3.40 \(\int \csc (a+b x) \sin (2 a+2 b x) \, dx\)
Optimal. Leaf size=11 \[ \frac {2 \sin (a+b x)}{b} \]
[Out]
2*sin(b*x+a)/b
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used =
{4288, 2637} \[ \frac {2 \sin (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]*Sin[2*a + 2*b*x],x]
[Out]
(2*Sin[a + b*x])/b
Rule 2637
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]
Rubi steps
\begin {align*} \int \csc (a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cos (a+b x) \, dx\\ &=\frac {2 \sin (a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.01, size = 23, normalized size = 2.09 \[ 2 \left (\frac {\sin (a) \cos (b x)}{b}+\frac {\cos (a) \sin (b x)}{b}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x],x]
[Out]
2*((Cos[b*x]*Sin[a])/b + (Cos[a]*Sin[b*x])/b)
________________________________________________________________________________________
fricas [A] time = 0.47, size = 11, normalized size = 1.00 \[ \frac {2 \, \sin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(2*b*x+2*a),x, algorithm="fricas")
[Out]
2*sin(b*x + a)/b
________________________________________________________________________________________
giac [A] time = 0.36, size = 11, normalized size = 1.00 \[ \frac {2 \, \sin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(2*b*x+2*a),x, algorithm="giac")
[Out]
2*sin(b*x + a)/b
________________________________________________________________________________________
maple [A] time = 0.34, size = 12, normalized size = 1.09 \[ \frac {2 \sin \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)*sin(2*b*x+2*a),x)
[Out]
2*sin(b*x+a)/b
________________________________________________________________________________________
maxima [A] time = 0.34, size = 11, normalized size = 1.00 \[ \frac {2 \, \sin \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(2*b*x+2*a),x, algorithm="maxima")
[Out]
2*sin(b*x + a)/b
________________________________________________________________________________________
mupad [B] time = 0.02, size = 11, normalized size = 1.00 \[ \frac {2\,\sin \left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(2*a + 2*b*x)/sin(a + b*x),x)
[Out]
(2*sin(a + b*x))/b
________________________________________________________________________________________
sympy [B] time = 21.21, size = 3636, normalized size = 330.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*sin(2*b*x+2*a),x)
[Out]
4*Piecewise((x, Eq(a, 0) & Eq(b, 0)), (0, Eq(b, 0)), (sin(b*x)/b, Eq(a, 0)), (2*log(tan(a/2) + tan(b*x/2))*tan
(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(
a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan
(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(a/2) + tan(b*x/2
))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*t
an(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(a/2) + tan(b*x/2))*tan(a/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan
(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - 1/tan(a
/2))*tan(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 +
2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)*
*2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2)
- 1/tan(a/2))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2
)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)/(b*tan(a/2)**4*tan(b*x
/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*tan(a/2)**
4*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 +
b*tan(b*x/2)**2 + b) - 4*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)
**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*tan(a/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*t
an(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2
+ b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b), True))*cos(a)**2 - 2
*Piecewise((x, Eq(a, 0) & Eq(b, 0)), (0, Eq(b, 0)), (sin(b*x)/b, Eq(a, 0)), (2*log(tan(a/2) + tan(b*x/2))*tan(
a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a
/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(
a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(a/2) + tan(b*x/2)
)*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*ta
n(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(a/2) + tan(b*x/2))*tan(a/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(
a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - 1/tan(a/
2))*tan(a/2)**3*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2
*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**
2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2)
- 1/tan(a/2))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)
**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)/(b*tan(a/2)**4*tan(b*x/
2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*tan(a/2)**4
*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b
*tan(b*x/2)**2 + b) - 4*tan(a/2)**3/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)*
*2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 4*tan(a/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*ta
n(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 +
b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b), True)) - 2*Piecewise(
(zoo*x, Eq(a, 0) & Eq(b, 0)), (x/sin(a), Eq(b, 0)), (log(tan(b*x/2))/b, Eq(a, 0)), (log(tan(a/2) + tan(b*x/2))
/b - log(tan(b*x/2) - 1/tan(a/2))/b, True))*sin(a)*cos(a) + 4*Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (x/sin(a
), Eq(b, 0)), (log(tan(b*x/2))*tan(b*x/2)**2/(b*tan(b*x/2)**2 + b) + log(tan(b*x/2))/(b*tan(b*x/2)**2 + b) + 2
/(b*tan(b*x/2)**2 + b), Eq(a, 0)), (log(tan(a/2) + tan(b*x/2))*tan(a/2)**4*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*
x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + log(tan(a/2
) + tan(b*x/2))*tan(a/2)**4/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b
*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(tan(a/2) + tan(b*x/2))*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**4*ta
n(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - 2*log(t
an(a/2) + tan(b*x/2))*tan(a/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2
+ 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + log(tan(a/2) + tan(b*x/2))*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)
**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + log(tan(a/2) +
tan(b*x/2))/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b
*tan(b*x/2)**2 + b) - log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**4*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*
tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - log(tan(b*x/2) - 1/tan(
a/2))*tan(a/2)**4/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)*
*2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2) - 1/tan(a/2))*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)
**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 2*log(tan(b*x/2
) - 1/tan(a/2))*tan(a/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b
*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - log(tan(b*x/2) - 1/tan(a/2))*tan(b*x/2)**2/(b*tan(a/2)**4*tan(b*x/2)**2
+ b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) - log(tan(b*x/2) - 1/
tan(a/2))/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*t
an(b*x/2)**2 + b) - 2*tan(a/2)**4/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2
+ 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 4*tan(a/2)**3*tan(b*x/2)/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)
**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) + 4*tan(a/2)*tan(b*x/2)/(b*tan(a/
2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/2)**2 + b) +
2/(b*tan(a/2)**4*tan(b*x/2)**2 + b*tan(a/2)**4 + 2*b*tan(a/2)**2*tan(b*x/2)**2 + 2*b*tan(a/2)**2 + b*tan(b*x/
2)**2 + b), True))*sin(a)*cos(a)
________________________________________________________________________________________