3.368 \(\int (c+d x)^4 \csc (a+b x) \sin (3 a+3 b x) \, dx\)

Optimal. Leaf size=198 \[ \frac {3 d^4 \sin (a+b x) \cos (a+b x)}{b^5}+\frac {3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}-\frac {9 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}-\frac {6 d^2 (c+d x)^2 \sin (a+b x) \cos (a+b x)}{b^3}-\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}+\frac {3 d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {2 (c+d x)^4 \sin (a+b x) \cos (a+b x)}{b}+\frac {3 d^4 x}{2 b^4}-\frac {d (c+d x)^3}{b^2}+\frac {(c+d x)^5}{5 d} \]

[Out]

3/2*d^4*x/b^4-d*(d*x+c)^3/b^2+1/5*(d*x+c)^5/d-9/2*d^3*(d*x+c)*cos(b*x+a)^2/b^4+3*d*(d*x+c)^3*cos(b*x+a)^2/b^2+
3*d^4*cos(b*x+a)*sin(b*x+a)/b^5-6*d^2*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)/b^3+2*(d*x+c)^4*cos(b*x+a)*sin(b*x+a)/b+
3/2*d^3*(d*x+c)*sin(b*x+a)^2/b^4-d*(d*x+c)^3*sin(b*x+a)^2/b^2

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Rubi [A]  time = 0.25, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4431, 3311, 32, 2635, 8} \[ \frac {3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}-\frac {9 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}-\frac {6 d^2 (c+d x)^2 \sin (a+b x) \cos (a+b x)}{b^3}-\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}+\frac {3 d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {3 d^4 \sin (a+b x) \cos (a+b x)}{b^5}+\frac {2 (c+d x)^4 \sin (a+b x) \cos (a+b x)}{b}-\frac {d (c+d x)^3}{b^2}+\frac {3 d^4 x}{2 b^4}+\frac {(c+d x)^5}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Csc[a + b*x]*Sin[3*a + 3*b*x],x]

[Out]

(3*d^4*x)/(2*b^4) - (d*(c + d*x)^3)/b^2 + (c + d*x)^5/(5*d) - (9*d^3*(c + d*x)*Cos[a + b*x]^2)/(2*b^4) + (3*d*
(c + d*x)^3*Cos[a + b*x]^2)/b^2 + (3*d^4*Cos[a + b*x]*Sin[a + b*x])/b^5 - (6*d^2*(c + d*x)^2*Cos[a + b*x]*Sin[
a + b*x])/b^3 + (2*(c + d*x)^4*Cos[a + b*x]*Sin[a + b*x])/b + (3*d^3*(c + d*x)*Sin[a + b*x]^2)/(2*b^4) - (d*(c
+ d*x)^3*Sin[a + b*x]^2)/b^2

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 4431

Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && M
emberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d,
1]

Rubi steps

\begin {align*} \int (c+d x)^4 \csc (a+b x) \sin (3 a+3 b x) \, dx &=\int \left (3 (c+d x)^4 \cos ^2(a+b x)-(c+d x)^4 \sin ^2(a+b x)\right ) \, dx\\ &=3 \int (c+d x)^4 \cos ^2(a+b x) \, dx-\int (c+d x)^4 \sin ^2(a+b x) \, dx\\ &=\frac {3 d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {2 (c+d x)^4 \cos (a+b x) \sin (a+b x)}{b}-\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {1}{2} \int (c+d x)^4 \, dx+\frac {3}{2} \int (c+d x)^4 \, dx+\frac {\left (3 d^2\right ) \int (c+d x)^2 \sin ^2(a+b x) \, dx}{b^2}-\frac {\left (9 d^2\right ) \int (c+d x)^2 \cos ^2(a+b x) \, dx}{b^2}\\ &=\frac {(c+d x)^5}{5 d}-\frac {9 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}+\frac {3 d (c+d x)^3 \cos ^2(a+b x)}{b^2}-\frac {6 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^3}+\frac {2 (c+d x)^4 \cos (a+b x) \sin (a+b x)}{b}+\frac {3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}+\frac {\left (3 d^2\right ) \int (c+d x)^2 \, dx}{2 b^2}-\frac {\left (9 d^2\right ) \int (c+d x)^2 \, dx}{2 b^2}-\frac {\left (3 d^4\right ) \int \sin ^2(a+b x) \, dx}{2 b^4}+\frac {\left (9 d^4\right ) \int \cos ^2(a+b x) \, dx}{2 b^4}\\ &=-\frac {d (c+d x)^3}{b^2}+\frac {(c+d x)^5}{5 d}-\frac {9 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}+\frac {3 d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {3 d^4 \cos (a+b x) \sin (a+b x)}{b^5}-\frac {6 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^3}+\frac {2 (c+d x)^4 \cos (a+b x) \sin (a+b x)}{b}+\frac {3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}-\frac {\left (3 d^4\right ) \int 1 \, dx}{4 b^4}+\frac {\left (9 d^4\right ) \int 1 \, dx}{4 b^4}\\ &=\frac {3 d^4 x}{2 b^4}-\frac {d (c+d x)^3}{b^2}+\frac {(c+d x)^5}{5 d}-\frac {9 d^3 (c+d x) \cos ^2(a+b x)}{2 b^4}+\frac {3 d (c+d x)^3 \cos ^2(a+b x)}{b^2}+\frac {3 d^4 \cos (a+b x) \sin (a+b x)}{b^5}-\frac {6 d^2 (c+d x)^2 \cos (a+b x) \sin (a+b x)}{b^3}+\frac {2 (c+d x)^4 \cos (a+b x) \sin (a+b x)}{b}+\frac {3 d^3 (c+d x) \sin ^2(a+b x)}{2 b^4}-\frac {d (c+d x)^3 \sin ^2(a+b x)}{b^2}\\ \end {align*}

