3.198 \(\int \csc ^4(c+b x) \sin (a+b x) \, dx\)
Optimal. Leaf size=67 \[ -\frac {\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{2 b}-\frac {\sin (a-c) \csc ^3(b x+c)}{3 b}-\frac {\cos (a-c) \cot (b x+c) \csc (b x+c)}{2 b} \]
[Out]
-1/2*arctanh(cos(b*x+c))*cos(a-c)/b-1/2*cos(a-c)*cot(b*x+c)*csc(b*x+c)/b-1/3*csc(b*x+c)^3*sin(a-c)/b
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used =
{4582, 2606, 30, 3768, 3770} \[ -\frac {\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{2 b}-\frac {\sin (a-c) \csc ^3(b x+c)}{3 b}-\frac {\cos (a-c) \cot (b x+c) \csc (b x+c)}{2 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[c + b*x]^4*Sin[a + b*x],x]
[Out]
-(ArcTanh[Cos[c + b*x]]*Cos[a - c])/(2*b) - (Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(2*b) - (Csc[c + b*x]^3*Sin
[a - c])/(3*b)
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
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 3768
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4582
Int[Csc[w_]^(n_.)*Sin[v_], x_Symbol] :> Dist[Sin[v - w], Int[Cot[w]*Csc[w]^(n - 1), x], x] + Dist[Cos[v - w],
Int[Csc[w]^(n - 1), x], x] /; GtQ[n, 0] && FreeQ[v - w, x] && NeQ[w, v]
Rubi steps
\begin {align*} \int \csc ^4(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \csc ^3(c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc ^3(c+b x) \, dx\\ &=-\frac {\cos (a-c) \cot (c+b x) \csc (c+b x)}{2 b}+\frac {1}{2} \cos (a-c) \int \csc (c+b x) \, dx-\frac {\sin (a-c) \operatorname {Subst}\left (\int x^2 \, dx,x,\csc (c+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{2 b}-\frac {\cos (a-c) \cot (c+b x) \csc (c+b x)}{2 b}-\frac {\csc ^3(c+b x) \sin (a-c)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.58, size = 67, normalized size = 1.00 \[ -\frac {2 \sin (a-c) \csc ^3(b x+c)+3 \cos (a-c) \cot (b x+c) \csc (b x+c)+6 \cos (a-c) \tanh ^{-1}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right )}{6 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[c + b*x]^4*Sin[a + b*x],x]
[Out]
-1/6*(6*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + 3*Cos[a - c]*Cot[c + b*x]*Csc[c + b*x] + 2*Csc[c +
b*x]^3*Sin[a - c])/b
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fricas [B] time = 0.65, size = 141, normalized size = 2.10 \[ \frac {6 \, \cos \left (b x + c\right ) \cos \left (-a + c\right ) \sin \left (b x + c\right ) - 3 \, {\left (\cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) - \cos \left (-a + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) + 3 \, {\left (\cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) - \cos \left (-a + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 4 \, \sin \left (-a + c\right )}{12 \, {\left (b \cos \left (b x + c\right )^{2} - b\right )} \sin \left (b x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^4*sin(b*x+a),x, algorithm="fricas")
[Out]
1/12*(6*cos(b*x + c)*cos(-a + c)*sin(b*x + c) - 3*(cos(b*x + c)^2*cos(-a + c) - cos(-a + c))*log(1/2*cos(b*x +
