∫ csc4(c + bx) sin(a + bx) dx [198]

3.2.98 \(\int \csc ^4(c+b x) \sin (a+b x) \, dx\) [198]

Optimal. Leaf size=67 \[ -\frac {\tanh ^{-1}(\cos (c+b x)) \cos (a-c)}{2 b}-\frac {\cos (a-c) \cot (c+b x) \csc (c+b x)}{2 b}-\frac {\csc ^3(c+b x) \sin (a-c)}{3 b} \]

[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 = 0.03, 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 = {4678, 2686, 30, 3853, 3855} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[Csc[c + b*x]^4*Sin[a + b*x],x]

[Out]

-1/2*(ArcTanh[Cos[c + b*x]]*Cos[a - c])/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 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 ^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) \text {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 = 0.61, size = 67, normalized size = 1.00 \begin {gather*} -\frac {6 \tanh ^{-1}\left (\cos (c)-\sin (c) \tan \left (\frac {b x}{2}\right )\right ) \cos (a-c)+3 \cos (a-c) \cot (c+b x) \csc (c+b x)+2 \csc ^3(c+b x) \sin (a-c)}{6 b} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1747\) vs. \(2(61)=122\).
time = 1.88, size = 1748, normalized size = 26.09


method	result	size

risch	\(\frac {-3 \,{\mathrm e}^{i \left (5 b x +7 a +4 c \right )}-3 \,{\mathrm e}^{i \left (5 b x +5 a +6 c \right )}-8 \,{\mathrm e}^{i \left (3 b x +7 a +2 c \right )}+8 \,{\mathrm e}^{i \left (3 b x +5 a +4 c \right )}+3 \,{\mathrm e}^{i \left (b x +7 a \right )}+3 \,{\mathrm e}^{i \left (b x +5 a +2 c \right )}}{6 b \left (-{\mathrm e}^{2 i \left (b x +a +c \right )}+{\mathrm e}^{2 i a}\right )^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+{\mathrm e}^{i \left (a -c \right )}\right ) \cos \left (a -c \right )}{2 b}\)	\(189\)
default	\(\text {Expression too large to display}\)	\(1748\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+c)^4*sin(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b*(32*(1/32*(cos(a)*cos(c)+sin(a)*sin(c))*(cos(c)^2*sin(a)^2-2*cos(a)*cos(c)*sin(a)*sin(c)+cos(a)^2*sin(c)^2

)/(cos(a)^4*cos(c)^4+2*cos(a)^2*cos(c)^4*sin(a)^2+cos(c)^4*sin(a)^4+2*cos(a)^4*cos(c)^2*sin(c)^2+4*cos(a)^2*co

s(c)^2*sin(a)^2*sin(c)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*

sin(a)^4)*tan(1/2*b*x+1/2*a)^5-1/32*(2*cos(a)^4*cos(c)^4-cos(a)^2*cos(c)^4*sin(a)^2+2*cos(c)^4*sin(a)^4+10*cos

(a)^3*cos(c)^3*sin(a)*sin(c)-10*cos(c)^3*sin(c)*cos(a)*sin(a)^3-cos(a)^4*cos(c)^2*sin(c)^2+28*cos(a)^2*cos(c)^

2*sin(a)^2*sin(c)^2-sin(a)^4*sin(c)^2*cos(c)^2-10*cos(c)*sin(c)^3*cos(a)^3*sin(a)+10*sin(a)^3*cos(a)*sin(c)^3*

cos(c)+2*sin(c)^4*cos(a)^4-sin(a)^2*cos(a)^2*sin(c)^4+2*sin(c)^4*sin(a)^4)/(sin(a)*cos(c)-cos(a)*sin(c))/(cos(

a)^4*cos(c)^4+2*cos(a)^2*cos(c)^4*sin(a)^2+cos(c)^4*sin(a)^4+2*cos(a)^4*cos(c)^2*sin(c)^2+4*cos(a)^2*cos(c)^2*

sin(a)^2*sin(c)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*sin(a)^

4)*tan(1/2*b*x+1/2*a)^4-1/48*(cos(a)*cos(c)+sin(a)*sin(c))*(2*cos(a)^4*cos(c)^4-7*cos(a)^2*cos(c)^4*sin(a)^2+6

*cos(c)^4*sin(a)^4+22*cos(a)^3*cos(c)^3*sin(a)*sin(c)-38*cos(c)^3*sin(c)*cos(a)*sin(a)^3-7*cos(a)^4*cos(c)^2*s

in(c)^2+76*cos(a)^2*cos(c)^2*sin(a)^2*sin(c)^2-7*sin(a)^4*sin(c)^2*cos(c)^2-38*cos(c)*sin(c)^3*cos(a)^3*sin(a)