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Mathematica [A]  time = 0.68, size = 128, normalized size = 0.65 \[ \frac {d (c+d x) \cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-3 d^2\right )}{b^4}+\frac {\sin (2 (a+b x)) \left (2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right )}{2 b^5}+c^4 x+2 c^3 d x^2+2 c^2 d^2 x^3+c d^3 x^4+\frac {d^4 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*Csc[a + b*x]*Sin[3*a + 3*b*x],x]

[Out]

c^4*x + 2*c^3*d*x^2 + 2*c^2*d^2*x^3 + c*d^3*x^4 + (d^4*x^5)/5 + (d*(c + d*x)*(-3*d^2 + 2*b^2*(c + d*x)^2)*Cos[
2*(a + b*x)])/b^4 + ((3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Sin[2*(a + b*x)])/(2*b^5)

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fricas [A]  time = 0.45, size = 283, normalized size = 1.43 \[ \frac {b^{5} d^{4} x^{5} + 5 \, b^{5} c d^{3} x^{4} + 10 \, {\left (b^{5} c^{2} d^{2} - b^{3} d^{4}\right )} x^{3} + 10 \, {\left (b^{5} c^{3} d - 3 \, b^{3} c d^{3}\right )} x^{2} + 10 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 2 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (2 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + 5 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 2 \, b^{4} c^{4} - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 6 \, {\left (2 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 4 \, {\left (2 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 5 \, {\left (b^{5} c^{4} - 6 \, b^{3} c^{2} d^{2} + 3 \, b d^{4}\right )} x}{5 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*csc(b*x+a)*sin(3*b*x+3*a),x, algorithm="fricas")

[Out]

1/5*(b^5*d^4*x^5 + 5*b^5*c*d^3*x^4 + 10*(b^5*c^2*d^2 - b^3*d^4)*x^3 + 10*(b^5*c^3*d - 3*b^3*c*d^3)*x^2 + 10*(2
*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 2*b^3*c^3*d - 3*b*c*d^3 + 3*(2*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)^2 + 5*(2*
b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 2*b^4*c^4 - 6*b^2*c^2*d^2 + 3*d^4 + 6*(2*b^4*c^2*d^2 - b^2*d^4)*x^2 + 4*(2*b^4
*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)*sin(b*x + a) + 5*(b^5*c^4 - 6*b^3*c^2*d^2 + 3*b*d^4)*x)/b^5

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giac [B]  time = 0.55, size = 4684, normalized size = 23.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*csc(b*x+a)*sin(3*b*x+3*a),x, algorithm="giac")

[Out]

