3.200 \(\int \csc ^6(c+b x) \sin (a+b x) \, dx\)
Optimal. Leaf size=94 \[ -\frac {3 \cos (a-c) \tanh ^{-1}(\cos (b x+c))}{8 b}-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}-\frac {\cos (a-c) \cot (b x+c) \csc ^3(b x+c)}{4 b}-\frac {3 \cos (a-c) \cot (b x+c) \csc (b x+c)}{8 b} \]
[Out]
-3/8*arctanh(cos(b*x+c))*cos(a-c)/b-3/8*cos(a-c)*cot(b*x+c)*csc(b*x+c)/b-1/4*cos(a-c)*cot(b*x+c)*csc(b*x+c)^3/
b-1/5*csc(b*x+c)^5*sin(a-c)/b
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Rubi [A] time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00,
number of steps used = 6, 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} \[ -\frac {3 \cos (a-c) \tanh ^{-1}(\cos (b x+c))}{8 b}-\frac {\sin (a-c) \csc ^5(b x+c)}{5 b}-\frac {\cos (a-c) \cot (b x+c) \csc ^3(b x+c)}{4 b}-\frac {3 \cos (a-c) \cot (b x+c) \csc (b x+c)}{8 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[c + b*x]^6*Sin[a + b*x],x]
[Out]
(-3*ArcTanh[Cos[c + b*x]]*Cos[a - c])/(8*b) - (3*Cos[a - c]*Cot[c + b*x]*Csc[c + b*x])/(8*b) - (Cos[a - c]*Cot
[c + b*x]*Csc[c + b*x]^3)/(4*b) - (Csc[c + b*x]^5*Sin[a - c])/(5*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 ^6(c+b x) \sin (a+b x) \, dx &=\cos (a-c) \int \csc ^5(c+b x) \, dx+\sin (a-c) \int \cot (c+b x) \csc ^5(c+b x) \, dx\\ &=-\frac {\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}+\frac {1}{4} (3 \cos (a-c)) \int \csc ^3(c+b x) \, dx-\frac {\sin (a-c) \operatorname {Subst}\left (\int x^4 \, dx,x,\csc (c+b x)\right )}{b}\\ &=-\frac {3 \cos (a-c) \cot (c+b x) \csc (c+b x)}{8 b}-\frac {\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}-\frac {\csc ^5(c+b x) \sin (a-c)}{5 b}+\frac {1}{8} (3 \cos (a-c)) \int \csc (c+b x) \, dx\\ &=-\frac {3 \tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{8 b}-\frac {3 \cos (a-c) \cot (c+b x) \csc (c+b x)}{8 b}-\frac {\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}-\frac {\csc ^5(c+b x) \sin (a-c)}{5 b}\\ \end {align*}
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Mathematica [A] time = 1.16, size = 79, normalized size = 0.84 \[ -\frac {2 \csc ^5(b x+c) (5 \cos (a-c) (14 \sin (2 (b x+c))-3 \sin (4 (b x+c)))+64 \sin (a-c))+480 \cos (a-c) \tanh ^{-1}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right )}{640 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[c + b*x]^6*Sin[a + b*x],x]
[Out]
-1/640*(480*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + 2*Csc[c + b*x]^5*(64*Sin[a - c] + 5*Cos[a - c]*
(14*Sin[2*(c + b*x)] - 3*Sin[4*(c + b*x)])))/b
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fricas [B] time = 0.49, size = 197, normalized size = 2.10 \[ -\frac {15 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 15 \, {\left (\cos \left (b x + c\right )^{4} \cos \left (-a + c\right ) - 2 \, \cos \left (b x + c\right )^{2} \cos \left (-a + c\right ) + \cos \left (-a + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + c\right ) + \frac {1}{2}\right ) \sin \left (b x + c\right ) - 10 \, {\left (3 \, \cos \left (b x + c\right )^{3} \cos \left (-a + c\right ) - 5 \, \cos \left (b x + c\right ) \cos \left (-a + c\right )\right )} \sin \left (b x + c\right ) - 16 \, \sin \left (-a + c\right )}{80 \, {\left (b \cos \left (b x + c\right )^{4} - 2 \, b \cos \left (b x + c\right )^{2} + b\right )} \sin \left (b x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="fricas")
[Out]
-1/80*(15*(cos(b*x + c)^4*cos(-a + c) - 2*cos(b*x + c)^2*cos(-a + c) + cos(-a + c))*log(1/2*cos(b*x + c) + 1/2
)*sin(b*x + c) - 15*(cos(b*x + c)^4*cos(-a + c) - 2*cos(b*x + c)^2*cos(-a + c) + cos(-a + c))*log(-1/2*cos(b*x
