3.53 \(\int \csc ^2(a+b x) \csc (2 a+2 b x) \, dx\)
Optimal. Leaf size=30 \[ \frac{\log (\tan (a+b x))}{2 b}-\frac{\cot ^2(a+b x)}{4 b} \]
[Out]
-Cot[a + b*x]^2/(4*b) + Log[Tan[a + b*x]]/(2*b)
________________________________________________________________________________________
Rubi [A] time = 0.0420328, antiderivative size = 30, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used =
{4288, 2620, 14} \[ \frac{\log (\tan (a+b x))}{2 b}-\frac{\cot ^2(a+b x)}{4 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^2*Csc[2*a + 2*b*x],x]
[Out]
-Cot[a + b*x]^2/(4*b) + Log[Tan[a + b*x]]/(2*b)
Rule 4288
Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \csc (2 a+2 b x) \, dx &=\frac{1}{2} \int \csc ^3(a+b x) \sec (a+b x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^3} \, dx,x,\tan (a+b x)\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x^3}+\frac{1}{x}\right ) \, dx,x,\tan (a+b x)\right )}{2 b}\\ &=-\frac{\cot ^2(a+b x)}{4 b}+\frac{\log (\tan (a+b x))}{2 b}\\ \end{align*}
Mathematica [A] time = 0.051582, size = 34, normalized size = 1.13 \[ -\frac{\csc ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))}{4 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x],x]
[Out]
-(Csc[a + b*x]^2 + 2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]])/(4*b)
________________________________________________________________________________________
Maple [A] time = 0.03, size = 27, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}}}+{\frac{\ln \left ( \tan \left ( bx+a \right ) \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^2*csc(2*b*x+2*a),x)
[Out]
-1/4/b/sin(b*x+a)^2+1/2*ln(tan(b*x+a))/b
________________________________________________________________________________________
Maxima [B] time = 1.22505, size = 886, normalized size = 29.53 \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)^2*csc(2*b*x+2*a),x, algorithm="maxima")
[Out]
1/4*(4*cos(4*b*x + 4*a)*cos(2*b*x + 2*a) - 8*cos(2*b*x + 2*a)^2 + (2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a)
- cos(4*b*x + 4*a)^2 - 4*cos(2*b*x + 2*a)^2 - sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*si
n(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a) + cos(2*a)^2 + sin(2*b*x)^
2 - 2*sin(2*b*x)*sin(2*a) + sin(2*a)^2) - (2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - cos(4*b*x + 4*a)^2 -
4*cos(2*b*x + 2*a)^2 - sin(4*b*x + 4*a)^2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x + 2*a)^2 + 4*cos
(2*b*x + 2*a) - 1)*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)
- (2*(2*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - cos(4*b*x + 4*a)^2 - 4*cos(2*b*x + 2*a)^2 - sin(4*b*x + 4*a)^
2 + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 4*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a) - 1)*log(cos(b*x)^2 - 2*co
s(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(a)^2) + 4*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) -
8*sin(2*b*x + 2*a)^2 + 4*cos(2*b*x + 2*a))/(b*cos(4*b*x + 4*a)^2 + 4*b*cos(2*b*x + 2*a)^2 + b*sin(4*b*x + 4*a)
