3.145 \(\int \csc (a+b x) \csc (c+b x) \, dx\)

Optimal. Leaf size=36 \[ \frac{\csc (a-c) \log (\sin (b x+c))}{b}-\frac{\csc (a-c) \log (\sin (a+b x))}{b} \]

[Out]

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

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Rubi [A]  time = 0.0184961, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4611, 3475} \[ \frac{\csc (a-c) \log (\sin (b x+c))}{b}-\frac{\csc (a-c) \log (\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]*Csc[c + b*x],x]

[Out]

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

Rule 4611

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

Rule 3475

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

Rubi steps

\begin{align*} \int \csc (a+b x) \csc (c+b x) \, dx &=-(\csc (a-c) \int \cot (a+b x) \, dx)+\csc (a-c) \int \cot (c+b x) \, dx\\ &=-\frac{\csc (a-c) \log (\sin (a+b x))}{b}+\frac{\csc (a-c) \log (\sin (c+b x))}{b}\\ \end{align*}

Mathematica [A]  time = 0.241302, size = 28, normalized size = 0.78 \[ -\frac{\csc (a-c) (\log (\sin (a+b x))-\log (\sin (b x+c)))}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]*Csc[c + b*x],x]

[Out]

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

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Maple [B]  time = 0.234, size = 169, normalized size = 4.7 \begin{align*}{\frac{\ln \left ( \tan \left ( bx+a \right ) \right ) }{b \left ( \cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) }}-{\frac{\ln \left ( \tan \left ( bx+a \right ) \cos \left ( a \right ) \cos \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \sin \left ( c \right ) +\cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) \cos \left ( a \right ) \cos \left ( c \right ) }{b \left ( \cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) \left ( \cos \left ( a \right ) \cos \left ( c \right ) +\sin \left ( a \right ) \sin \left ( c \right ) \right ) }}-{\frac{\ln \left ( \tan \left ( bx+a \right ) \cos \left ( a \right ) \cos \left ( c \right ) +\tan \left ( bx+a \right ) \sin \left ( a \right ) \sin \left ( c \right ) +\cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) \sin \left ( a \right ) \sin \left ( c \right ) }{b \left ( \cos \left ( a \right ) \sin \left ( c \right ) -\sin \left ( a \right ) \cos \left ( c \right ) \right ) \left ( \cos \left ( a \right ) \cos \left ( c \right ) +\sin \left ( a \right ) \sin \left ( c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)*csc(b*x+c),x)

[Out]

1/b/(cos(a)*sin(c)-sin(a)*cos(c))*ln(tan(b*x+a))-1/b/(cos(a)*sin(c)-sin(a)*cos(c))/(cos(a)*cos(c)+sin(a)*sin(c
))*ln(tan(b*x+a)*cos(a)*cos(c)+tan(b*x+a)*sin(a)*sin(c)+cos(a)*sin(c)-sin(a)*cos(c))*cos(a)*cos(c)-1/b/(cos(a)
*sin(c)-sin(a)*cos(c))/(cos(a)*cos(c)+sin(a)*sin(c))*ln(tan(b*x+a)*cos(a)*cos(c)+tan(b*x+a)*sin(a)*sin(c)+cos(
a)*sin(c)-sin(a)*cos(c))*sin(a)*sin(c)

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Maxima [B]  time = 1.11522, size = 761, normalized size = 21.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*csc(b*x+c),x, algorithm="maxima")

[Out]

