∫ csc6(c + bx) sin(a + bx) dx [200]

3.2.100 \(\int \csc ^6(c+b x) \sin (a+b x) \, dx\) [200]

Optimal. Leaf size=94 \[ -\frac {3 \tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{8 b}-\frac {3 \cos (a-c) \cot (c+b x) \csc (c+b x)}{8 b}-\frac {\cos (a-c) \cot (c+b x) \csc ^3(c+b x)}{4 b}-\frac {\csc ^5(c+b x) \sin (a-c)}{5 b} \]

[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 = 0.04, 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 = {4678, 2686, 30, 3853, 3855} \begin {gather*} -\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} \end {gather*}

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 2686

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 3853

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4678

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) \text {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 = 1.23, size = 79, normalized size = 0.84 \begin {gather*} -\frac {480 \tanh ^{-1}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a-c)+2 \csc ^5(c+b x) (64 \sin (a-c)+5 \cos (a-c) (14 \sin (2 (c+b x))-3 \sin (4 (c+b x))))}{640 b} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6745\) vs. \(2(86)=172\).
time = 6.25, size = 6746, normalized size = 71.77


method	result	size

risch	\(\frac {-15 \,{\mathrm e}^{i \left (9 b x +11 a +8 c \right )}-15 \,{\mathrm e}^{i \left (9 b x +9 a +10 c \right )}+70 \,{\mathrm e}^{i \left (7 b x +11 a +6 c \right )}+70 \,{\mathrm e}^{i \left (7 b x +9 a +8 c \right )}+128 \,{\mathrm e}^{i \left (5 b x +11 a +4 c \right )}-128 \,{\mathrm e}^{i \left (5 b x +9 a +6 c \right )}-70 \,{\mathrm e}^{i \left (3 b x +11 a +2 c \right )}-70 \,{\mathrm e}^{i \left (3 b x +9 a +4 c \right )}+15 \,{\mathrm e}^{i \left (b x +11 a \right )}+15 \,{\mathrm e}^{i \left (b x +9 a +2 c \right )}}{40 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{5}}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{8 b}\)	\(257\)
default	\(\text {Expression too large to display}\)	\(6746\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+c)^6*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3879 vs. \(2 (86) = 172\).
time = 0.54, size = 3879, normalized size = 41.27 \begin {gather*} \text {Too large to display} \end {gather*}

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 +...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (86) = 172\).
time = 2.91, size = 197, normalized size = 2.10 \begin {gather*} -\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 )} \end {gather*}

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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8035 vs. \(2 (86) = 172\).
time = 0.48, size = 8035, normalized size = 85.48 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/320*(120*(tan(1/2*a)^2*tan(1/2*c)^2 - tan(1/2*a)^2 + 4*tan(1/2*a)*tan(1/2*c) - tan(1/2*c)^2 + 1)*log(abs(tan

(1/2*b*x + 1/2*c)))/(tan(1/2*a)^2*tan(1/2*c)^2 + tan(1/2*a)^2 + tan(1/2*c)^2 + 1) - (4*tan(1/2*b*x + 1/2*c)^5*

tan(1/2*a)^10*tan(1/2*c)^9 - 4*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(1/2*c)^10 - 5*tan(1/2*b*x + 1/2*c)^4*ta

n(1/2*a)^10*tan(1/2*c)^10 + 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^7 - 12*tan(1/2*b*x + 1/2*c)^5*t

an(1/2*a)^9*tan(1/2*c)^8 - 15*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^8 + 12*tan(1/2*b*x + 1/2*c)^5*ta

n(1/2*a)^8*tan(1/2*c)^9 - 20*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^9 + 20*tan(1/2*b*x + 1/2*c)^3*tan(

1/2*a)^10*tan(1/2*c)^9 - 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^10 - 15*tan(1/2*b*x + 1/2*c)^4*tan(

1/2*a)^8*tan(1/2*c)^10 - 20*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^10 - 40*tan(1/2*b*x + 1/2*c)^2*tan(

1/2*a)^10*tan(1/2*c)^10 + 24*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^5 - 8*tan(1/2*b*x + 1/2*c)^5*tan(

1/2*a)^9*tan(1/2*c)^6 - 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^6 + 48*tan(1/2*b*x + 1/2*c)^5*tan(1

/2*a)^8*tan(1/2*c)^7 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^7 + 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2