+ c) + 1/2)*sin(b*x + c) - 10*(3*cos(b*x + c)^3*cos(-a + c) - 5*cos(b*x + c)*cos(-a + c))*sin(b*x + c) - 16*s
in(-a + c))/((b*cos(b*x + c)^4 - 2*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)^6*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*((-1048576/5*tan((b*x+c)/2)^5*tan(a/2)^10*tan(c/2)^9-4194304/5*tan((b*x+c)/2)^5*tan(a/2)^10*tan(c/2)^7
-6291456/5*tan((b*x+c)/2)^5*tan(a/2)^10*tan(c/2)^5-4194304/5*tan((b*x+c)/2)^5*tan(a/2)^10*tan(c/2)^3-1048576/5
*tan((b*x+c)/2)^5*tan(a/2)^10*tan(c/2)+1048576/5*tan((b*x+c)/2)^5*tan(a/2)^9*tan(c/2)^10+3145728/5*tan((b*x+c)
/2)^5*tan(a/2)^9*tan(c/2)^8+2097152/5*tan((b*x+c)/2)^5*tan(a/2)^9*tan(c/2)^6-2097152/5*tan((b*x+c)/2)^5*tan(a/
2)^9*tan(c/2)^4-3145728/5*tan((b*x+c)/2)^5*tan(a/2)^9*tan(c/2)^2-1048576/5*tan((b*x+c)/2)^5*tan(a/2)^9-3145728
/5*tan((b*x+c)/2)^5*tan(a/2)^8*tan(c/2)^9-12582912/5*tan((b*x+c)/2)^5*tan(a/2)^8*tan(c/2)^7-18874368/5*tan((b*
x+c)/2)^5*tan(a/2)^8*tan(c/2)^5-12582912/5*tan((b*x+c)/2)^5*tan(a/2)^8*tan(c/2)^3-3145728/5*tan((b*x+c)/2)^5*t
an(a/2)^8*tan(c/2)+4194304/5*tan((b*x+c)/2)^5*tan(a/2)^7*tan(c/2)^10+12582912/5*tan((b*x+c)/2)^5*tan(a/2)^7*ta
n(c/2)^8+8388608/5*tan((b*x+c)/2)^5*tan(a/2)^7*tan(c/2)^6-8388608/5*tan((b*x+c)/2)^5*tan(a/2)^7*tan(c/2)^4-125
82912/5*tan((b*x+c)/2)^5*tan(a/2)^7*tan(c/2)^2-4194304/5*tan((b*x+c)/2)^5*tan(a/2)^7-2097152/5*tan((b*x+c)/2)^
5*tan(a/2)^6*tan(c/2)^9-8388608/5*tan((b*x+c)/2)^5*tan(a/2)^6*tan(c/2)^7-12582912/5*tan((b*x+c)/2)^5*tan(a/2)^
6*tan(c/2)^5-8388608/5*tan((b*x+c)/2)^5*tan(a/2)^6*tan(c/2)^3-2097152/5*tan((b*x+c)/2)^5*tan(a/2)^6*tan(c/2)+6
291456/5*tan((b*x+c)/2)^5*tan(a/2)^5*tan(c/2)^10+18874368/5*tan((b*x+c)/2)^5*tan(a/2)^5*tan(c/2)^8+12582912/5*
tan((b*x+c)/2)^5*tan(a/2)^5*tan(c/2)^6-12582912/5*tan((b*x+c)/2)^5*tan(a/2)^5*tan(c/2)^4-18874368/5*tan((b*x+c
)/2)^5*tan(a/2)^5*tan(c/2)^2-6291456/5*tan((b*x+c)/2)^5*tan(a/2)^5+2097152/5*tan((b*x+c)/2)^5*tan(a/2)^4*tan(c
/2)^9+8388608/5*tan((b*x+c)/2)^5*tan(a/2)^4*tan(c/2)^7+12582912/5*tan((b*x+c)/2)^5*tan(a/2)^4*tan(c/2)^5+83886
08/5*tan((b*x+c)/2)^5*tan(a/2)^4*tan(c/2)^3+2097152/5*tan((b*x+c)/2)^5*tan(a/2)^4*tan(c/2)+4194304/5*tan((b*x+
c)/2)^5*tan(a/2)^3*tan(c/2)^10+12582912/5*tan((b*x+c)/2)^5*tan(a/2)^3*tan(c/2)^8+8388608/5*tan((b*x+c)/2)^5*ta
n(a/2)^3*tan(c/2)^6-8388608/5*tan((b*x+c)/2)^5*tan(a/2)^3*tan(c/2)^4-12582912/5*tan((b*x+c)/2)^5*tan(a/2)^3*ta
n(c/2)^2-4194304/5*tan((b*x+c)/2)^5*tan(a/2)^3+3145728/5*tan((b*x+c)/2)^5*tan(a/2)^2*tan(c/2)^9+12582912/5*tan
((b*x+c)/2)^5*tan(a/2)^2*tan(c/2)^7+18874368/5*tan((b*x+c)/2)^5*tan(a/2)^2*tan(c/2)^5+12582912/5*tan((b*x+c)/2
)^5*tan(a/2)^2*tan(c/2)^3+3145728/5*tan((b*x+c)/2)^5*tan(a/2)^2*tan(c/2)+1048576/5*tan((b*x+c)/2)^5*tan(a/2)*t
an(c/2)^10+3145728/5*tan((b*x+c)/2)^5*tan(a/2)*tan(c/2)^8+2097152/5*tan((b*x+c)/2)^5*tan(a/2)*tan(c/2)^6-20971
52/5*tan((b*x+c)/2)^5*tan(a/2)*tan(c/2)^4-3145728/5*tan((b*x+c)/2)^5*tan(a/2)*tan(c/2)^2-1048576/5*tan((b*x+c)
/2)^5*tan(a/2)+1048576/5*tan((b*x+c)/2)^5*tan(c/2)^9+4194304/5*tan((b*x+c)/2)^5*tan(c/2)^7+6291456/5*tan((b*x+
c)/2)^5*tan(c/2)^5+4194304/5*tan((b*x+c)/2)^5*tan(c/2)^3+1048576/5*tan((b*x+c)/2)^5*tan(c/2)+262144*tan((b*x+c
)/2)^4*tan(a/2)^10*tan(c/2)^10+786432*tan((b*x+c)/2)^4*tan(a/2)^10*tan(c/2)^8+524288*tan((b*x+c)/2)^4*tan(a/2)
^10*tan(c/2)^6-524288*tan((b*x+c)/2)^4*tan(a/2)^10*tan(c/2)^4-786432*tan((b*x+c)/2)^4*tan(a/2)^10*tan(c/2)^2-2
62144*tan((b*x+c)/2)^4*tan(a/2)^10+1048576*tan((b*x+c)/2)^4*tan(a/2)^9*tan(c/2)^9+4194304*tan((b*x+c)/2)^4*tan
(a/2)^9*tan(c/2)^7+6291456*tan((b*x+c)/2)^4*tan(a/2)^9*tan(c/2)^5+4194304*tan((b*x+c)/2)^4*tan(a/2)^9*tan(c/2)