-(2*((cos(2*a) - cos(2*c))*cos(a + c) + (sin(2*a) - sin(2*c))*sin(a + c))*arctan2(sin(b*x) + sin(a), cos(b*x)
- cos(a)) + 2*((cos(2*a) - cos(2*c))*cos(a + c) + (sin(2*a) - sin(2*c))*sin(a + c))*arctan2(sin(b*x) - sin(a),
cos(b*x) + cos(a)) - 2*((cos(2*a) - cos(2*c))*cos(a + c) + (sin(2*a) - sin(2*c))*sin(a + c))*arctan2(sin(b*x)
+ sin(c), cos(b*x) - cos(c)) - 2*((cos(2*a) - cos(2*c))*cos(a + c) + (sin(2*a) - sin(2*c))*sin(a + c))*arctan
2(sin(b*x) - sin(c), cos(b*x) + cos(c)) - ((sin(2*a) - sin(2*c))*cos(a + c) - (cos(2*a) - cos(2*c))*sin(a + c)
)*log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - ((sin(2*a) - si
n(2*c))*cos(a + c) - (cos(2*a) - cos(2*c))*sin(a + c))*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x
)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + ((sin(2*a) - sin(2*c))*cos(a + c) - (cos(2*a) - cos(2*c))*sin(a + c))*lo
g(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) + ((sin(2*a) - sin(2*
c))*cos(a + c) - (cos(2*a) - cos(2*c))*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*b*cos(2*a)*cos(2*c) - b*cos(2*c)^2 + 2*b*sin(2*a)*sin(2*c) - b*sin(2*c)^2
- (cos(2*a)^2 + sin(2*a)^2)*b)

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Fricas [B]  time = 2.75295, size = 281, normalized size = 7.81 \begin{align*} -\frac{\log \left (-\frac{1}{4} \, \cos \left (b x + c\right )^{2} + \frac{1}{4}\right ) - \log \left (-\frac{2 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) \sin \left (-a + c\right ) +{\left (2 \, \cos \left (-a + c\right )^{2} - 1\right )} \cos \left (b x + c\right )^{2} - \cos \left (-a + c\right )^{2}}{\cos \left (-a + c\right )^{2} + 2 \, \cos \left (-a + c\right ) + 1}\right )}{2 \, b \sin \left (-a + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*csc(b*x+c),x, algorithm="fricas")

[Out]

-1/2*(log(-1/4*cos(b*x + c)^2 + 1/4) - log(-(2*cos(b*x + c)*cos(-a + c)*sin(b*x + c)*sin(-a + c) + (2*cos(-a +
c)^2 - 1)*cos(b*x + c)^2 - cos(-a + c)^2)/(cos(-a + c)^2 + 2*cos(-a + c) + 1)))/(b*sin(-a + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*csc(b*x+c),x)

[Out]

Integral(csc(a + b*x)*csc(b*x + c), x)

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Giac [B]  time = 1.25554, size = 535, normalized size = 14.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*csc(b*x+c),x, algorithm="giac")

[Out]

1/2*((tan(1/2*a)^4*tan(1/2*c)^4 + 4*tan(1/2*a)^3*tan(1/2*c)^3 - tan(1/2*a)^4 + 4*tan(1/2*a)^3*tan(1/2*c) + 4*t
an(1/2*a)*tan(1/2*c)^3 - tan(1/2*c)^4 + 4*tan(1/2*a)*tan(1/2*c) + 1)*log(abs(tan(b*x + a)*tan(1/2*a)^2*tan(1/2
*c)^2 - tan(b*x + a)*tan(1/2*a)^2 + 4*tan(b*x + a)*tan(1/2*a)*tan(1/2*c) - 2*tan(1/2*a)^2*tan(1/2*c) - tan(b*x
+ a)*tan(1/2*c)^2 + 2*tan(1/2*a)*tan(1/2*c)^2 + tan(b*x + a) - 2*tan(1/2*a) + 2*tan(1/2*c)))/(tan(1/2*a)^4*ta
n(1/2*c)^3 - tan(1/2*a)^3*tan(1/2*c)^4 - tan(1/2*a)^4*tan(1/2*c) + 6*tan(1/2*a)^3*tan(1/2*c)^2 - 6*tan(1/2*a)^
2*tan(1/2*c)^3 + tan(1/2*a)*tan(1/2*c)^4 - tan(1/2*a)^3 + 6*tan(1/2*a)^2*tan(1/2*c) - 6*tan(1/2*a)*tan(1/2*c)^
2 + tan(1/2*c)^3 + tan(1/2*a) - tan(1/2*c)) - (tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1)*lo
g(abs(tan(b*x + a)))/(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