1/5*(b^5*d^4*x^5*tan(1/2*b*x)^4*tan(1/2*a)^4 + 5*b^5*c*d^3*x^4*tan(1/2*b*x)^4*tan(1/2*a)^4 + 2*b^5*d^4*x^5*tan
(1/2*b*x)^4*tan(1/2*a)^2 + 2*b^5*d^4*x^5*tan(1/2*b*x)^2*tan(1/2*a)^4 + 10*b^5*c^2*d^2*x^3*tan(1/2*b*x)^4*tan(1
/2*a)^4 + 10*b^5*c*d^3*x^4*tan(1/2*b*x)^4*tan(1/2*a)^2 - 20*b^4*d^4*x^4*tan(1/2*b*x)^4*tan(1/2*a)^3 + 10*b^5*c
*d^3*x^4*tan(1/2*b*x)^2*tan(1/2*a)^4 - 20*b^4*d^4*x^4*tan(1/2*b*x)^3*tan(1/2*a)^4 + 10*b^5*c^3*d*x^2*tan(1/2*b
*x)^4*tan(1/2*a)^4 + b^5*d^4*x^5*tan(1/2*b*x)^4 + 4*b^5*d^4*x^5*tan(1/2*b*x)^2*tan(1/2*a)^2 + 20*b^5*c^2*d^2*x
^3*tan(1/2*b*x)^4*tan(1/2*a)^2 - 80*b^4*c*d^3*x^3*tan(1/2*b*x)^4*tan(1/2*a)^3 + b^5*d^4*x^5*tan(1/2*a)^4 + 20*
b^5*c^2*d^2*x^3*tan(1/2*b*x)^2*tan(1/2*a)^4 - 80*b^4*c*d^3*x^3*tan(1/2*b*x)^3*tan(1/2*a)^4 + 5*b^5*c^4*x*tan(1
/2*b*x)^4*tan(1/2*a)^4 + 10*b^3*d^4*x^3*tan(1/2*b*x)^4*tan(1/2*a)^4 + 5*b^5*c*d^3*x^4*tan(1/2*b*x)^4 + 20*b^4*
d^4*x^4*tan(1/2*b*x)^4*tan(1/2*a) + 20*b^5*c*d^3*x^4*tan(1/2*b*x)^2*tan(1/2*a)^2 + 120*b^4*d^4*x^4*tan(1/2*b*x
)^3*tan(1/2*a)^2 + 20*b^5*c^3*d*x^2*tan(1/2*b*x)^4*tan(1/2*a)^2 + 120*b^4*d^4*x^4*tan(1/2*b*x)^2*tan(1/2*a)^3
- 120*b^4*c^2*d^2*x^2*tan(1/2*b*x)^4*tan(1/2*a)^3 + 5*b^5*c*d^3*x^4*tan(1/2*a)^4 + 20*b^4*d^4*x^4*tan(1/2*b*x)
*tan(1/2*a)^4 + 20*b^5*c^3*d*x^2*tan(1/2*b*x)^2*tan(1/2*a)^4 - 120*b^4*c^2*d^2*x^2*tan(1/2*b*x)^3*tan(1/2*a)^4
+ 30*b^3*c*d^3*x^2*tan(1/2*b*x)^4*tan(1/2*a)^4 + 2*b^5*d^4*x^5*tan(1/2*b*x)^2 + 10*b^5*c^2*d^2*x^3*tan(1/2*b*
x)^4 + 80*b^4*c*d^3*x^3*tan(1/2*b*x)^4*tan(1/2*a) + 2*b^5*d^4*x^5*tan(1/2*a)^2 + 40*b^5*c^2*d^2*x^3*tan(1/2*b*
x)^2*tan(1/2*a)^2 + 480*b^4*c*d^3*x^3*tan(1/2*b*x)^3*tan(1/2*a)^2 + 10*b^5*c^4*x*tan(1/2*b*x)^4*tan(1/2*a)^2 -
60*b^3*d^4*x^3*tan(1/2*b*x)^4*tan(1/2*a)^2 + 480*b^4*c*d^3*x^3*tan(1/2*b*x)^2*tan(1/2*a)^3 - 160*b^3*d^4*x^3*
tan(1/2*b*x)^3*tan(1/2*a)^3 - 80*b^4*c^3*d*x*tan(1/2*b*x)^4*tan(1/2*a)^3 + 10*b^5*c^2*d^2*x^3*tan(1/2*a)^4 + 8
0*b^4*c*d^3*x^3*tan(1/2*b*x)*tan(1/2*a)^4 + 10*b^5*c^4*x*tan(1/2*b*x)^2*tan(1/2*a)^4 - 60*b^3*d^4*x^3*tan(1/2*
b*x)^2*tan(1/2*a)^4 - 80*b^4*c^3*d*x*tan(1/2*b*x)^3*tan(1/2*a)^4 + 30*b^3*c^2*d^2*x*tan(1/2*b*x)^4*tan(1/2*a)^
4 + 10*b^5*c*d^3*x^4*tan(1/2*b*x)^2 - 20*b^4*d^4*x^4*tan(1/2*b*x)^3 + 10*b^5*c^3*d*x^2*tan(1/2*b*x)^4 - 120*b^
4*d^4*x^4*tan(1/2*b*x)^2*tan(1/2*a) + 120*b^4*c^2*d^2*x^2*tan(1/2*b*x)^4*tan(1/2*a) + 10*b^5*c*d^3*x^4*tan(1/2
