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

Optimal. Leaf size=171 \[ \frac {3 d^3 \sin ^2(a+b x)}{8 b^4}-\frac {9 d^3 \cos ^2(a+b x)}{8 b^4}-\frac {3 d^2 (c+d x) \sin (a+b x) \cos (a+b x)}{b^3}-\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}+\frac {9 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}+\frac {2 (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b}-\frac {3 c d^2 x}{2 b^2}-\frac {3 d^3 x^2}{4 b^2}+\frac {(c+d x)^4}{4 d} \]

[Out]

-3/2*c*d^2*x/b^2-3/4*d^3*x^2/b^2+1/4*(d*x+c)^4/d-9/8*d^3*cos(b*x+a)^2/b^4+9/4*d*(d*x+c)^2*cos(b*x+a)^2/b^2-3*d
^2*(d*x+c)*cos(b*x+a)*sin(b*x+a)/b^3+2*(d*x+c)^3*cos(b*x+a)*sin(b*x+a)/b+3/8*d^3*sin(b*x+a)^2/b^4-3/4*d*(d*x+c
)^2*sin(b*x+a)^2/b^2

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Rubi [A]  time = 0.18, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4431, 3311, 32, 3310} \[ -\frac {3 d^2 (c+d x) \sin (a+b x) \cos (a+b x)}{b^3}-\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}+\frac {9 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}+\frac {3 d^3 \sin ^2(a+b x)}{8 b^4}-\frac {9 d^3 \cos ^2(a+b x)}{8 b^4}+\frac {2 (c+d x)^3 \sin (a+b x) \cos (a+b x)}{b}-\frac {3 c d^2 x}{2 b^2}-\frac {3 d^3 x^2}{4 b^2}+\frac {(c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*c*d^2*x)/(2*b^2) - (3*d^3*x^2)/(4*b^2) + (c + d*x)^4/(4*d) - (9*d^3*Cos[a + b*x]^2)/(8*b^4) + (9*d*(c + d*
x)^2*Cos[a + b*x]^2)/(4*b^2) - (3*d^2*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/b^3 + (2*(c + d*x)^3*Cos[a + b*x]*S
in[a + b*x])/b + (3*d^3*Sin[a + b*x]^2)/(8*b^4) - (3*d*(c + d*x)^2*Sin[a + b*x]^2)/(4*b^2)