+22*sin(a)^3*cos(a)*sin(c)^3*cos(c)+6*sin(c)^4*cos(a)^4-7*sin(a)^2*cos(a)^2*sin(c)^4+2*sin(c)^4*sin(a)^4)/(cos

(a)^4*cos(c)^4+2*cos(a)^2*cos(c)^4*sin(a)^2+cos(c)^4*sin(a)^4+2*cos(a)^4*cos(c)^2*sin(c)^2+4*cos(a)^2*cos(c)^2

*sin(a)^2*sin(c)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*sin(a)

^4)/(sin(a)*cos(c)-cos(a)*sin(c))^2*tan(1/2*b*x+1/2*a)^3+1/16*(cos(a)^3*cos(c)^3-4*cos(c)^3*cos(a)*sin(a)^2+11

*cos(a)^2*cos(c)^2*sin(a)*sin(c)-4*cos(c)^2*sin(c)*sin(a)^3-4*cos(c)*sin(c)^2*cos(a)^3+11*cos(c)*sin(c)^2*cos(

a)*sin(a)^2-4*sin(c)^3*cos(a)^2*sin(a)+sin(c)^3*sin(a)^3)/(sin(a)*cos(c)-cos(a)*sin(c))*(cos(a)*cos(c)+sin(a)*

sin(c))/(cos(a)^4*cos(c)^4+2*cos(a)^2*cos(c)^4*sin(a)^2+cos(c)^4*sin(a)^4+2*cos(a)^4*cos(c)^2*sin(c)^2+4*cos(a

)^2*cos(c)^2*sin(a)^2*sin(c)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin

(c)^4*sin(a)^4)*tan(1/2*b*x+1/2*a)^2-1/32*(cos(a)*cos(c)+sin(a)*sin(c))*(2*cos(a)^2*cos(c)^2-3*cos(c)^2*sin(a)

^2+10*cos(a)*cos(c)*sin(a)*sin(c)-3*cos(a)^2*sin(c)^2+2*sin(a)^2*sin(c)^2)/(cos(a)^4*cos(c)^4+2*cos(a)^2*cos(c

)^4*sin(a)^2+cos(c)^4*sin(a)^4+2*cos(a)^4*cos(c)^2*sin(c)^2+4*cos(a)^2*cos(c)^2*sin(a)^2*sin(c)^2+2*sin(a)^4*s

in(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(a)^2*sin(c)^4+sin(c)^4*sin(a)^4)*tan(1/2*b*x+1/2*a)+1/96*(co

s(a)^2*cos(c)^2-2*cos(c)^2*sin(a)^2+6*cos(a)*cos(c)*sin(a)*sin(c)-2*cos(a)^2*sin(c)^2+sin(a)^2*sin(c)^2)*(sin(

a)*cos(c)-cos(a)*sin(c))/(cos(a)^4*cos(c)^4+2*cos(a)^2*cos(c)^4*sin(a)^2+cos(c)^4*sin(a)^4+2*cos(a)^4*cos(c)^2

*sin(c)^2+4*cos(a)^2*cos(c)^2*sin(a)^2*sin(c)^2+2*sin(a)^4*sin(c)^2*cos(c)^2+sin(c)^4*cos(a)^4+2*sin(a)^2*cos(

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

(1/2*b*x+1/2*a)*cos(a)*cos(c)+2*tan(1/2*b*x+1/2*a)*sin(a)*sin(c)-sin(a)*cos(c)+cos(a)*sin(c))^3+8*(cos(a)*cos(

c)+sin(a)*sin(c))/(8*cos(a)^4*cos(c)^4+16*cos(a)^2*cos(c)^4*sin(a)^2+8*cos(c)^4*sin(a)^4+16*cos(a)^4*cos(c)^2*

sin(c)^2+32*cos(a)^2*cos(c)^2*sin(a)^2*sin(c)^2+16*sin(a)^4*sin(c)^2*cos(c)^2+8*sin(c)^4*cos(a)^4+16*sin(a)^2*

cos(a)^2*sin(c)^4+8*sin(c)^4*sin(a)^4)/(-cos(c)^2*sin(a)^2-cos(a)^2*cos(c)^2-sin(a)^2*sin(c)^2-cos(a)^2*sin(c)

^2)^(1/2)*arctan(1/2*(2*(sin(a)*cos(c)-cos(a)*sin(c))*tan(1/2*b*x+1/2*a)+2*cos(a)*cos(c)+2*sin(a)*sin(c))/(-co

s(c)^2*sin(a)^2-cos(a)^2*cos(c)^2-sin(a)^2*sin(c)^2-cos(a)^2*sin(c)^2)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1773 vs. \(2 (61) = 122\).
time = 0.35, size = 1773, normalized size = 26.46 \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)^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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (61) = 122\).
time = 3.45, size = 141, normalized size = 2.10 \begin {gather*} \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 )} \end {gather*}

