-sin(a)*cos(c))*tan(1/2*b*x+1/2*a)*sin(a)^2*sin(
c)^2-1/b/(-1/2*cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+1/2*cos(c)*sin(a)*tan(1/2*b*x+1/2*a)^2+tan(1/2*b*x+1/2*a)*co
s(a)*cos(c)+tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+1/2*cos(a)*sin(c)-1/2*sin(a)*cos(c))/(cos(a)^2*cos(c)^2+cos(a)^2*
sin(c)^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)*cos(a)*cos(c)-1/b/(-1/2*cos(a)*sin(c)*tan(1/2*b*x+1/2*a)^2+1/2*c
os(c)*sin(a)*tan(1/2*b*x+1/2*a)^2+tan(1/2*b*x+1/2*a)*cos(a)*cos(c)+tan(1/2*b*x+1/2*a)*sin(a)*sin(c)+1/2*cos(a)
*sin(c)-1/2*sin(a)*cos(c))/(cos(a)^2*cos(c)^2+cos(a)^2*sin(c)^2+cos(c)^2*sin(a)^2+sin(a)^2*sin(c)^2)*sin(a)*si
n(c)+4/b/(2*cos(a)^2*cos(c)^2+2*cos(a)^2*sin(c)^2+2*cos(c)^2*sin(a)^2+2*sin(a)^2*sin(c)^2)/(-cos(a)^2*cos(c)^2
-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2)*arctan(1/2*(2*(sin(a)*cos(c)-cos(a)*sin(c))*tan(
1/2*b*x+1/2*a)+2*cos(a)*cos(c)+2*sin(a)*sin(c))/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)
^2*sin(c)^2)^(1/2))*cos(a)*sin(c)-4/b/(2*cos(a)^2*cos(c)^2+2*cos(a)^2*sin(c)^2+2*cos(c)^2*sin(a)^2+2*sin(a)^2*
sin(c)^2)/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2)*arctan(1/2*(2*(sin(
a)*cos(c)-cos(a)*sin(c))*tan(1/2*b*x+1/2*a)+2*cos(a)*cos(c)+2*sin(a)*sin(c))/(-cos(a)^2*cos(c)^2-cos(a)^2*sin(
c)^2-cos(c)^2*sin(a)^2-sin(a)^2*sin(c)^2)^(1/2))*sin(a)*cos(c)

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Maxima [B]  time = 1.3102, size = 608, normalized size = 17.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+c)^2,x, algorithm="maxima")

[Out]

1/2*(2*(sin(b*x + 2*a) + sin(b*x + 2*c))*cos(2*b*x + a + 2*c) - (cos(2*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(2*
b*x + a + 2*c)*cos(a)*sin(-a + c) + sin(2*b*x + a + 2*c)^2*sin(-a + c) - 2*sin(2*b*x + a + 2*c)*sin(a)*sin(-a
+ c) + (cos(a)^2 + sin(a)^2)*sin(-a + c))*log(cos(b*x)^2 + 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 - 2*sin(b
*x)*sin(c) + sin(c)^2) + (cos(2*b*x + a + 2*c)^2*sin(-a + c) - 2*cos(2*b*x + a + 2*c)*cos(a)*sin(-a + c) + sin
(2*b*x + a + 2*c)^2*sin(-a + c) - 2*sin(2*b*x + a + 2*c)*sin(a)*sin(-a + c) + (cos(a)^2 + sin(a)^2)*sin(-a + c
))*log(cos(b*x)^2 - 2*cos(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) - 2*(cos(b*x + 2
*a) + cos(b*x + 2*c))*sin(2*b*x + a + 2*c) - 2*cos(a)*sin(b*x + 2*a) - 2*cos(a)*sin(b*x + 2*c) + 2*cos(b*x + 2
*a)*sin(a) + 2*cos(b*x + 2*c)*sin(a))/(b*cos(2*b*x + a + 2*c)^2 - 2*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(2*b*
x + a + 2*c)^2 - 2*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*b)

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Fricas [B]  time = 0.517224, size = 203, normalized size = 5.8 \begin{align*} -\frac{\log \left (\frac{1}{2} \, \cos \left (b x + c\right ) + \frac{1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) - \log \left (-\frac{1}{2} \, \cos \left (b x + c\right ) + \frac{1}{2}\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) + 2 \, \cos \left (-a + c\right )}{2 \, b \sin \left (b x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+c)^2,x, algorithm="fricas")