^2 - 4*b*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b*sin(2*b*x + 2*a)^2 - 2*(2*b*cos(2*b*x + 2*a) - b)*cos(4*b*x +
4*a) - 4*b*cos(2*b*x + 2*a) + b)
________________________________________________________________________________________
Fricas [B] time = 0.499717, size = 176, normalized size = 5.87 \begin{align*} -\frac{{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) -{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac{1}{4} \, \cos \left (b x + a\right )^{2} + \frac{1}{4}\right ) - 1}{4 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*csc(2*b*x+2*a),x, algorithm="fricas")
[Out]
-1/4*((cos(b*x + a)^2 - 1)*log(cos(b*x + a)^2) - (cos(b*x + a)^2 - 1)*log(-1/4*cos(b*x + a)^2 + 1/4) - 1)/(b*c
os(b*x + a)^2 - b)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{2}{\left (a + b x \right )} \csc{\left (2 a + 2 b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**2*csc(2*b*x+2*a),x)
[Out]
Integral(csc(a + b*x)**2*csc(2*a + 2*b*x), x)
________________________________________________________________________________________
Giac [B] time = 1.47037, size = 1098, normalized size = 36.6 \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)^2*csc(2*b*x+2*a),x, algorithm="giac")
[Out]
-1/4*((3*tan(b*x + 4*a)^2*tan(1/2*a)^24 - 180*tan(b*x + 4*a)^2*tan(1/2*a)^22 + 24*tan(b*x + 4*a)*tan(1/2*a)^23
+ tan(1/2*a)^24 + 4230*tan(b*x + 4*a)^2*tan(1/2*a)^20 - 1592*tan(b*x + 4*a)*tan(1/2*a)^21 + 48*tan(1/2*a)^22
- 48612*tan(b*x + 4*a)^2*tan(1/2*a)^18 + 31704*tan(b*x + 4*a)*tan(1/2*a)^19 - 3846*tan(1/2*a)^20 + 277965*tan(
b*x + 4*a)^2*tan(1/2*a)^16 - 259128*tan(b*x + 4*a)*tan(1/2*a)^17 + 51632*tan(1/2*a)^18 - 737640*tan(b*x + 4*a)
^2*tan(1/2*a)^14 + 982192*tan(b*x + 4*a)*tan(1/2*a)^15 - 274545*tan(1/2*a)^16 + 1008468*tan(b*x + 4*a)^2*tan(1
/2*a)^12 - 1871088*tan(b*x + 4*a)*tan(1/2*a)^13 + 733728*tan(1/2*a)^14 - 737640*tan(b*x + 4*a)^2*tan(1/2*a)^10
+ 1871088*tan(b*x + 4*a)*tan(1/2*a)^11 - 1018132*tan(1/2*a)^12 + 277965*tan(b*x + 4*a)^2*tan(1/2*a)^8 - 98219
2*tan(b*x + 4*a)*tan(1/2*a)^9 + 733728*tan(1/2*a)^10 - 48612*tan(b*x + 4*a)^2*tan(1/2*a)^6 + 259128*tan(b*x +
4*a)*tan(1/2*a)^7 - 274545*tan(1/2*a)^8 + 4230*tan(b*x + 4*a)^2*tan(1/2*a)^4 - 31704*tan(b*x + 4*a)*tan(1/2*a)
^5 + 51632*tan(1/2*a)^6 - 180*tan(b*x + 4*a)^2*tan(1/2*a)^2 + 1592*tan(b*x + 4*a)*tan(1/2*a)^3 - 3846*tan(1/2*
a)^4 + 3*tan(b*x + 4*a)^2 - 24*tan(b*x + 4*a)*tan(1/2*a) + 48*tan(1/2*a)^2 + 1)/((tan(1/2*a)^12 - 30*tan(1/2*a
)^10 + 255*tan(1/2*a)^8 - 452*tan(1/2*a)^6 + 255*tan(1/2*a)^4 - 30*tan(1/2*a)^2 + 1)*(tan(b*x + 4*a)*tan(1/2*a
)^6 - 15*tan(b*x + 4*a)*tan(1/2*a)^4 + 6*tan(1/2*a)^5 + 15*tan(b*x + 4*a)*tan(1/2*a)^2 - 20*tan(1/2*a)^3 - tan
(b*x + 4*a) + 6*tan(1/2*a))^2) - 2*log(abs(tan(b*x + 4*a)*tan(1/2*a)^6 - 15*tan(b*x + 4*a)*tan(1/2*a)^4 + 6*ta
n(1/2*a)^5 + 15*tan(b*x + 4*a)*tan(1/2*a)^2 - 20*tan(1/2*a)^3 - tan(b*x + 4*a) + 6*tan(1/2*a))) + 2*log(abs(6*
tan(b*x + 4*a)*tan(1/2*a)^5 - tan(1/2*a)^6 - 20*tan(b*x + 4*a)*tan(1/2*a)^3 + 15*tan(1/2*a)^4 + 6*tan(b*x + 4*
a)*tan(1/2*a) - 15*tan(1/2*a)^2 + 1)))/b