c) + 1/2)*sin(b*x + c) + 3*(cos(b*x + c)^2*cos(-a + c) - cos(-a + c))*log(-1/2*cos(b*x + c) + 1/2)*sin(b*x +
c) - 4*sin(-a + c))/((b*cos(b*x + c)^2 - b)*sin(b*x + c))
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^4*sin(b*x+a),x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)2/b*((-64/3*tan((b*x+c)/2)^3*tan(a/2)^6*tan(c/2)^5-128/3*tan((b*x+c)/2)^3*tan(a/2)^6*tan(c/2)^3-64/3*tan((
b*x+c)/2)^3*tan(a/2)^6*tan(c/2)+64/3*tan((b*x+c)/2)^3*tan(a/2)^5*tan(c/2)^6+64/3*tan((b*x+c)/2)^3*tan(a/2)^5*t
an(c/2)^4-64/3*tan((b*x+c)/2)^3*tan(a/2)^5*tan(c/2)^2-64/3*tan((b*x+c)/2)^3*tan(a/2)^5-64/3*tan((b*x+c)/2)^3*t
an(a/2)^4*tan(c/2)^5-128/3*tan((b*x+c)/2)^3*tan(a/2)^4*tan(c/2)^3-64/3*tan((b*x+c)/2)^3*tan(a/2)^4*tan(c/2)+12
8/3*tan((b*x+c)/2)^3*tan(a/2)^3*tan(c/2)^6+128/3*tan((b*x+c)/2)^3*tan(a/2)^3*tan(c/2)^4-128/3*tan((b*x+c)/2)^3
*tan(a/2)^3*tan(c/2)^2-128/3*tan((b*x+c)/2)^3*tan(a/2)^3+64/3*tan((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^5+128/3*tan
((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^3+64/3*tan((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)+64/3*tan((b*x+c)/2)^3*tan(a/2)*t
an(c/2)^6+64/3*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2)^4-64/3*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2)^2-64/3*tan((b*x+c)
/2)^3*tan(a/2)+64/3*tan((b*x+c)/2)^3*tan(c/2)^5+128/3*tan((b*x+c)/2)^3*tan(c/2)^3+64/3*tan((b*x+c)/2)^3*tan(c/
2)+32*tan((b*x+c)/2)^2*tan(a/2)^6*tan(c/2)^6+32*tan((b*x+c)/2)^2*tan(a/2)^6*tan(c/2)^4-32*tan((b*x+c)/2)^2*tan
(a/2)^6*tan(c/2)^2-32*tan((b*x+c)/2)^2*tan(a/2)^6+128*tan((b*x+c)/2)^2*tan(a/2)^5*tan(c/2)^5+256*tan((b*x+c)/2
)^2*tan(a/2)^5*tan(c/2)^3+128*tan((b*x+c)/2)^2*tan(a/2)^5*tan(c/2)+32*tan((b*x+c)/2)^2*tan(a/2)^4*tan(c/2)^6+3
2*tan((b*x+c)/2)^2*tan(a/2)^4*tan(c/2)^4-32*tan((b*x+c)/2)^2*tan(a/2)^4*tan(c/2)^2-32*tan((b*x+c)/2)^2*tan(a/2
)^4+256*tan((b*x+c)/2)^2*tan(a/2)^3*tan(c/2)^5+512*tan((b*x+c)/2)^2*tan(a/2)^3*tan(c/2)^3+256*tan((b*x+c)/2)^2
*tan(a/2)^3*tan(c/2)-32*tan((b*x+c)/2)^2*tan(a/2)^2*tan(c/2)^6-32*tan((b*x+c)/2)^2*tan(a/2)^2*tan(c/2)^4+32*ta
n((b*x+c)/2)^2*tan(a/2)^2*tan(c/2)^2+32*tan((b*x+c)/2)^2*tan(a/2)^2+128*tan((b*x+c)/2)^2*tan(a/2)*tan(c/2)^5+2
56*tan((b*x+c)/2)^2*tan(a/2)*tan(c/2)^3+128*tan((b*x+c)/2)^2*tan(a/2)*tan(c/2)-32*tan((b*x+c)/2)^2*tan(c/2)^6-
32*tan((b*x+c)/2)^2*tan(c/2)^4+32*tan((b*x+c)/2)^2*tan(c/2)^2+32*tan((b*x+c)/2)^2-64*tan((b*x+c)/2)*tan(a/2)^6
*tan(c/2)^5-128*tan((b*x+c)/2)*tan(a/2)^6*tan(c/2)^3-64*tan((b*x+c)/2)*tan(a/2)^6*tan(c/2)+64*tan((b*x+c)/2)*t
an(a/2)^5*tan(c/2)^6+64*tan((b*x+c)/2)*tan(a/2)^5*tan(c/2)^4-64*tan((b*x+c)/2)*tan(a/2)^5*tan(c/2)^2-64*tan((b
*x+c)/2)*tan(a/2)^5-64*tan((b*x+c)/2)*tan(a/2)^4*tan(c/2)^5-128*tan((b*x+c)/2)*tan(a/2)^4*tan(c/2)^3-64*tan((b
*x+c)/2)*tan(a/2)^4*tan(c/2)+128*tan((b*x+c)/2)*tan(a/2)^3*tan(c/2)^6+128*tan((b*x+c)/2)*tan(a/2)^3*tan(c/2)^4
-128*tan((b*x+c)/2)*tan(a/2)^3*tan(c/2)^2-128*tan((b*x+c)/2)*tan(a/2)^3+64*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^