*a)^2 - 120*b^4*d^4*x^4*tan(1/2*b*x)*tan(1/2*a)^2 + 40*b^5*c^3*d*x^2*tan(1/2*b*x)^2*tan(1/2*a)^2 + 720*b^4*c^2
*d^2*x^2*tan(1/2*b*x)^3*tan(1/2*a)^2 - 180*b^3*c*d^3*x^2*tan(1/2*b*x)^4*tan(1/2*a)^2 - 20*b^4*d^4*x^4*tan(1/2*
a)^3 + 720*b^4*c^2*d^2*x^2*tan(1/2*b*x)^2*tan(1/2*a)^3 - 480*b^3*c*d^3*x^2*tan(1/2*b*x)^3*tan(1/2*a)^3 - 20*b^
4*c^4*tan(1/2*b*x)^4*tan(1/2*a)^3 + 60*b^2*d^4*x^2*tan(1/2*b*x)^4*tan(1/2*a)^3 + 10*b^5*c^3*d*x^2*tan(1/2*a)^4
+ 120*b^4*c^2*d^2*x^2*tan(1/2*b*x)*tan(1/2*a)^4 - 180*b^3*c*d^3*x^2*tan(1/2*b*x)^2*tan(1/2*a)^4 - 20*b^4*c^4*
tan(1/2*b*x)^3*tan(1/2*a)^4 + 60*b^2*d^4*x^2*tan(1/2*b*x)^3*tan(1/2*a)^4 + 10*b^3*c^3*d*tan(1/2*b*x)^4*tan(1/2
*a)^4 + b^5*d^4*x^5 + 20*b^5*c^2*d^2*x^3*tan(1/2*b*x)^2 - 80*b^4*c*d^3*x^3*tan(1/2*b*x)^3 + 5*b^5*c^4*x*tan(1/
2*b*x)^4 + 10*b^3*d^4*x^3*tan(1/2*b*x)^4 - 480*b^4*c*d^3*x^3*tan(1/2*b*x)^2*tan(1/2*a) + 160*b^3*d^4*x^3*tan(1
/2*b*x)^3*tan(1/2*a) + 80*b^4*c^3*d*x*tan(1/2*b*x)^4*tan(1/2*a) + 20*b^5*c^2*d^2*x^3*tan(1/2*a)^2 - 480*b^4*c*
d^3*x^3*tan(1/2*b*x)*tan(1/2*a)^2 + 20*b^5*c^4*x*tan(1/2*b*x)^2*tan(1/2*a)^2 + 360*b^3*d^4*x^3*tan(1/2*b*x)^2*
tan(1/2*a)^2 + 480*b^4*c^3*d*x*tan(1/2*b*x)^3*tan(1/2*a)^2 - 180*b^3*c^2*d^2*x*tan(1/2*b*x)^4*tan(1/2*a)^2 - 8
0*b^4*c*d^3*x^3*tan(1/2*a)^3 + 160*b^3*d^4*x^3*tan(1/2*b*x)*tan(1/2*a)^3 + 480*b^4*c^3*d*x*tan(1/2*b*x)^2*tan(
1/2*a)^3 - 480*b^3*c^2*d^2*x*tan(1/2*b*x)^3*tan(1/2*a)^3 + 120*b^2*c*d^3*x*tan(1/2*b*x)^4*tan(1/2*a)^3 + 5*b^5
*c^4*x*tan(1/2*a)^4 + 10*b^3*d^4*x^3*tan(1/2*a)^4 + 80*b^4*c^3*d*x*tan(1/2*b*x)*tan(1/2*a)^4 - 180*b^3*c^2*d^2
*x*tan(1/2*b*x)^2*tan(1/2*a)^4 + 120*b^2*c*d^3*x*tan(1/2*b*x)^3*tan(1/2*a)^4 - 15*b*d^4*x*tan(1/2*b*x)^4*tan(1
/2*a)^4 + 5*b^5*c*d^3*x^4 + 20*b^4*d^4*x^4*tan(1/2*b*x) + 20*b^5*c^3*d*x^2*tan(1/2*b*x)^2 - 120*b^4*c^2*d^2*x^
2*tan(1/2*b*x)^3 + 30*b^3*c*d^3*x^2*tan(1/2*b*x)^4 + 20*b^4*d^4*x^4*tan(1/2*a) - 720*b^4*c^2*d^2*x^2*tan(1/2*b
*x)^2*tan(1/2*a) + 480*b^3*c*d^3*x^2*tan(1/2*b*x)^3*tan(1/2*a) + 20*b^4*c^4*tan(1/2*b*x)^4*tan(1/2*a) - 60*b^2
*d^4*x^2*tan(1/2*b*x)^4*tan(1/2*a) + 20*b^5*c^3*d*x^2*tan(1/2*a)^2 - 720*b^4*c^2*d^2*x^2*tan(1/2*b*x)*tan(1/2*
a)^2 + 1080*b^3*c*d^3*x^2*tan(1/2*b*x)^2*tan(1/2*a)^2 + 120*b^4*c^4*tan(1/2*b*x)^3*tan(1/2*a)^2 - 360*b^2*d^4*