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 3310

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

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)^3 \csc (a+b x) \sin (3 a+3 b x) \, dx &=\int \left (3 (c+d x)^3 \cos ^2(a+b x)-(c+d x)^3 \sin ^2(a+b x)\right ) \, dx\\ &=3 \int (c+d x)^3 \cos ^2(a+b x) \, dx-\int (c+d x)^3 \sin ^2(a+b x) \, dx\\ &=\frac {9 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}+\frac {2 (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b}-\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {1}{2} \int (c+d x)^3 \, dx+\frac {3}{2} \int (c+d x)^3 \, dx+\frac {\left (3 d^2\right ) \int (c+d x) \sin ^2(a+b x) \, dx}{2 b^2}-\frac {\left (9 d^2\right ) \int (c+d x) \cos ^2(a+b x) \, dx}{2 b^2}\\ &=\frac {(c+d x)^4}{4 d}-\frac {9 d^3 \cos ^2(a+b x)}{8 b^4}+\frac {9 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \cos (a+b x) \sin (a+b x)}{b^3}+\frac {2 (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b}+\frac {3 d^3 \sin ^2(a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}+\frac {\left (3 d^2\right ) \int (c+d x) \, dx}{4 b^2}-\frac {\left (9 d^2\right ) \int (c+d x) \, dx}{4 b^2}\\ &=-\frac {3 c d^2 x}{2 b^2}-\frac {3 d^3 x^2}{4 b^2}+\frac {(c+d x)^4}{4 d}-\frac {9 d^3 \cos ^2(a+b x)}{8 b^4}+\frac {9 d (c+d x)^2 \cos ^2(a+b x)}{4 b^2}-\frac {3 d^2 (c+d x) \cos (a+b x) \sin (a+b x)}{b^3}+\frac {2 (c+d x)^3 \cos (a+b x) \sin (a+b x)}{b}+\frac {3 d^3 \sin ^2(a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 105, normalized size = 0.61 \[ \frac {2 b (c+d x) \sin (2 (a+b x)) \left (2 b^2 (c+d x)^2-3 d^2\right )+3 d \cos (2 (a+b x)) \left (2 b^2 (c+d x)^2-d^2\right )+b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )}{4 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 188, normalized size = 1.10 \[ \frac {b^{4} d^{3} x^{4} + 4 \, b^{4} c d^{2} x^{3} + 6 \, {\left (b^{4} c^{2} d - b^{2} d^{3}\right )} x^{2} + 6 \, {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2} + 4 \, {\left (2 \, b^{3} d^{3} x^{3} + 6 \, b^{3} c d^{2} x^{2} + 2 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (2 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 4 \, {\left (b^{4} c^{3} - 3 \, b^{2} c d^{2}\right )} x}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 4.43, size = 3139, normalized size = 18.36 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^4*d^3*x^4*tan(1/2*b*x)^4*tan(1/2*a)^4 + 4*b^4*c*d^2*x^3*tan(1/2*b*x)^4*tan(1/2*a)^4 + 2*b^4*d^3*x^4*tan
