3.1.52 \(\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx\) [52]
Optimal. Leaf size=12 \[ \frac {2 \log (\sin (a+b x))}{b} \]
[Out]
2*ln(sin(b*x+a))/b
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Rubi [A]
time = 0.02, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4373, 3556}
\begin {gather*} \frac {2 \log (\sin (a+b x))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^2*Sin[2*a + 2*b*x],x]
[Out]
(2*Log[Sin[a + b*x]])/b
Rule 3556
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4373
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 ^2(a+b x) \sin (2 a+2 b x) \, dx &=2 \int \cot (a+b x) \, dx\\ &=\frac {2 \log (\sin (a+b x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.67 \begin {gather*} \frac {2 (\log (\cos (a+b x))+\log (\tan (a+b x)))}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x],x]
[Out]
(2*(Log[Cos[a + b*x]] + Log[Tan[a + b*x]]))/b
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Maple [A]
time = 0.06, size = 13, normalized size = 1.08
| | |
| method |
result |
size |
| | |
| default |
\(\frac {2 \ln \left (\sin \left (x b +a \right )\right )}{b}\) |
\(13\) |
| risch |
\(-2 i x -\frac {4 i a}{b}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}{b}\) |
\(30\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^2*sin(2*b*x+2*a),x,method=_RETURNVERBOSE)
[Out]
2*ln(sin(b*x+a))/b
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (12) = 24\).
time = 0.27, size = 81, normalized size = 6.75 \begin {gather*} \frac {\log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*sin(2*b*x+2*a),x, algorithm="maxima")
[Out]
(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) + 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))/b
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Fricas [A]
time = 3.92, size = 14, normalized size = 1.17 \begin {gather*} \frac {2 \, \log \left (\frac {1}{2} \, \sin \left (b x + a\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*sin(2*b*x+2*a),x, algorithm="fricas")
[Out]
2*log(1/2*sin(b*x + a))/b
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 6307 vs.
\(2 (10) = 20\).
time = 155.40, size = 18894, normalized size = 1574.50 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**2*sin(2*b*x+2*a),x)
[Out]
4*Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0)), (log(sin(b*x))/b, Eq(a, 0)), (0, Eq(b, 0)), (-4*b*x*tan(a/2)**5*tan(
b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*
b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/
2) - b*tan(b*x/2)) - 4*b*x*tan(a/2)**4*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 -
b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*ta
n(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 4*b*x*tan(a/2)**4/(b*tan(a/2)**6*tan(b*x/2)
+ b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*
b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 8*b*x*tan(a
/2)**3*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(
b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2
- b*tan(a/2) - b*tan(b*x/2)) + 4*b*x*tan(a/2)**2*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(
b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan
(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 4*b*x*tan(a/2)**2/(b*tan(a/2)**6
*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x
/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) -
4*b*x*tan(a/2)*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)
**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b
*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(a/2) + tan(b*x/2))*tan(a/2)**6*tan(b*x/2)/(b*tan(a/2)**6*tan(b
*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2
- 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(ta
n(a/2) + tan(b*x/2))*tan(a/2)**5*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan
(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/
2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(a/2) + tan(b*x/2))*tan(a/2)**5/(b*tan(a/2
)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan
(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)
) - 7*log(tan(a/2) + tan(b*x/2))*tan(a/2)**4*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**
2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2
*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - 6*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3
*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x
/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 -
b*tan(a/2) - b*tan(b*x/2)) + 6*log(tan(a/2) + tan(b*x/2))*tan(a/2)**3/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**
5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 -
b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + 7*log(tan(a/2) + tan(b*x/2
))*tan(a/2)**2*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)
**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b
*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(a/2) + tan(b*x/2))*tan(a/2)*tan(b*x/2)**2/(b*tan(a/2)**6*tan(b
*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2
- 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(ta
n(a/2) + tan(b*x/2))*tan(a/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(
a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*t
an(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) - log(tan(a/2) + tan(b*x/2))*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) +
b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5 + b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*t
an(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*tan(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2)) + log(tan(b*x/2)
- 1/tan(a/2))*tan(a/2)**6*tan(b*x/2)/(b*tan(a/2)**6*tan(b*x/2) + b*tan(a/2)**5*tan(b*x/2)**2 - b*tan(a/2)**5
+ b*tan(a/2)**4*tan(b*x/2) + 2*b*tan(a/2)**3*tan(b*x/2)**2 - 2*b*tan(a/2)**3 - b*tan(a/2)**2*tan(b*x/2) + b*ta
n(a/2)*tan(b*x/2)**2 - b*tan(a/2) - b*tan(b*x/2...
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Giac [A]
time = 0.42, size = 13, normalized size = 1.08 \begin {gather*} \frac {2 \, \log \left ({\left | \sin \left (b x + a\right ) \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*sin(2*b*x+2*a),x, algorithm="giac")
[Out]
2*log(abs(sin(b*x + a)))/b
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Mupad [B]
time = 0.13, size = 13, normalized size = 1.08 \begin {gather*} \frac {\ln \left ({\sin \left (a+b\,x\right )}^2\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(2*a + 2*b*x)/sin(a + b*x)^2,x)
[Out]
log(sin(a + b*x)^2)/b
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