x^2*tan(1/2*b*x)^3*tan(1/2*a)^2 - 60*b^3*c^3*d*tan(1/2*b*x)^4*tan(1/2*a)^2 - 120*b^4*c^2*d^2*x^2*tan(1/2*a)^3
+ 480*b^3*c*d^3*x^2*tan(1/2*b*x)*tan(1/2*a)^3 + 120*b^4*c^4*tan(1/2*b*x)^2*tan(1/2*a)^3 - 360*b^2*d^4*x^2*tan(
1/2*b*x)^2*tan(1/2*a)^3 - 160*b^3*c^3*d*tan(1/2*b*x)^3*tan(1/2*a)^3 + 60*b^2*c^2*d^2*tan(1/2*b*x)^4*tan(1/2*a)
^3 + 30*b^3*c*d^3*x^2*tan(1/2*a)^4 + 20*b^4*c^4*tan(1/2*b*x)*tan(1/2*a)^4 - 60*b^2*d^4*x^2*tan(1/2*b*x)*tan(1/
2*a)^4 - 60*b^3*c^3*d*tan(1/2*b*x)^2*tan(1/2*a)^4 + 60*b^2*c^2*d^2*tan(1/2*b*x)^3*tan(1/2*a)^4 - 15*b*c*d^3*ta
n(1/2*b*x)^4*tan(1/2*a)^4 + 10*b^5*c^2*d^2*x^3 + 80*b^4*c*d^3*x^3*tan(1/2*b*x) + 10*b^5*c^4*x*tan(1/2*b*x)^2 -
60*b^3*d^4*x^3*tan(1/2*b*x)^2 - 80*b^4*c^3*d*x*tan(1/2*b*x)^3 + 30*b^3*c^2*d^2*x*tan(1/2*b*x)^4 + 80*b^4*c*d^
3*x^3*tan(1/2*a) - 160*b^3*d^4*x^3*tan(1/2*b*x)*tan(1/2*a) - 480*b^4*c^3*d*x*tan(1/2*b*x)^2*tan(1/2*a) + 480*b
^3*c^2*d^2*x*tan(1/2*b*x)^3*tan(1/2*a) - 120*b^2*c*d^3*x*tan(1/2*b*x)^4*tan(1/2*a) + 10*b^5*c^4*x*tan(1/2*a)^2
- 60*b^3*d^4*x^3*tan(1/2*a)^2 - 480*b^4*c^3*d*x*tan(1/2*b*x)*tan(1/2*a)^2 + 1080*b^3*c^2*d^2*x*tan(1/2*b*x)^2
*tan(1/2*a)^2 - 720*b^2*c*d^3*x*tan(1/2*b*x)^3*tan(1/2*a)^2 + 90*b*d^4*x*tan(1/2*b*x)^4*tan(1/2*a)^2 - 80*b^4*
c^3*d*x*tan(1/2*a)^3 + 480*b^3*c^2*d^2*x*tan(1/2*b*x)*tan(1/2*a)^3 - 720*b^2*c*d^3*x*tan(1/2*b*x)^2*tan(1/2*a)
^3 + 240*b*d^4*x*tan(1/2*b*x)^3*tan(1/2*a)^3 + 30*b^3*c^2*d^2*x*tan(1/2*a)^4 - 120*b^2*c*d^3*x*tan(1/2*b*x)*ta
n(1/2*a)^4 + 90*b*d^4*x*tan(1/2*b*x)^2*tan(1/2*a)^4 + 10*b^5*c^3*d*x^2 + 120*b^4*c^2*d^2*x^2*tan(1/2*b*x) - 18
0*b^3*c*d^3*x^2*tan(1/2*b*x)^2 - 20*b^4*c^4*tan(1/2*b*x)^3 + 60*b^2*d^4*x^2*tan(1/2*b*x)^3 + 10*b^3*c^3*d*tan(
1/2*b*x)^4 + 120*b^4*c^2*d^2*x^2*tan(1/2*a) - 480*b^3*c*d^3*x^2*tan(1/2*b*x)*tan(1/2*a) - 120*b^4*c^4*tan(1/2*
b*x)^2*tan(1/2*a) + 360*b^2*d^4*x^2*tan(1/2*b*x)^2*tan(1/2*a) + 160*b^3*c^3*d*tan(1/2*b*x)^3*tan(1/2*a) - 60*b
^2*c^2*d^2*tan(1/2*b*x)^4*tan(1/2*a) - 180*b^3*c*d^3*x^2*tan(1/2*a)^2 - 120*b^4*c^4*tan(1/2*b*x)*tan(1/2*a)^2
+ 360*b^2*d^4*x^2*tan(1/2*b*x)*tan(1/2*a)^2 + 360*b^3*c^3*d*tan(1/2*b*x)^2*tan(1/2*a)^2 - 360*b^2*c^2*d^2*tan(
1/2*b*x)^3*tan(1/2*a)^2 + 90*b*c*d^3*tan(1/2*b*x)^4*tan(1/2*a)^2 - 20*b^4*c^4*tan(1/2*a)^3 + 60*b^2*d^4*x^2*ta
n(1/2*a)^3 + 160*b^3*c^3*d*tan(1/2*b*x)*tan(1/2*a)^3 - 360*b^2*c^2*d^2*tan(1/2*b*x)^2*tan(1/2*a)^3 + 240*b*c*d