*a)^10*tan(1/2*c)^7 - 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^8 - 45*tan(1/2*b*x + 1/2*c)^4*tan(1/2*

a)^8*tan(1/2*c)^8 - 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^8 - 120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a

)^10*tan(1/2*c)^8 + 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^6*tan(1/2*c)^9 - 80*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^

7*tan(1/2*c)^9 + 60*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^8*tan(1/2*c)^9 - 160*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^9

*tan(1/2*c)^9 + 40*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^10*tan(1/2*c)^9 - 24*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*ta

n(1/2*c)^10 - 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^6*tan(1/2*c)^10 - 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*ta

n(1/2*c)^10 - 120*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^10 - 40*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan

(1/2*c)^10 + 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c)^3 + 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(

1/2*c)^4 + 10*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^4 + 72*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^8*tan(1

/2*c)^5 - 120*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^5 + 120*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^10*tan(

1/2*c)^5 - 32*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^6 - 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8*tan(1/

2*c)^6 - 40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^6 - 80*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^10*tan(1/2

*c)^6 + 32*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^6*tan(1/2*c)^7 - 320*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^7*tan(1/2*

c)^7 + 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^8*tan(1/2*c)^7 - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^9*tan(1/2*

c)^7 + 160*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^10*tan(1/2*c)^7 - 72*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*tan(1/2*c)

^8 - 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^6*tan(1/2*c)^8 - 240*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*tan(1/2*c)^

8 - 360*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^8 - 120*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan(1/2*c)^8

- 8*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^4*tan(1/2*c)^9 - 120*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^5*tan(1/2*c)^9 +

40*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^6*tan(1/2*c)^9 - 640*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c)^9 + 1

20*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^8*tan(1/2*c)^9 - 16*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^3*tan(1/2*c)^10 + 10*

tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^4*tan(1/2*c)^10 - 120*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^10 - 80

*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^10 - 160*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^7*tan(1/2*c)^10 + 4*t

an(1/2*b*x + 1/2*c)^5*tan(1/2*a)^10*tan(1/2*c) + 12*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^9*tan(1/2*c)^2 + 15*tan(

1/2*b*x + 1/2*c)^4*tan(1/2*a)^10*tan(1/2*c)^2 + 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^8*tan(1/2*c)^3 - 80*tan(1

/2*b*x + 1/2*c)^4*tan(1/2*a)^9*tan(1/2*c)^3 + 80*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^10*tan(1/2*c)^3 + 32*tan(1/

2*b*x + 1/2*c)^5*tan(1/2*a)^7*tan(1/2*c)^4 + 30*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^8*tan(1/2*c)^4 + 40*tan(1/2*

b*x + 1/2*c)^3*tan(1/2*a)^9*tan(1/2*c)^4 + 80*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^10*tan(1/2*c)^4 + 48*tan(1/2*b

*x + 1/2*c)^5*tan(1/2*a)^6*tan(1/2*c)^5 - 480*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^7*tan(1/2*c)^5 + 360*tan(1/2*b

*x + 1/2*c)^3*tan(1/2*a)^8*tan(1/2*c)^5 - 960*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^9*tan(1/2*c)^5 + 240*tan(1/2*b

*x + 1/2*c)*tan(1/2*a)^10*tan(1/2*c)^5 - 48*tan(1/2*b*x + 1/2*c)^5*tan(1/2*a)^5*tan(1/2*c)^6 - 20*tan(1/2*b*x

+ 1/2*c)^4*tan(1/2*a)^6*tan(1/2*c)^6 - 160*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^7*tan(1/2*c)^6 - 240*tan(1/2*b*x

+ 1/2*c)^2*tan(1/2*a)^8*tan(1/2*c)^6 - 80*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^9*tan(1/2*c)^6 - 32*tan(1/2*b*x + 1/

2*c)^5*tan(1/2*a)^4*tan(1/2*c)^7 - 480*tan(1/2*b*x + 1/2*c)^4*tan(1/2*a)^5*tan(1/2*c)^7 + 160*tan(1/2*b*x + 1/

2*c)^3*tan(1/2*a)^6*tan(1/2*c)^7 - 2560*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^7*tan(1/2*c)^7 + 480*tan(1/2*b*x + 1

/2*c)*tan(1/2*a)^8*tan(1/2*c)^7 - 48*tan(1/2*b*...

________________________________________________________________________________________

Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}

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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