^3+1048576*tan((b*x+c)/2)^4*tan(a/2)^9*tan(c/2)+786432*tan((b*x+c)/2)^4*tan(a/2)^8*tan(c/2)^10+2359296*tan((b*
x+c)/2)^4*tan(a/2)^8*tan(c/2)^8+1572864*tan((b*x+c)/2)^4*tan(a/2)^8*tan(c/2)^6-1572864*tan((b*x+c)/2)^4*tan(a/
2)^8*tan(c/2)^4-2359296*tan((b*x+c)/2)^4*tan(a/2)^8*tan(c/2)^2-786432*tan((b*x+c)/2)^4*tan(a/2)^8+4194304*tan(
(b*x+c)/2)^4*tan(a/2)^7*tan(c/2)^9+16777216*tan((b*x+c)/2)^4*tan(a/2)^7*tan(c/2)^7+25165824*tan((b*x+c)/2)^4*t
an(a/2)^7*tan(c/2)^5+16777216*tan((b*x+c)/2)^4*tan(a/2)^7*tan(c/2)^3+4194304*tan((b*x+c)/2)^4*tan(a/2)^7*tan(c
/2)+524288*tan((b*x+c)/2)^4*tan(a/2)^6*tan(c/2)^10+1572864*tan((b*x+c)/2)^4*tan(a/2)^6*tan(c/2)^8+1048576*tan(
(b*x+c)/2)^4*tan(a/2)^6*tan(c/2)^6-1048576*tan((b*x+c)/2)^4*tan(a/2)^6*tan(c/2)^4-1572864*tan((b*x+c)/2)^4*tan
(a/2)^6*tan(c/2)^2-524288*tan((b*x+c)/2)^4*tan(a/2)^6+6291456*tan((b*x+c)/2)^4*tan(a/2)^5*tan(c/2)^9+25165824*
tan((b*x+c)/2)^4*tan(a/2)^5*tan(c/2)^7+37748736*tan((b*x+c)/2)^4*tan(a/2)^5*tan(c/2)^5+25165824*tan((b*x+c)/2)
^4*tan(a/2)^5*tan(c/2)^3+6291456*tan((b*x+c)/2)^4*tan(a/2)^5*tan(c/2)-524288*tan((b*x+c)/2)^4*tan(a/2)^4*tan(c
/2)^10-1572864*tan((b*x+c)/2)^4*tan(a/2)^4*tan(c/2)^8-1048576*tan((b*x+c)/2)^4*tan(a/2)^4*tan(c/2)^6+1048576*t
an((b*x+c)/2)^4*tan(a/2)^4*tan(c/2)^4+1572864*tan((b*x+c)/2)^4*tan(a/2)^4*tan(c/2)^2+524288*tan((b*x+c)/2)^4*t
an(a/2)^4+4194304*tan((b*x+c)/2)^4*tan(a/2)^3*tan(c/2)^9+16777216*tan((b*x+c)/2)^4*tan(a/2)^3*tan(c/2)^7+25165
824*tan((b*x+c)/2)^4*tan(a/2)^3*tan(c/2)^5+16777216*tan((b*x+c)/2)^4*tan(a/2)^3*tan(c/2)^3+4194304*tan((b*x+c)
/2)^4*tan(a/2)^3*tan(c/2)-786432*tan((b*x+c)/2)^4*tan(a/2)^2*tan(c/2)^10-2359296*tan((b*x+c)/2)^4*tan(a/2)^2*t
an(c/2)^8-1572864*tan((b*x+c)/2)^4*tan(a/2)^2*tan(c/2)^6+1572864*tan((b*x+c)/2)^4*tan(a/2)^2*tan(c/2)^4+235929
6*tan((b*x+c)/2)^4*tan(a/2)^2*tan(c/2)^2+786432*tan((b*x+c)/2)^4*tan(a/2)^2+1048576*tan((b*x+c)/2)^4*tan(a/2)*
tan(c/2)^9+4194304*tan((b*x+c)/2)^4*tan(a/2)*tan(c/2)^7+6291456*tan((b*x+c)/2)^4*tan(a/2)*tan(c/2)^5+4194304*t
an((b*x+c)/2)^4*tan(a/2)*tan(c/2)^3+1048576*tan((b*x+c)/2)^4*tan(a/2)*tan(c/2)-262144*tan((b*x+c)/2)^4*tan(c/2
)^10-786432*tan((b*x+c)/2)^4*tan(c/2)^8-524288*tan((b*x+c)/2)^4*tan(c/2)^6+524288*tan((b*x+c)/2)^4*tan(c/2)^4+
786432*tan((b*x+c)/2)^4*tan(c/2)^2+262144*tan((b*x+c)/2)^4-1048576*tan((b*x+c)/2)^3*tan(a/2)^10*tan(c/2)^9-419
4304*tan((b*x+c)/2)^3*tan(a/2)^10*tan(c/2)^7-6291456*tan((b*x+c)/2)^3*tan(a/2)^10*tan(c/2)^5-4194304*tan((b*x+
c)/2)^3*tan(a/2)^10*tan(c/2)^3-1048576*tan((b*x+c)/2)^3*tan(a/2)^10*tan(c/2)+1048576*tan((b*x+c)/2)^3*tan(a/2)
^9*tan(c/2)^10+3145728*tan((b*x+c)/2)^3*tan(a/2)^9*tan(c/2)^8+2097152*tan((b*x+c)/2)^3*tan(a/2)^9*tan(c/2)^6-2
097152*tan((b*x+c)/2)^3*tan(a/2)^9*tan(c/2)^4-3145728*tan((b*x+c)/2)^3*tan(a/2)^9*tan(c/2)^2-1048576*tan((b*x+
c)/2)^3*tan(a/2)^9-3145728*tan((b*x+c)/2)^3*tan(a/2)^8*tan(c/2)^9-12582912*tan((b*x+c)/2)^3*tan(a/2)^8*tan(c/2
)^7-18874368*tan((b*x+c)/2)^3*tan(a/2)^8*tan(c/2)^5-12582912*tan((b*x+c)/2)^3*tan(a/2)^8*tan(c/2)^3-3145728*ta
n((b*x+c)/2)^3*tan(a/2)^8*tan(c/2)+4194304*tan((b*x+c)/2)^3*tan(a/2)^7*tan(c/2)^10+12582912*tan((b*x+c)/2)^3*t
an(a/2)^7*tan(c/2)^8+8388608*tan((b*x+c)/2)^3*tan(a/2)^7*tan(c/2)^6-8388608*tan((b*x+c)/2)^3*tan(a/2)^7*tan(c/
2)^4-12582912*tan((b*x+c)/2)^3*tan(a/2)^7*tan(c/2)^2-4194304*tan((b*x+c)/2)^3*tan(a/2)^7-2097152*tan((b*x+c)/2
)^3*tan(a/2)^6*tan(c/2)^9-8388608*tan((b*x+c)/2)^3*tan(a/2)^6*tan(c/2)^7-12582912*tan((b*x+c)/2)^3*tan(a/2)^6*
tan(c/2)^5-8388608*tan((b*x+c)/2)^3*tan(a/2)^6*tan(c/2)^3-2097152*tan((b*x+c)/2)^3*tan(a/2)^6*tan(c/2)+6291456