^3*tan(1/2*b*x)^3*tan(1/2*a)^3 - 30*d^4*tan(1/2*b*x)^4*tan(1/2*a)^3 + 10*b^3*c^3*d*tan(1/2*a)^4 - 60*b^2*c^2*d
^2*tan(1/2*b*x)*tan(1/2*a)^4 + 90*b*c*d^3*tan(1/2*b*x)^2*tan(1/2*a)^4 - 30*d^4*tan(1/2*b*x)^3*tan(1/2*a)^4 + 5
*b^5*c^4*x + 10*b^3*d^4*x^3 + 80*b^4*c^3*d*x*tan(1/2*b*x) - 180*b^3*c^2*d^2*x*tan(1/2*b*x)^2 + 120*b^2*c*d^3*x
*tan(1/2*b*x)^3 - 15*b*d^4*x*tan(1/2*b*x)^4 + 80*b^4*c^3*d*x*tan(1/2*a) - 480*b^3*c^2*d^2*x*tan(1/2*b*x)*tan(1
/2*a) + 720*b^2*c*d^3*x*tan(1/2*b*x)^2*tan(1/2*a) - 240*b*d^4*x*tan(1/2*b*x)^3*tan(1/2*a) - 180*b^3*c^2*d^2*x*
tan(1/2*a)^2 + 720*b^2*c*d^3*x*tan(1/2*b*x)*tan(1/2*a)^2 - 540*b*d^4*x*tan(1/2*b*x)^2*tan(1/2*a)^2 + 120*b^2*c
*d^3*x*tan(1/2*a)^3 - 240*b*d^4*x*tan(1/2*b*x)*tan(1/2*a)^3 - 15*b*d^4*x*tan(1/2*a)^4 + 30*b^3*c*d^3*x^2 + 20*
b^4*c^4*tan(1/2*b*x) - 60*b^2*d^4*x^2*tan(1/2*b*x) - 60*b^3*c^3*d*tan(1/2*b*x)^2 + 60*b^2*c^2*d^2*tan(1/2*b*x)
^3 - 15*b*c*d^3*tan(1/2*b*x)^4 + 20*b^4*c^4*tan(1/2*a) - 60*b^2*d^4*x^2*tan(1/2*a) - 160*b^3*c^3*d*tan(1/2*b*x
)*tan(1/2*a) + 360*b^2*c^2*d^2*tan(1/2*b*x)^2*tan(1/2*a) - 240*b*c*d^3*tan(1/2*b*x)^3*tan(1/2*a) + 30*d^4*tan(
1/2*b*x)^4*tan(1/2*a) - 60*b^3*c^3*d*tan(1/2*a)^2 + 360*b^2*c^2*d^2*tan(1/2*b*x)*tan(1/2*a)^2 - 540*b*c*d^3*ta
n(1/2*b*x)^2*tan(1/2*a)^2 + 180*d^4*tan(1/2*b*x)^3*tan(1/2*a)^2 + 60*b^2*c^2*d^2*tan(1/2*a)^3 - 240*b*c*d^3*ta
n(1/2*b*x)*tan(1/2*a)^3 + 180*d^4*tan(1/2*b*x)^2*tan(1/2*a)^3 - 15*b*c*d^3*tan(1/2*a)^4 + 30*d^4*tan(1/2*b*x)*
tan(1/2*a)^4 + 30*b^3*c^2*d^2*x - 120*b^2*c*d^3*x*tan(1/2*b*x) + 90*b*d^4*x*tan(1/2*b*x)^2 - 120*b^2*c*d^3*x*t
an(1/2*a) + 240*b*d^4*x*tan(1/2*b*x)*tan(1/2*a) + 90*b*d^4*x*tan(1/2*a)^2 + 10*b^3*c^3*d - 60*b^2*c^2*d^2*tan(
1/2*b*x) + 90*b*c*d^3*tan(1/2*b*x)^2 - 30*d^4*tan(1/2*b*x)^3 - 60*b^2*c^2*d^2*tan(1/2*a) + 240*b*c*d^3*tan(1/2
*b*x)*tan(1/2*a) - 180*d^4*tan(1/2*b*x)^2*tan(1/2*a) + 90*b*c*d^3*tan(1/2*a)^2 - 180*d^4*tan(1/2*b*x)*tan(1/2*
a)^2 - 30*d^4*tan(1/2*a)^3 - 15*b*d^4*x - 15*b*c*d^3 + 30*d^4*tan(1/2*b*x) + 30*d^4*tan(1/2*a))/(b^5*tan(1/2*b
*x)^4*tan(1/2*a)^4 + 2*b^5*tan(1/2*b*x)^4*tan(1/2*a)^2 + 2*b^5*tan(1/2*b*x)^2*tan(1/2*a)^4 + b^5*tan(1/2*b*x)^
4 + 4*b^5*tan(1/2*b*x)^2*tan(1/2*a)^2 + b^5*tan(1/2*a)^4 + 2*b^5*tan(1/2*b*x)^2 + 2*b^5*tan(1/2*a)^2 + b^5)