(1/2*b*x)^4*tan(1/2*a)^2 + 2*b^4*d^3*x^4*tan(1/2*b*x)^2*tan(1/2*a)^4 + 6*b^4*c^2*d*x^2*tan(1/2*b*x)^4*tan(1/2*
a)^4 + 8*b^4*c*d^2*x^3*tan(1/2*b*x)^4*tan(1/2*a)^2 - 16*b^3*d^3*x^3*tan(1/2*b*x)^4*tan(1/2*a)^3 + 8*b^4*c*d^2*
x^3*tan(1/2*b*x)^2*tan(1/2*a)^4 - 16*b^3*d^3*x^3*tan(1/2*b*x)^3*tan(1/2*a)^4 + 4*b^4*c^3*x*tan(1/2*b*x)^4*tan(
1/2*a)^4 + b^4*d^3*x^4*tan(1/2*b*x)^4 + 4*b^4*d^3*x^4*tan(1/2*b*x)^2*tan(1/2*a)^2 + 12*b^4*c^2*d*x^2*tan(1/2*b
*x)^4*tan(1/2*a)^2 - 48*b^3*c*d^2*x^2*tan(1/2*b*x)^4*tan(1/2*a)^3 + b^4*d^3*x^4*tan(1/2*a)^4 + 12*b^4*c^2*d*x^
2*tan(1/2*b*x)^2*tan(1/2*a)^4 - 48*b^3*c*d^2*x^2*tan(1/2*b*x)^3*tan(1/2*a)^4 + 6*b^2*d^3*x^2*tan(1/2*b*x)^4*ta
n(1/2*a)^4 + 4*b^4*c*d^2*x^3*tan(1/2*b*x)^4 + 16*b^3*d^3*x^3*tan(1/2*b*x)^4*tan(1/2*a) + 16*b^4*c*d^2*x^3*tan(
1/2*b*x)^2*tan(1/2*a)^2 + 96*b^3*d^3*x^3*tan(1/2*b*x)^3*tan(1/2*a)^2 + 8*b^4*c^3*x*tan(1/2*b*x)^4*tan(1/2*a)^2
+ 96*b^3*d^3*x^3*tan(1/2*b*x)^2*tan(1/2*a)^3 - 48*b^3*c^2*d*x*tan(1/2*b*x)^4*tan(1/2*a)^3 + 4*b^4*c*d^2*x^3*t
an(1/2*a)^4 + 16*b^3*d^3*x^3*tan(1/2*b*x)*tan(1/2*a)^4 + 8*b^4*c^3*x*tan(1/2*b*x)^2*tan(1/2*a)^4 - 48*b^3*c^2*
d*x*tan(1/2*b*x)^3*tan(1/2*a)^4 + 12*b^2*c*d^2*x*tan(1/2*b*x)^4*tan(1/2*a)^4 + 2*b^4*d^3*x^4*tan(1/2*b*x)^2 +
6*b^4*c^2*d*x^2*tan(1/2*b*x)^4 + 48*b^3*c*d^2*x^2*tan(1/2*b*x)^4*tan(1/2*a) + 2*b^4*d^3*x^4*tan(1/2*a)^2 + 24*
b^4*c^2*d*x^2*tan(1/2*b*x)^2*tan(1/2*a)^2 + 288*b^3*c*d^2*x^2*tan(1/2*b*x)^3*tan(1/2*a)^2 - 36*b^2*d^3*x^2*tan
(1/2*b*x)^4*tan(1/2*a)^2 + 288*b^3*c*d^2*x^2*tan(1/2*b*x)^2*tan(1/2*a)^3 - 96*b^2*d^3*x^2*tan(1/2*b*x)^3*tan(1
/2*a)^3 - 16*b^3*c^3*tan(1/2*b*x)^4*tan(1/2*a)^3 + 6*b^4*c^2*d*x^2*tan(1/2*a)^4 + 48*b^3*c*d^2*x^2*tan(1/2*b*x
)*tan(1/2*a)^4 - 36*b^2*d^3*x^2*tan(1/2*b*x)^2*tan(1/2*a)^4 - 16*b^3*c^3*tan(1/2*b*x)^3*tan(1/2*a)^4 + 6*b^2*c
^2*d*tan(1/2*b*x)^4*tan(1/2*a)^4 + 8*b^4*c*d^2*x^3*tan(1/2*b*x)^2 - 16*b^3*d^3*x^3*tan(1/2*b*x)^3 + 4*b^4*c^3*
x*tan(1/2*b*x)^4 - 96*b^3*d^3*x^3*tan(1/2*b*x)^2*tan(1/2*a) + 48*b^3*c^2*d*x*tan(1/2*b*x)^4*tan(1/2*a) + 8*b^4
*c*d^2*x^3*tan(1/2*a)^2 - 96*b^3*d^3*x^3*tan(1/2*b*x)*tan(1/2*a)^2 + 16*b^4*c^3*x*tan(1/2*b*x)^2*tan(1/2*a)^2
+ 288*b^3*c^2*d*x*tan(1/2*b*x)^3*tan(1/2*a)^2 - 72*b^2*c*d^2*x*tan(1/2*b*x)^4*tan(1/2*a)^2 - 16*b^3*d^3*x^3*ta
n(1/2*a)^3 + 288*b^3*c^2*d*x*tan(1/2*b*x)^2*tan(1/2*a)^3 - 192*b^2*c*d^2*x*tan(1/2*b*x)^3*tan(1/2*a)^3 + 24*b*
d^3*x*tan(1/2*b*x)^4*tan(1/2*a)^3 + 4*b^4*c^3*x*tan(1/2*a)^4 + 48*b^3*c^2*d*x*tan(1/2*b*x)*tan(1/2*a)^4 - 72*b