5+128*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^3+64*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)+64*tan((b*x+c)/2)*tan(a/2)*ta
n(c/2)^6+64*tan((b*x+c)/2)*tan(a/2)*tan(c/2)^4-64*tan((b*x+c)/2)*tan(a/2)*tan(c/2)^2-64*tan((b*x+c)/2)*tan(a/2
)+64*tan((b*x+c)/2)*tan(c/2)^5+128*tan((b*x+c)/2)*tan(c/2)^3+64*tan((b*x+c)/2)*tan(c/2))/(512*tan(a/2)^6*tan(c
/2)^6+1536*tan(a/2)^6*tan(c/2)^4+1536*tan(a/2)^6*tan(c/2)^2+512*tan(a/2)^6+1536*tan(a/2)^4*tan(c/2)^6+4608*tan
(a/2)^4*tan(c/2)^4+4608*tan(a/2)^4*tan(c/2)^2+1536*tan(a/2)^4+1536*tan(a/2)^2*tan(c/2)^6+4608*tan(a/2)^2*tan(c
/2)^4+4608*tan(a/2)^2*tan(c/2)^2+1536*tan(a/2)^2+512*tan(c/2)^6+1536*tan(c/2)^4+1536*tan(c/2)^2+512)+(-22*tan(
(b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^2+22*tan((b*x+c)/2)^3*tan(a/2)^2-88*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2)+22*tan
((b*x+c)/2)^3*tan(c/2)^2-22*tan((b*x+c)/2)^3-6*tan((b*x+c)/2)^2*tan(a/2)^2*tan(c/2)+6*tan((b*x+c)/2)^2*tan(a/2
)*tan(c/2)^2-6*tan((b*x+c)/2)^2*tan(a/2)+6*tan((b*x+c)/2)^2*tan(c/2)-3*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^2+3*
tan((b*x+c)/2)*tan(a/2)^2-12*tan((b*x+c)/2)*tan(a/2)*tan(c/2)+3*tan((b*x+c)/2)*tan(c/2)^2-3*tan((b*x+c)/2)-2*t
an(a/2)^2*tan(c/2)+2*tan(a/2)*tan(c/2)^2-2*tan(a/2)+2*tan(c/2))/(48*tan(a/2)^2*tan(c/2)^2+48*tan(a/2)^2+48*tan
(c/2)^2+48)/tan((b*x+c)/2)^3+(tan(a/2)^2*tan(c/2)^2-tan(a/2)^2+4*tan(a/2)*tan(c/2)-tan(c/2)^2+1)/(4*tan(a/2)^2
*tan(c/2)^2+4*tan(a/2)^2+4*tan(c/2)^2+4)*ln(abs(tan((b*x+c)/2))))
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maple [B] time = 4.90, size = 14880, normalized size = 222.09 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+c)^4*sin(b*x+a),x)
[Out]
result too large to display
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maxima [B] time = 0.42, size = 1773, normalized size = 26.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^4*sin(b*x+a),x, algorithm="maxima")
[Out]
1/12*(2*(3*cos(5*b*x + 2*a + 4*c) + 3*cos(5*b*x + 6*c) + 8*cos(3*b*x + 2*a + 2*c) - 8*cos(3*b*x + 4*c) - 3*cos
(b*x + 2*a) - 3*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) - 6*(3*cos(4*b*x + a + 4*c) - 3*cos(2*b*x + a + 2*c) + co
s(a))*cos(5*b*x + 2*a + 4*c) - 6*(3*cos(4*b*x + a + 4*c) - 3*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b*x + 6*c) -
6*(8*cos(3*b*x + 2*a + 2*c) - 8*cos(3*b*x + 4*c) - 3*cos(b*x + 2*a) - 3*cos(b*x + 2*c))*cos(4*b*x + a + 4*c)
+ 16*(3*cos(2*b*x + a + 2*c) - cos(a))*cos(3*b*x + 2*a + 2*c) - 16*(3*cos(2*b*x + a + 2*c) - cos(a))*cos(3*b*x
+ 4*c) - 18*(cos(b*x + 2*a) + cos(b*x + 2*c))*cos(2*b*x + a + 2*c) + 6*cos(b*x + 2*a)*cos(a) + 6*cos(b*x + 2*
c)*cos(a) - 3*(cos(6*b*x + a + 6*c)^2*cos(-a + c) + 9*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 9*cos(2*b*x + a + 2
*c)^2*cos(-a + c) - 6*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 9*cos(-a
+ c)*sin(4*b*x + a + 4*c)^2 + 9*cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 6*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a)