*tan((b*x+c)/2)^3*tan(a/2)^5*tan(c/2)^10+18874368*tan((b*x+c)/2)^3*tan(a/2)^5*tan(c/2)^8+12582912*tan((b*x+c)/
2)^3*tan(a/2)^5*tan(c/2)^6-12582912*tan((b*x+c)/2)^3*tan(a/2)^5*tan(c/2)^4-18874368*tan((b*x+c)/2)^3*tan(a/2)^
5*tan(c/2)^2-6291456*tan((b*x+c)/2)^3*tan(a/2)^5+2097152*tan((b*x+c)/2)^3*tan(a/2)^4*tan(c/2)^9+8388608*tan((b
*x+c)/2)^3*tan(a/2)^4*tan(c/2)^7+12582912*tan((b*x+c)/2)^3*tan(a/2)^4*tan(c/2)^5+8388608*tan((b*x+c)/2)^3*tan(
a/2)^4*tan(c/2)^3+2097152*tan((b*x+c)/2)^3*tan(a/2)^4*tan(c/2)+4194304*tan((b*x+c)/2)^3*tan(a/2)^3*tan(c/2)^10
+12582912*tan((b*x+c)/2)^3*tan(a/2)^3*tan(c/2)^8+8388608*tan((b*x+c)/2)^3*tan(a/2)^3*tan(c/2)^6-8388608*tan((b
*x+c)/2)^3*tan(a/2)^3*tan(c/2)^4-12582912*tan((b*x+c)/2)^3*tan(a/2)^3*tan(c/2)^2-4194304*tan((b*x+c)/2)^3*tan(
a/2)^3+3145728*tan((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^9+12582912*tan((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^7+18874368
*tan((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^5+12582912*tan((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^3+3145728*tan((b*x+c)/2)
^3*tan(a/2)^2*tan(c/2)+1048576*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2)^10+3145728*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2
)^8+2097152*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2)^6-2097152*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2)^4-3145728*tan((b*x
+c)/2)^3*tan(a/2)*tan(c/2)^2-1048576*tan((b*x+c)/2)^3*tan(a/2)+1048576*tan((b*x+c)/2)^3*tan(c/2)^9+4194304*tan
((b*x+c)/2)^3*tan(c/2)^7+6291456*tan((b*x+c)/2)^3*tan(c/2)^5+4194304*tan((b*x+c)/2)^3*tan(c/2)^3+1048576*tan((
b*x+c)/2)^3*tan(c/2)+2097152*tan((b*x+c)/2)^2*tan(a/2)^10*tan(c/2)^10+6291456*tan((b*x+c)/2)^2*tan(a/2)^10*tan
(c/2)^8+4194304*tan((b*x+c)/2)^2*tan(a/2)^10*tan(c/2)^6-4194304*tan((b*x+c)/2)^2*tan(a/2)^10*tan(c/2)^4-629145
6*tan((b*x+c)/2)^2*tan(a/2)^10*tan(c/2)^2-2097152*tan((b*x+c)/2)^2*tan(a/2)^10+8388608*tan((b*x+c)/2)^2*tan(a/
2)^9*tan(c/2)^9+33554432*tan((b*x+c)/2)^2*tan(a/2)^9*tan(c/2)^7+50331648*tan((b*x+c)/2)^2*tan(a/2)^9*tan(c/2)^
5+33554432*tan((b*x+c)/2)^2*tan(a/2)^9*tan(c/2)^3+8388608*tan((b*x+c)/2)^2*tan(a/2)^9*tan(c/2)+6291456*tan((b*
x+c)/2)^2*tan(a/2)^8*tan(c/2)^10+18874368*tan((b*x+c)/2)^2*tan(a/2)^8*tan(c/2)^8+12582912*tan((b*x+c)/2)^2*tan
(a/2)^8*tan(c/2)^6-12582912*tan((b*x+c)/2)^2*tan(a/2)^8*tan(c/2)^4-18874368*tan((b*x+c)/2)^2*tan(a/2)^8*tan(c/
2)^2-6291456*tan((b*x+c)/2)^2*tan(a/2)^8+33554432*tan((b*x+c)/2)^2*tan(a/2)^7*tan(c/2)^9+134217728*tan((b*x+c)
/2)^2*tan(a/2)^7*tan(c/2)^7+201326592*tan((b*x+c)/2)^2*tan(a/2)^7*tan(c/2)^5+134217728*tan((b*x+c)/2)^2*tan(a/
2)^7*tan(c/2)^3+33554432*tan((b*x+c)/2)^2*tan(a/2)^7*tan(c/2)+4194304*tan((b*x+c)/2)^2*tan(a/2)^6*tan(c/2)^10+
12582912*tan((b*x+c)/2)^2*tan(a/2)^6*tan(c/2)^8+8388608*tan((b*x+c)/2)^2*tan(a/2)^6*tan(c/2)^6-8388608*tan((b*
x+c)/2)^2*tan(a/2)^6*tan(c/2)^4-12582912*tan((b*x+c)/2)^2*tan(a/2)^6*tan(c/2)^2-4194304*tan((b*x+c)/2)^2*tan(a
/2)^6+50331648*tan((b*x+c)/2)^2*tan(a/2)^5*tan(c/2)^9+201326592*tan((b*x+c)/2)^2*tan(a/2)^5*tan(c/2)^7+3019898
88*tan((b*x+c)/2)^2*tan(a/2)^5*tan(c/2)^5+201326592*tan((b*x+c)/2)^2*tan(a/2)^5*tan(c/2)^3+50331648*tan((b*x+c
)/2)^2*tan(a/2)^5*tan(c/2)-4194304*tan((b*x+c)/2)^2*tan(a/2)^4*tan(c/2)^10-12582912*tan((b*x+c)/2)^2*tan(a/2)^
4*tan(c/2)^8-8388608*tan((b*x+c)/2)^2*tan(a/2)^4*tan(c/2)^6+8388608*tan((b*x+c)/2)^2*tan(a/2)^4*tan(c/2)^4+125
82912*tan((b*x+c)/2)^2*tan(a/2)^4*tan(c/2)^2+4194304*tan((b*x+c)/2)^2*tan(a/2)^4+33554432*tan((b*x+c)/2)^2*tan