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maple [B]  time = 0.06, size = 1000, normalized size = 5.05 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*csc(b*x+a)*sin(3*b*x+3*a),x)

[Out]

-c^4*x-1/5*d^4*x^5+4*c^4/b*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-2*c^3*d*x^2-2*c^2*d^2*x^3-c*d^3*x^4+4*d^4
/b^5*((b*x+a)^4*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)+(b*x+a)^3*cos(b*x+a)^2-3*(b*x+a)^2*(1/2*cos(b*x+a)*s
in(b*x+a)+1/2*b*x+1/2*a)-3/2*(b*x+a)*cos(b*x+a)^2+3/4*cos(b*x+a)*sin(b*x+a)+3/4*b*x+3/4*a+(b*x+a)^3-2/5*(b*x+a
)^5-4*a*((b*x+a)^3*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)+3/4*(b*x+a)^2*cos(b*x+a)^2-3/2*(b*x+a)*(1/2*cos(b
*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)+3/8*(b*x+a)^2+3/8*sin(b*x+a)^2-3/8*(b*x+a)^4)+6*a^2*((b*x+a)^2*(1/2*cos(b*x+a)
*sin(b*x+a)+1/2*b*x+1/2*a)+1/2*(b*x+a)*cos(b*x+a)^2-1/4*cos(b*x+a)*sin(b*x+a)-1/4*b*x-1/4*a-1/3*(b*x+a)^3)-4*a
^3*((b*x+a)*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/4*(b*x+a)^2-1/4*sin(b*x+a)^2)+a^4*(1/2*cos(b*x+a)*sin(
b*x+a)+1/2*b*x+1/2*a))+16*c^3*d/b^2*((b*x+a)*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/4*(b*x+a)^2-1/4*sin(b
*x+a)^2-a*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a))+24*c^2*d^2/b^3*((b*x+a)^2*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*
b*x+1/2*a)+1/2*(b*x+a)*cos(b*x+a)^2-1/4*cos(b*x+a)*sin(b*x+a)-1/4*b*x-1/4*a-1/3*(b*x+a)^3-2*a*((b*x+a)*(1/2*co
s(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/4*(b*x+a)^2-1/4*sin(b*x+a)^2)+a^2*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*
a))+16*d^3*c/b^4*((b*x+a)^3*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)+3/4*(b*x+a)^2*cos(b*x+a)^2-3/2*(b*x+a)*(
1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)+3/8*(b*x+a)^2+3/8*sin(b*x+a)^2-3/8*(b*x+a)^4-3*a*((b*x+a)^2*(1/2*cos(
b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)+1/2*(b*x+a)*cos(b*x+a)^2-1/4*cos(b*x+a)*sin(b*x+a)-1/4*b*x-1/4*a-1/3*(b*x+a)^
3)+3*a^2*((b*x+a)*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/4*(b*x+a)^2-1/4*sin(b*x+a)^2)-a^3*(1/2*cos(b*x+a
)*sin(b*x+a)+1/2*b*x+1/2*a))