- 2*(3*cos(4*b*x + a + 4*c)*cos(-a + c) - 3*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*b*x
+ a + 6*c) - 6*(3*cos(2*b*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(4*b*x + a + 4*c) + (cos(a)^2 + si
n(a)^2)*cos(-a + c) - 2*(3*cos(-a + c)*sin(4*b*x + a + 4*c) - 3*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)
*sin(a))*sin(6*b*x + a + 6*c) - 6*(3*cos(-a + c)*sin(2*b*x + a + 2*c) - cos(-a + c)*sin(a))*sin(4*b*x + a + 4*
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) + 3*(cos(6*b*x
+ a + 6*c)^2*cos(-a + c) + 9*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 9*cos(2*b*x + a + 2*c)^2*cos(-a + c) - 6*cos
(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 9*cos(-a + c)*sin(4*b*x + a + 4*c)
^2 + 9*cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 6*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) - 2*(3*cos(4*b*x + a + 4
*c)*cos(-a + c) - 3*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*b*x + a + 6*c) - 6*(3*cos(2*b
*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(4*b*x + a + 4*c) + (cos(a)^2 + sin(a)^2)*cos(-a + c) - 2*(
3*cos(-a + c)*sin(4*b*x + a + 4*c) - 3*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(6*b*x + a +
6*c) - 6*(3*cos(-a + c)*sin(2*b*x + a + 2*c) - cos(-a + c)*sin(a))*sin(4*b*x + a + 4*c))*log(cos(b*x)^2 - 2*co
s(b*x)*cos(c) + cos(c)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(c) + sin(c)^2) + 2*(3*sin(5*b*x + 2*a + 4*c) + 3*sin(5*
b*x + 6*c) + 8*sin(3*b*x + 2*a + 2*c) - 8*sin(3*b*x + 4*c) - 3*sin(b*x + 2*a) - 3*sin(b*x + 2*c))*sin(6*b*x +
a + 6*c) - 6*(3*sin(4*b*x + a + 4*c) - 3*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 2*a + 4*c) - 6*(3*sin(4*b*
x + a + 4*c) - 3*sin(2*b*x + a + 2*c) + sin(a))*sin(5*b*x + 6*c) - 6*(8*sin(3*b*x + 2*a + 2*c) - 8*sin(3*b*x +
4*c) - 3*sin(b*x + 2*a) - 3*sin(b*x + 2*c))*sin(4*b*x + a + 4*c) + 16*(3*sin(2*b*x + a + 2*c) - sin(a))*sin(3
*b*x + 2*a + 2*c) - 16*(3*sin(2*b*x + a + 2*c) - sin(a))*sin(3*b*x + 4*c) - 18*(sin(b*x + 2*a) + sin(b*x + 2*c
))*sin(2*b*x + a + 2*c) + 6*sin(b*x + 2*a)*sin(a) + 6*sin(b*x + 2*c)*sin(a))/(b*cos(6*b*x + a + 6*c)^2 + 9*b*c
os(4*b*x + a + 4*c)^2 + 9*b*cos(2*b*x + a + 2*c)^2 - 6*b*cos(2*b*x + a + 2*c)*cos(a) + b*sin(6*b*x + a + 6*c)^
2 + 9*b*sin(4*b*x + a + 4*c)^2 + 9*b*sin(2*b*x + a + 2*c)^2 - 6*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + si
n(a)^2)*b - 2*(3*b*cos(4*b*x + a + 4*c) - 3*b*cos(2*b*x + a + 2*c) + b*cos(a))*cos(6*b*x + a + 6*c) - 6*(3*b*c
os(2*b*x + a + 2*c) - b*cos(a))*cos(4*b*x + a + 4*c) - 2*(3*b*sin(4*b*x + a + 4*c) - 3*b*sin(2*b*x + a + 2*c)
+ b*sin(a))*sin(6*b*x + a + 6*c) - 6*(3*b*sin(2*b*x + a + 2*c) - b*sin(a))*sin(4*b*x + a + 4*c))
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(a + b*x)/sin(c + b*x)^4,x)
[Out]
\text{Hanged}
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)**4*sin(b*x+a),x)
[Out]
Timed out
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