(a/2)^3*tan(c/2)^9+134217728*tan((b*x+c)/2)^2*tan(a/2)^3*tan(c/2)^7+201326592*tan((b*x+c)/2)^2*tan(a/2)^3*tan(
c/2)^5+134217728*tan((b*x+c)/2)^2*tan(a/2)^3*tan(c/2)^3+33554432*tan((b*x+c)/2)^2*tan(a/2)^3*tan(c/2)-6291456*
tan((b*x+c)/2)^2*tan(a/2)^2*tan(c/2)^10-18874368*tan((b*x+c)/2)^2*tan(a/2)^2*tan(c/2)^8-12582912*tan((b*x+c)/2
)^2*tan(a/2)^2*tan(c/2)^6+12582912*tan((b*x+c)/2)^2*tan(a/2)^2*tan(c/2)^4+18874368*tan((b*x+c)/2)^2*tan(a/2)^2
*tan(c/2)^2+6291456*tan((b*x+c)/2)^2*tan(a/2)^2+8388608*tan((b*x+c)/2)^2*tan(a/2)*tan(c/2)^9+33554432*tan((b*x
+c)/2)^2*tan(a/2)*tan(c/2)^7+50331648*tan((b*x+c)/2)^2*tan(a/2)*tan(c/2)^5+33554432*tan((b*x+c)/2)^2*tan(a/2)*
tan(c/2)^3+8388608*tan((b*x+c)/2)^2*tan(a/2)*tan(c/2)-2097152*tan((b*x+c)/2)^2*tan(c/2)^10-6291456*tan((b*x+c)
/2)^2*tan(c/2)^8-4194304*tan((b*x+c)/2)^2*tan(c/2)^6+4194304*tan((b*x+c)/2)^2*tan(c/2)^4+6291456*tan((b*x+c)/2
)^2*tan(c/2)^2+2097152*tan((b*x+c)/2)^2-2097152*tan((b*x+c)/2)*tan(a/2)^10*tan(c/2)^9-8388608*tan((b*x+c)/2)*t
an(a/2)^10*tan(c/2)^7-12582912*tan((b*x+c)/2)*tan(a/2)^10*tan(c/2)^5-8388608*tan((b*x+c)/2)*tan(a/2)^10*tan(c/
2)^3-2097152*tan((b*x+c)/2)*tan(a/2)^10*tan(c/2)+2097152*tan((b*x+c)/2)*tan(a/2)^9*tan(c/2)^10+6291456*tan((b*
x+c)/2)*tan(a/2)^9*tan(c/2)^8+4194304*tan((b*x+c)/2)*tan(a/2)^9*tan(c/2)^6-4194304*tan((b*x+c)/2)*tan(a/2)^9*t
an(c/2)^4-6291456*tan((b*x+c)/2)*tan(a/2)^9*tan(c/2)^2-2097152*tan((b*x+c)/2)*tan(a/2)^9-6291456*tan((b*x+c)/2
)*tan(a/2)^8*tan(c/2)^9-25165824*tan((b*x+c)/2)*tan(a/2)^8*tan(c/2)^7-37748736*tan((b*x+c)/2)*tan(a/2)^8*tan(c
/2)^5-25165824*tan((b*x+c)/2)*tan(a/2)^8*tan(c/2)^3-6291456*tan((b*x+c)/2)*tan(a/2)^8*tan(c/2)+8388608*tan((b*
x+c)/2)*tan(a/2)^7*tan(c/2)^10+25165824*tan((b*x+c)/2)*tan(a/2)^7*tan(c/2)^8+16777216*tan((b*x+c)/2)*tan(a/2)^
7*tan(c/2)^6-16777216*tan((b*x+c)/2)*tan(a/2)^7*tan(c/2)^4-25165824*tan((b*x+c)/2)*tan(a/2)^7*tan(c/2)^2-83886
08*tan((b*x+c)/2)*tan(a/2)^7-4194304*tan((b*x+c)/2)*tan(a/2)^6*tan(c/2)^9-16777216*tan((b*x+c)/2)*tan(a/2)^6*t
an(c/2)^7-25165824*tan((b*x+c)/2)*tan(a/2)^6*tan(c/2)^5-16777216*tan((b*x+c)/2)*tan(a/2)^6*tan(c/2)^3-4194304*
tan((b*x+c)/2)*tan(a/2)^6*tan(c/2)+12582912*tan((b*x+c)/2)*tan(a/2)^5*tan(c/2)^10+37748736*tan((b*x+c)/2)*tan(
a/2)^5*tan(c/2)^8+25165824*tan((b*x+c)/2)*tan(a/2)^5*tan(c/2)^6-25165824*tan((b*x+c)/2)*tan(a/2)^5*tan(c/2)^4-
37748736*tan((b*x+c)/2)*tan(a/2)^5*tan(c/2)^2-12582912*tan((b*x+c)/2)*tan(a/2)^5+4194304*tan((b*x+c)/2)*tan(a/
2)^4*tan(c/2)^9+16777216*tan((b*x+c)/2)*tan(a/2)^4*tan(c/2)^7+25165824*tan((b*x+c)/2)*tan(a/2)^4*tan(c/2)^5+16
777216*tan((b*x+c)/2)*tan(a/2)^4*tan(c/2)^3+4194304*tan((b*x+c)/2)*tan(a/2)^4*tan(c/2)+8388608*tan((b*x+c)/2)*
tan(a/2)^3*tan(c/2)^10+25165824*tan((b*x+c)/2)*tan(a/2)^3*tan(c/2)^8+16777216*tan((b*x+c)/2)*tan(a/2)^3*tan(c/
2)^6-16777216*tan((b*x+c)/2)*tan(a/2)^3*tan(c/2)^4-25165824*tan((b*x+c)/2)*tan(a/2)^3*tan(c/2)^2-8388608*tan((
b*x+c)/2)*tan(a/2)^3+6291456*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^9+25165824*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^
7+37748736*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^5+25165824*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^3+6291456*tan((b*x
+c)/2)*tan(a/2)^2*tan(c/2)+2097152*tan((b*x+c)/2)*tan(a/2)*tan(c/2)^10+6291456*tan((b*x+c)/2)*tan(a/2)*tan(c/2
)^8+4194304*tan((b*x+c)/2)*tan(a/2)*tan(c/2)^6-4194304*tan((b*x+c)/2)*tan(a/2)*tan(c/2)^4-6291456*tan((b*x+c)/
2)*tan(a/2)*tan(c/2)^2-2097152*tan((b*x+c)/2)*tan(a/2)+2097152*tan((b*x+c)/2)*tan(c/2)^9+8388608*tan((b*x+c)/2
)*tan(c/2)^7+12582912*tan((b*x+c)/2)*tan(c/2)^5+8388608*tan((b*x+c)/2)*tan(c/2)^3+2097152*tan((b*x+c)/2)*tan(c