[Out]

-1/2*(log(1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin(-a + c) - log(-1/2*cos(b*x + c) + 1/2)*sin(b*x + c)*sin(-a
+ c) + 2*cos(-a + c))/(b*sin(b*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+c)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.3323, size = 1326, normalized size = 37.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)/sin(b*x+c)^2,x, algorithm="giac")

[Out]

-1/2*(4*(tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^3 + 2*tan(1/2*a)^2*tan(1/2*c) - 2*tan(1/2*a)*tan(
1/2*c)^2 + tan(1/2*a) - tan(1/2*c))*log(abs(tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c) + tan(1/2*b*x + 1/2*a)
- tan(1/2*a) + tan(1/2*c)))/(tan(1/2*a)^3*tan(1/2*c)^3 + tan(1/2*a)^3*tan(1/2*c) + tan(1/2*a)^2*tan(1/2*c)^2 +
tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)^2 + tan(1/2*a)*tan(1/2*c) + tan(1/2*c)^2 + 1) - 4*(tan(1/2*a)^3*tan(1/2*
c) - 2*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)*tan(1/2*c)^3 + tan(1/2*a)^2 - 2*tan(1/2*a)*tan(1/2*c) + tan(1/2*
c)^2)*log(abs(tan(1/2*b*x + 1/2*a)*tan(1/2*a) - tan(1/2*b*x + 1/2*a)*tan(1/2*c) + tan(1/2*a)*tan(1/2*c) + 1))/
(tan(1/2*a)^3*tan(1/2*c)^2 - tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*a)^3 - tan(1/2*a)^2*tan(1/2*c) + tan(1/2*a)*t
an(1/2*c)^2 - tan(1/2*c)^3 + tan(1/2*a) - tan(1/2*c)) - (tan(1/2*b*x + 1/2*a)*tan(1/2*a)^4*tan(1/2*c)^4 - 2*ta
n(1/2*b*x + 1/2*a)*tan(1/2*a)^4*tan(1/2*c)^2 + 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^3*tan(1/2*c)^3 - 2*tan(1/2*a)
^4*tan(1/2*c)^3 - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^4 + 2*tan(1/2*a)^3*tan(1/2*c)^4 + tan(1/2*b*x
+ 1/2*a)*tan(1/2*a)^4 - 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^3*tan(1/2*c) + 2*tan(1/2*a)^4*tan(1/2*c) + 20*tan(1
/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^2 - 12*tan(1/2*a)^3*tan(1/2*c)^2 - 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*t
an(1/2*c)^3 + 12*tan(1/2*a)^2*tan(1/2*c)^3 + tan(1/2*b*x + 1/2*a)*tan(1/2*c)^4 - 2*tan(1/2*a)*tan(1/2*c)^4 - 2
*tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2 + 2*tan(1/2*a)^3 + 8*tan(1/2*b*x + 1/2*a)*tan(1/2*a)*tan(1/2*c) - 12*tan(1/
2*a)^2*tan(1/2*c) - 2*tan(1/2*b*x + 1/2*a)*tan(1/2*c)^2 + 12*tan(1/2*a)*tan(1/2*c)^2 - 2*tan(1/2*c)^3 + tan(1/
2*b*x + 1/2*a) - 2*tan(1/2*a) + 2*tan(1/2*c))/((tan(1/2*b*x + 1/2*a)^2*tan(1/2*a)^2*tan(1/2*c) - tan(1/2*b*x +
1/2*a)^2*tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*b*x + 1/2*a)^2*ta
n(1/2*a) - tan(1/2*b*x + 1/2*a)*tan(1/2*a)^2 - tan(1/2*b*x + 1/2*a)^2*tan(1/2*c) + 4*tan(1/2*b*x + 1/2*a)*tan(
1/2*a)*tan(1/2*c) - tan(1/2*a)^2*tan(1/2*c) - tan(1/2*b*x + 1/2*a)*tan(1/2*c)^2 + tan(1/2*a)*tan(1/2*c)^2 + ta
n(1/2*b*x + 1/2*a) - tan(1/2*a) + tan(1/2*c))*(tan(1/2*a)^2*tan(1/2*c) - tan(1/2*a)*tan(1/2*c)^2 + tan(1/2*a)
- tan(1/2*c))))/b