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maxima [A]  time = 0.40, size = 244, normalized size = 1.23 \[ \frac {{\left (b x + \sin \left (2 \, b x + 2 \, a\right )\right )} c^{4}}{b} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, b x \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} c^{3} d}{b^{2}} + \frac {{\left (2 \, b^{3} x^{3} + 6 \, b x \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (2 \, b^{2} x^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c^{2} d^{2}}{b^{3}} + \frac {{\left (b^{4} x^{4} + 3 \, {\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{3}}{b^{4}} + \frac {{\left (2 \, b^{5} x^{5} + 10 \, {\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \cos \left (2 \, b x + 2 \, a\right ) + 5 \, {\left (2 \, b^{4} x^{4} - 6 \, b^{2} x^{2} + 3\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{4}}{10 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*csc(b*x+a)*sin(3*b*x+3*a),x, algorithm="maxima")

[Out]

(b*x + sin(2*b*x + 2*a))*c^4/b + 2*(b^2*x^2 + 2*b*x*sin(2*b*x + 2*a) + cos(2*b*x + 2*a))*c^3*d/b^2 + (2*b^3*x^
3 + 6*b*x*cos(2*b*x + 2*a) + 3*(2*b^2*x^2 - 1)*sin(2*b*x + 2*a))*c^2*d^2/b^3 + (b^4*x^4 + 3*(2*b^2*x^2 - 1)*co
s(2*b*x + 2*a) + 2*(2*b^3*x^3 - 3*b*x)*sin(2*b*x + 2*a))*c*d^3/b^4 + 1/10*(2*b^5*x^5 + 10*(2*b^3*x^3 - 3*b*x)*
cos(2*b*x + 2*a) + 5*(2*b^4*x^4 - 6*b^2*x^2 + 3)*sin(2*b*x + 2*a))*d^4/b^5

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mupad [B]  time = 0.65, size = 344, normalized size = 1.74 \[ \frac {\frac {3\,d^4\,\sin \left (2\,a+2\,b\,x\right )}{2}+b^5\,c^4\,x+b^4\,c^4\,\sin \left (2\,a+2\,b\,x\right )+\frac {b^5\,d^4\,x^5}{5}+2\,b^3\,c^3\,d\,\cos \left (2\,a+2\,b\,x\right )+2\,b^5\,c^3\,d\,x^2+b^5\,c\,d^3\,x^4-3\,b^2\,c^2\,d^2\,\sin \left (2\,a+2\,b\,x\right )+2\,b^3\,d^4\,x^3\,\cos \left (2\,a+2\,b\,x\right )+2\,b^5\,c^2\,d^2\,x^3-3\,b^2\,d^4\,x^2\,\sin \left (2\,a+2\,b\,x\right )+b^4\,d^4\,x^4\,\sin \left (2\,a+2\,b\,x\right )-3\,b\,c\,d^3\,\cos \left (2\,a+2\,b\,x\right )-3\,b\,d^4\,x\,\cos \left (2\,a+2\,b\,x\right )+6\,b^4\,c^2\,d^2\,x^2\,\sin \left (2\,a+2\,b\,x\right )-6\,b^2\,c\,d^3\,x\,\sin \left (2\,a+2\,b\,x\right )+4\,b^4\,c^3\,d\,x\,\sin \left (2\,a+2\,b\,x\right )+6\,b^3\,c^2\,d^2\,x\,\cos \left (2\,a+2\,b\,x\right )+6\,b^3\,c\,d^3\,x^2\,\cos \left (2\,a+2\,b\,x\right )+4\,b^4\,c\,d^3\,x^3\,\sin \left (2\,a+2\,b\,x\right )}{b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(3*a + 3*b*x)*(c + d*x)^4)/sin(a + b*x),x)

[Out]

((3*d^4*sin(2*a + 2*b*x))/2 + b^5*c^4*x + b^4*c^4*sin(2*a + 2*b*x) + (b^5*d^4*x^5)/5 + 2*b^3*c^3*d*cos(2*a + 2
*b*x) + 2*b^5*c^3*d*x^2 + b^5*c*d^3*x^4 - 3*b^2*c^2*d^2*sin(2*a + 2*b*x) + 2*b^3*d^4*x^3*cos(2*a + 2*b*x) + 2*
b^5*c^2*d^2*x^3 - 3*b^2*d^4*x^2*sin(2*a + 2*b*x) + b^4*d^4*x^4*sin(2*a + 2*b*x) - 3*b*c*d^3*cos(2*a + 2*b*x) -
3*b*d^4*x*cos(2*a + 2*b*x) + 6*b^4*c^2*d^2*x^2*sin(2*a + 2*b*x) - 6*b^2*c*d^3*x*sin(2*a + 2*b*x) + 4*b^4*c^3*
d*x*sin(2*a + 2*b*x) + 6*b^3*c^2*d^2*x*cos(2*a + 2*b*x) + 6*b^3*c*d^3*x^2*cos(2*a + 2*b*x) + 4*b^4*c*d^3*x^3*s
in(2*a + 2*b*x))/b^5

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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((d*x+c)**4*csc(b*x+a)*sin(3*b*x+3*a),x)

[Out]

Timed out

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