/2))/(33554432*tan(a/2)^10*tan(c/2)^10+167772160*tan(a/2)^10*tan(c/2)^8+335544320*tan(a/2)^10*tan(c/2)^6+33554
4320*tan(a/2)^10*tan(c/2)^4+167772160*tan(a/2)^10*tan(c/2)^2+33554432*tan(a/2)^10+167772160*tan(a/2)^8*tan(c/2
)^10+838860800*tan(a/2)^8*tan(c/2)^8+1677721600*tan(a/2)^8*tan(c/2)^6+1677721600*tan(a/2)^8*tan(c/2)^4+8388608
00*tan(a/2)^8*tan(c/2)^2+167772160*tan(a/2)^8+335544320*tan(a/2)^6*tan(c/2)^10+1677721600*tan(a/2)^6*tan(c/2)^
8+3355443200*tan(a/2)^6*tan(c/2)^6+3355443200*tan(a/2)^6*tan(c/2)^4+1677721600*tan(a/2)^6*tan(c/2)^2+335544320
*tan(a/2)^6+335544320*tan(a/2)^4*tan(c/2)^10+1677721600*tan(a/2)^4*tan(c/2)^8+3355443200*tan(a/2)^4*tan(c/2)^6
+3355443200*tan(a/2)^4*tan(c/2)^4+1677721600*tan(a/2)^4*tan(c/2)^2+335544320*tan(a/2)^4+167772160*tan(a/2)^2*t
an(c/2)^10+838860800*tan(a/2)^2*tan(c/2)^8+1677721600*tan(a/2)^2*tan(c/2)^6+1677721600*tan(a/2)^2*tan(c/2)^4+8
38860800*tan(a/2)^2*tan(c/2)^2+167772160*tan(a/2)^2+33554432*tan(c/2)^10+167772160*tan(c/2)^8+335544320*tan(c/
2)^6+335544320*tan(c/2)^4+167772160*tan(c/2)^2+33554432)+(-274*tan((b*x+c)/2)^5*tan(a/2)^2*tan(c/2)^2+274*tan(
(b*x+c)/2)^5*tan(a/2)^2-1096*tan((b*x+c)/2)^5*tan(a/2)*tan(c/2)+274*tan((b*x+c)/2)^5*tan(c/2)^2-274*tan((b*x+c
)/2)^5-40*tan((b*x+c)/2)^4*tan(a/2)^2*tan(c/2)+40*tan((b*x+c)/2)^4*tan(a/2)*tan(c/2)^2-40*tan((b*x+c)/2)^4*tan
(a/2)+40*tan((b*x+c)/2)^4*tan(c/2)-40*tan((b*x+c)/2)^3*tan(a/2)^2*tan(c/2)^2+40*tan((b*x+c)/2)^3*tan(a/2)^2-16
0*tan((b*x+c)/2)^3*tan(a/2)*tan(c/2)+40*tan((b*x+c)/2)^3*tan(c/2)^2-40*tan((b*x+c)/2)^3-20*tan((b*x+c)/2)^2*ta
n(a/2)^2*tan(c/2)+20*tan((b*x+c)/2)^2*tan(a/2)*tan(c/2)^2-20*tan((b*x+c)/2)^2*tan(a/2)+20*tan((b*x+c)/2)^2*tan
(c/2)-5*tan((b*x+c)/2)*tan(a/2)^2*tan(c/2)^2+5*tan((b*x+c)/2)*tan(a/2)^2-20*tan((b*x+c)/2)*tan(a/2)*tan(c/2)+5
*tan((b*x+c)/2)*tan(c/2)^2-5*tan((b*x+c)/2)-4*tan(a/2)^2*tan(c/2)+4*tan(a/2)*tan(c/2)^2-4*tan(a/2)+4*tan(c/2))
/(640*tan(a/2)^2*tan(c/2)^2+640*tan(a/2)^2+640*tan(c/2)^2+640)/tan((b*x+c)/2)^5+(3*tan(a/2)^2*tan(c/2)^2-3*tan
(a/2)^2+12*tan(a/2)*tan(c/2)-3*tan(c/2)^2+3)/(16*tan(a/2)^2*tan(c/2)^2+16*tan(a/2)^2+16*tan(c/2)^2+16)*ln(abs(
tan((b*x+c)/2))))
________________________________________________________________________________________
maple [B] time = 14.96, size = 97954, normalized size = 1042.06 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+c)^6*sin(b*x+a),x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [B] time = 0.57, size = 3879, normalized size = 41.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+c)^6*sin(b*x+a),x, algorithm="maxima")
[Out]
1/80*(2*(15*cos(9*b*x + 2*a + 8*c) + 15*cos(9*b*x + 10*c) - 70*cos(7*b*x + 2*a + 6*c) - 70*cos(7*b*x + 8*c) -
128*cos(5*b*x + 2*a + 4*c) + 128*cos(5*b*x + 6*c) + 70*cos(3*b*x + 2*a + 2*c) + 70*cos(3*b*x + 4*c) - 15*cos(b
*x + 2*a) - 15*cos(b*x + 2*c))*cos(10*b*x + a + 10*c) - 30*(5*cos(8*b*x + a + 8*c) - 10*cos(6*b*x + a + 6*c) +
10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 2*a + 8*c) - 30*(5*cos(8*b*x + a + 8*c
) - 10*cos(6*b*x + a + 6*c) + 10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(9*b*x + 10*c) + 1
0*(70*cos(7*b*x + 2*a + 6*c) + 70*cos(7*b*x + 8*c) + 128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*co
s(3*b*x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15*cos(b*x + 2*a) + 15*cos(b*x + 2*c))*cos(8*b*x + a + 8*c) - 140
*(10*cos(6*b*x + a + 6*c) - 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) - cos(a))*cos(7*b*x + 2*a + 6*c)
- 140*(10*cos(6*b*x + a + 6*c) - 10*cos(4*b*x + a + 4*c) + 5*cos(2*b*x + a + 2*c) - cos(a))*cos(7*b*x + 8*c) -