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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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)**4*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. 2221 vs. \(2 (61) = 122\).
time = 0.44, size = 2221, normalized size = 33.15 \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)^4*sin(b*x+a),x, algorithm="giac")

[Out]

1/24*(12*(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) - (2*tan(1/2*b*x + 1/2*c)^3*ta

n(1/2*a)^6*tan(1/2*c)^5 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^6 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/

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

^5*tan(1/2*c)^4 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^4 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*t

an(1/2*c)^5 - 12*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c)^5 + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/

2*c)^5 - 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^3*tan(1/2*c)^6 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^4*tan(1/2*c)

^6 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^5*tan(1/2*c)^6 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^6*tan(1/2*c) + 2*t

an(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5*tan(1/2*c)^2 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6*tan(1/2*c)^2 + 4*tan(1

/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c)^3 - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^5*tan(1/2*c)^3 + 12*tan(1/2

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

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

)^3*tan(1/2*a)^2*tan(1/2*c)^5 - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^3*tan(1/2*c)^5 + 6*tan(1/2*b*x + 1/2*c)*t

an(1/2*a)^4*tan(1/2*c)^5 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c)^6 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2

*a)^2*tan(1/2*c)^6 - 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)^6 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^5

+ 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^6 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^4*tan(1/2*c) - 12*tan(1/2*b*x +

1/2*c)^2*tan(1/2*a)^5*tan(1/2*c) + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^6*tan(1/2*c) + 4*tan(1/2*b*x + 1/2*c)^3*

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

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

)^3*tan(1/2*c)^3 + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^4*tan(1/2*c)^3 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan

(1/2*c)^4 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^4 - 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*

c)^4 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^5 - 12*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^5 - 6*tan(1/2*b

*x + 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^5 + 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c)^6 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2

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

*x + 1/2*c)*tan(1/2*a)^5 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c) - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/

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

/2*c)^2 - 3*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*c)^2 + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3*tan(1/2*c)

^2 - 4*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^3 - 24*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^3 - 12*tan(1/2*b*

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

a)*tan(1/2*c)^4 - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^5 + 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a) - 3*tan(1/2*b*x +

1/2*c)^2*tan(1/2*a)^2 + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)^3 - 2*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c) - 12*tan(1/

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

c)^2*tan(1/2*c)^2 + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1/2*c)^2 - 12*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^3 - 3*

tan(1/2*b*x + 1/2*c)^2 + 6*tan(1/2*b*x + 1/2*c)*tan(1/2*a) - 6*tan(1/2*b*x + 1/2*c)*tan(1/2*c))/(tan(1/2*a)^6*

tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^4 + 3*tan(1/2*a)^4*tan(1/2*c)^6 + 3*tan(1/2*a)^6*tan(1/2*c)^2 + 9*tan

(1/2*a)^4*tan(1/2*c)^4 + 3*tan(1/2*a)^2*tan(1/2*c)^6 + tan(1/2*a)^6 + 9*tan(1/2*a)^4*tan(1/2*c)^2 + 9*tan(1/2*

a)^2*tan(1/2*c)^4 + tan(1/2*c)^6 + 3*tan(1/2*a)^4 + 9*tan(1/2*a)^2*tan(1/2*c)^2 + 3*tan(1/2*c)^4 + 3*tan(1/2*a

)^2 + 3*tan(1/2*c)^2 + 1) - (22*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)^2*tan(1/2*c)^2 - 22*tan(1/2*b*x + 1/2*c)^3*t

an(1/2*a)^2 + 88*tan(1/2*b*x + 1/2*c)^3*tan(1/2*a)*tan(1/2*c) + 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)^2*tan(1/2*

c) - 22*tan(1/2*b*x + 1/2*c)^3*tan(1/2*c)^2 - 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a)*tan(1/2*c)^2 + 3*tan(1/2*b*x

+ 1/2*c)*tan(1/2*a)^2*tan(1/2*c)^2 + 22*tan(1/2*b*x + 1/2*c)^3 + 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*a) - 3*tan(

1/2*b*x + 1/2*c)*tan(1/2*a)^2 - 6*tan(1/2*b*x + 1/2*c)^2*tan(1/2*c) + 12*tan(1/2*b*x + 1/2*c)*tan(1/2*a)*tan(1

/2*c) + 2*tan(1/2*a)^2*tan(1/2*c) - 3*tan(1/2*b*x + 1/2*c)*tan(1/2*c)^2 - 2*tan(1/2*a)*tan(1/2*c)^2 + 3*tan(1/

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

an(1/2*b*x + 1/2*c)^3))/b

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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)^4,x)

[Out]

\text{Hanged}

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