20*(128*cos(5*b*x + 2*a + 4*c) - 128*cos(5*b*x + 6*c) - 70*cos(3*b*x + 2*a + 2*c) - 70*cos(3*b*x + 4*c) + 15*
cos(b*x + 2*a) + 15*cos(b*x + 2*c))*cos(6*b*x + a + 6*c) + 256*(10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*
c) + cos(a))*cos(5*b*x + 2*a + 4*c) - 256*(10*cos(4*b*x + a + 4*c) - 5*cos(2*b*x + a + 2*c) + cos(a))*cos(5*b*
x + 6*c) - 100*(14*cos(3*b*x + 2*a + 2*c) + 14*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) + 140*(5*cos(2*b*x + a + 2*c) - cos(a))*cos(3*b*x + 2*a + 2*c) + 140*(5*cos(2*b*x + a + 2*c) - c
os(a))*cos(3*b*x + 4*c) - 150*(cos(b*x + 2*a) + cos(b*x + 2*c))*cos(2*b*x + a + 2*c) + 30*cos(b*x + 2*a)*cos(a
) + 30*cos(b*x + 2*c)*cos(a) - 15*(cos(10*b*x + a + 10*c)^2*cos(-a + c) + 25*cos(8*b*x + a + 8*c)^2*cos(-a + c
) + 100*cos(6*b*x + a + 6*c)^2*cos(-a + c) + 100*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 25*cos(2*b*x + a + 2*c)^
2*cos(-a + c) - 10*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(10*b*x + a + 10*c)^2 + 25*cos(-a
+ c)*sin(8*b*x + a + 8*c)^2 + 100*cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 100*cos(-a + c)*sin(4*b*x + a + 4*c)^2
+ 25*cos(-a + c)*sin(2*b*x + a + 2*c)^2 - 10*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) - 2*(5*cos(8*b*x + a + 8*
c)*cos(-a + c) - 10*cos(6*b*x + a + 6*c)*cos(-a + c) + 10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a +
2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(10*b*x + a + 10*c) - 10*(10*cos(6*b*x + a + 6*c)*cos(-a + c) - 10*
cos(4*b*x + a + 4*c)*cos(-a + c) + 5*cos(2*b*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(8*b*x + a + 8*
c) - 20*(10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*
b*x + a + 6*c) - 20*(5*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*(5*cos(-a + c)*sin(8*b*x + a + 8*c) - 10*cos(-a + c)*sin(6*b*x + a + 6*c) + 10*co
s(-a + c)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(10*b*x + a + 10*
c) - 10*(10*cos(-a + c)*sin(6*b*x + a + 6*c) - 10*cos(-a + c)*sin(4*b*x + a + 4*c) + 5*cos(-a + c)*sin(2*b*x +
a + 2*c) - cos(-a + c)*sin(a))*sin(8*b*x + a + 8*c) - 20*(10*cos(-a + c)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)
*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(6*b*x + a + 6*c) - 20*(5*cos(-a + c)*sin(2*b*x + a + 2*c) - co
s(-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) + 15*(cos(10*b*x + a + 10*c)^2*cos(-a + c) + 25*cos(8*b*x + a + 8*c)^2*cos(-a + c) + 100*
cos(6*b*x + a + 6*c)^2*cos(-a + c) + 100*cos(4*b*x + a + 4*c)^2*cos(-a + c) + 25*cos(2*b*x + a + 2*c)^2*cos(-a
+ c) - 10*cos(2*b*x + a + 2*c)*cos(a)*cos(-a + c) + cos(-a + c)*sin(10*b*x + a + 10*c)^2 + 25*cos(-a + c)*sin
(8*b*x + a + 8*c)^2 + 100*cos(-a + c)*sin(6*b*x + a + 6*c)^2 + 100*cos(-a + c)*sin(4*b*x + a + 4*c)^2 + 25*cos
(-a + c)*sin(2*b*x + a + 2*c)^2 - 10*cos(-a + c)*sin(2*b*x + a + 2*c)*sin(a) - 2*(5*cos(8*b*x + a + 8*c)*cos(-
a + c) - 10*cos(6*b*x + a + 6*c)*cos(-a + c) + 10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a + 2*c)*co
s(-a + c) + cos(a)*cos(-a + c))*cos(10*b*x + a + 10*c) - 10*(10*cos(6*b*x + a + 6*c)*cos(-a + c) - 10*cos(4*b*
x + a + 4*c)*cos(-a + c) + 5*cos(2*b*x + a + 2*c)*cos(-a + c) - cos(a)*cos(-a + c))*cos(8*b*x + a + 8*c) - 20*
(10*cos(4*b*x + a + 4*c)*cos(-a + c) - 5*cos(2*b*x + a + 2*c)*cos(-a + c) + cos(a)*cos(-a + c))*cos(6*b*x + a
+ 6*c) - 20*(5*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*(5*cos(-a + c)*sin(8*b*x + a + 8*c) - 10*cos(-a + c)*sin(6*b*x + a + 6*c) + 10*cos(-a + c
)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)*sin(2*b*x + a + 2*c) + cos(-a + c)*sin(a))*sin(10*b*x + a + 10*c) - 10*
(10*cos(-a + c)*sin(6*b*x + a + 6*c) - 10*cos(-a + c)*sin(4*b*x + a + 4*c) + 5*cos(-a + c)*sin(2*b*x + a + 2*c
) - cos(-a + c)*sin(a))*sin(8*b*x + a + 8*c) - 20*(10*cos(-a + c)*sin(4*b*x + a + 4*c) - 5*cos(-a + c)*sin(2*b
*x + a + 2*c) + cos(-a + c)*sin(a))*sin(6*b*x + a + 6*c) - 20*(5*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) + 2*(15*sin(9*b*x + 2*a + 8*c) + 15*sin(9*b*x + 10*c) - 70*sin(7*b*x + 2*a + 6*c) - 70*sin(7*b*x
+ 8*c) - 128*sin(5*b*x + 2*a + 4*c) + 128*sin(5*b*x + 6*c) + 70*sin(3*b*x + 2*a + 2*c) + 70*sin(3*b*x + 4*c) -
15*sin(b*x + 2*a) - 15*sin(b*x + 2*c))*sin(10*b*x + a + 10*c) - 30*(5*sin(8*b*x + a + 8*c) - 10*sin(6*b*x + a
+ 6*c) + 10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x + a + 2*c) + sin(a))*sin(9*b*x + 2*a + 8*c) - 30*(5*sin(8*b*x
+ a + 8*c) - 10*sin(6*b*x + a + 6*c) + 10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x + a + 2*c) + sin(a))*sin(9*b*x +
10*c) + 10*(70*sin(7*b*x + 2*a + 6*c) + 70*sin(7*b*x + 8*c) + 128*sin(5*b*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c
) - 70*sin(3*b*x + 2*a + 2*c) - 70*sin(3*b*x + 4*c) + 15*sin(b*x + 2*a) + 15*sin(b*x + 2*c))*sin(8*b*x + a + 8
*c) - 140*(10*sin(6*b*x + a + 6*c) - 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) - sin(a))*sin(7*b*x + 2*
a + 6*c) - 140*(10*sin(6*b*x + a + 6*c) - 10*sin(4*b*x + a + 4*c) + 5*sin(2*b*x + a + 2*c) - sin(a))*sin(7*b*x
+ 8*c) - 20*(128*sin(5*b*x + 2*a + 4*c) - 128*sin(5*b*x + 6*c) - 70*sin(3*b*x + 2*a + 2*c) - 70*sin(3*b*x + 4
*c) + 15*sin(b*x + 2*a) + 15*sin(b*x + 2*c))*sin(6*b*x + a + 6*c) + 256*(10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x
+ a + 2*c) + sin(a))*sin(5*b*x + 2*a + 4*c) - 256*(10*sin(4*b*x + a + 4*c) - 5*sin(2*b*x + a + 2*c) + sin(a))
*sin(5*b*x + 6*c) - 100*(14*sin(3*b*x + 2*a + 2*c) + 14*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) + 140*(5*sin(2*b*x + a + 2*c) - sin(a))*sin(3*b*x + 2*a + 2*c) + 140*(5*sin(2*b*x + a +
2*c) - sin(a))*sin(3*b*x + 4*c) - 150*(sin(b*x + 2*a) + sin(b*x + 2*c))*sin(2*b*x + a + 2*c) + 30*sin(b*x + 2
*a)*sin(a) + 30*sin(b*x + 2*c)*sin(a))/(b*cos(10*b*x + a + 10*c)^2 + 25*b*cos(8*b*x + a + 8*c)^2 + 100*b*cos(6
*b*x + a + 6*c)^2 + 100*b*cos(4*b*x + a + 4*c)^2 + 25*b*cos(2*b*x + a + 2*c)^2 - 10*b*cos(2*b*x + a + 2*c)*cos
(a) + b*sin(10*b*x + a + 10*c)^2 + 25*b*sin(8*b*x + a + 8*c)^2 + 100*b*sin(6*b*x + a + 6*c)^2 + 100*b*sin(4*b*
x + a + 4*c)^2 + 25*b*sin(2*b*x + a + 2*c)^2 - 10*b*sin(2*b*x + a + 2*c)*sin(a) + (cos(a)^2 + sin(a)^2)*b - 2*
(5*b*cos(8*b*x + a + 8*c) - 10*b*cos(6*b*x + a + 6*c) + 10*b*cos(4*b*x + a + 4*c) - 5*b*cos(2*b*x + a + 2*c) +
b*cos(a))*cos(10*b*x + a + 10*c) - 10*(10*b*cos(6*b*x + a + 6*c) - 10*b*cos(4*b*x + a + 4*c) + 5*b*cos(2*b*x
+ a + 2*c) - b*cos(a))*cos(8*b*x + a + 8*c) - 20*(10*b*cos(4*b*x + a + 4*c) - 5*b*cos(2*b*x + a + 2*c) + b*cos
(a))*cos(6*b*x + a + 6*c) - 20*(5*b*cos(2*b*x + a + 2*c) - b*cos(a))*cos(4*b*x + a + 4*c) - 2*(5*b*sin(8*b*x +
a + 8*c) - 10*b*sin(6*b*x + a + 6*c) + 10*b*sin(4*b*x + a + 4*c) - 5*b*sin(2*b*x + a + 2*c) + b*sin(a))*sin(1
0*b*x + a + 10*c) - 10*(10*b*sin(6*b*x + a + 6*c) - 10*b*sin(4*b*x + a + 4*c) + 5*b*sin(2*b*x + a + 2*c) - b*s
in(a))*sin(8*b*x + a + 8*c) - 20*(10*b*sin(4*b*x + a + 4*c) - 5*b*sin(2*b*x + a + 2*c) + b*sin(a))*sin(6*b*x +
a + 6*c) - 20*(5*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)^6,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)**6*sin(b*x+a),x)
[Out]
Timed out
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