^2*c*d^2*x*tan(1/2*b*x)^2*tan(1/2*a)^4 + 24*b*d^3*x*tan(1/2*b*x)^3*tan(1/2*a)^4 + b^4*d^3*x^4 + 12*b^4*c^2*d*x
^2*tan(1/2*b*x)^2 - 48*b^3*c*d^2*x^2*tan(1/2*b*x)^3 + 6*b^2*d^3*x^2*tan(1/2*b*x)^4 - 288*b^3*c*d^2*x^2*tan(1/2
*b*x)^2*tan(1/2*a) + 96*b^2*d^3*x^2*tan(1/2*b*x)^3*tan(1/2*a) + 16*b^3*c^3*tan(1/2*b*x)^4*tan(1/2*a) + 12*b^4*
c^2*d*x^2*tan(1/2*a)^2 - 288*b^3*c*d^2*x^2*tan(1/2*b*x)*tan(1/2*a)^2 + 216*b^2*d^3*x^2*tan(1/2*b*x)^2*tan(1/2*
a)^2 + 96*b^3*c^3*tan(1/2*b*x)^3*tan(1/2*a)^2 - 36*b^2*c^2*d*tan(1/2*b*x)^4*tan(1/2*a)^2 - 48*b^3*c*d^2*x^2*ta
n(1/2*a)^3 + 96*b^2*d^3*x^2*tan(1/2*b*x)*tan(1/2*a)^3 + 96*b^3*c^3*tan(1/2*b*x)^2*tan(1/2*a)^3 - 96*b^2*c^2*d*
tan(1/2*b*x)^3*tan(1/2*a)^3 + 24*b*c*d^2*tan(1/2*b*x)^4*tan(1/2*a)^3 + 6*b^2*d^3*x^2*tan(1/2*a)^4 + 16*b^3*c^3
*tan(1/2*b*x)*tan(1/2*a)^4 - 36*b^2*c^2*d*tan(1/2*b*x)^2*tan(1/2*a)^4 + 24*b*c*d^2*tan(1/2*b*x)^3*tan(1/2*a)^4
- 3*d^3*tan(1/2*b*x)^4*tan(1/2*a)^4 + 4*b^4*c*d^2*x^3 + 16*b^3*d^3*x^3*tan(1/2*b*x) + 8*b^4*c^3*x*tan(1/2*b*x
)^2 - 48*b^3*c^2*d*x*tan(1/2*b*x)^3 + 12*b^2*c*d^2*x*tan(1/2*b*x)^4 + 16*b^3*d^3*x^3*tan(1/2*a) - 288*b^3*c^2*
d*x*tan(1/2*b*x)^2*tan(1/2*a) + 192*b^2*c*d^2*x*tan(1/2*b*x)^3*tan(1/2*a) - 24*b*d^3*x*tan(1/2*b*x)^4*tan(1/2*
a) + 8*b^4*c^3*x*tan(1/2*a)^2 - 288*b^3*c^2*d*x*tan(1/2*b*x)*tan(1/2*a)^2 + 432*b^2*c*d^2*x*tan(1/2*b*x)^2*tan
(1/2*a)^2 - 144*b*d^3*x*tan(1/2*b*x)^3*tan(1/2*a)^2 - 48*b^3*c^2*d*x*tan(1/2*a)^3 + 192*b^2*c*d^2*x*tan(1/2*b*
x)*tan(1/2*a)^3 - 144*b*d^3*x*tan(1/2*b*x)^2*tan(1/2*a)^3 + 12*b^2*c*d^2*x*tan(1/2*a)^4 - 24*b*d^3*x*tan(1/2*b
*x)*tan(1/2*a)^4 + 6*b^4*c^2*d*x^2 + 48*b^3*c*d^2*x^2*tan(1/2*b*x) - 36*b^2*d^3*x^2*tan(1/2*b*x)^2 - 16*b^3*c^
3*tan(1/2*b*x)^3 + 6*b^2*c^2*d*tan(1/2*b*x)^4 + 48*b^3*c*d^2*x^2*tan(1/2*a) - 96*b^2*d^3*x^2*tan(1/2*b*x)*tan(
1/2*a) - 96*b^3*c^3*tan(1/2*b*x)^2*tan(1/2*a) + 96*b^2*c^2*d*tan(1/2*b*x)^3*tan(1/2*a) - 24*b*c*d^2*tan(1/2*b*
x)^4*tan(1/2*a) - 36*b^2*d^3*x^2*tan(1/2*a)^2 - 96*b^3*c^3*tan(1/2*b*x)*tan(1/2*a)^2 + 216*b^2*c^2*d*tan(1/2*b
*x)^2*tan(1/2*a)^2 - 144*b*c*d^2*tan(1/2*b*x)^3*tan(1/2*a)^2 + 18*d^3*tan(1/2*b*x)^4*tan(1/2*a)^2 - 16*b^3*c^3
*tan(1/2*a)^3 + 96*b^2*c^2*d*tan(1/2*b*x)*tan(1/2*a)^3 - 144*b*c*d^2*tan(1/2*b*x)^2*tan(1/2*a)^3 + 48*d^3*tan(
1/2*b*x)^3*tan(1/2*a)^3 + 6*b^2*c^2*d*tan(1/2*a)^4 - 24*b*c*d^2*tan(1/2*b*x)*tan(1/2*a)^4 + 18*d^3*tan(1/2*b*x
)^2*tan(1/2*a)^4 + 4*b^4*c^3*x + 48*b^3*c^2*d*x*tan(1/2*b*x) - 72*b^2*c*d^2*x*tan(1/2*b*x)^2 + 24*b*d^3*x*tan(
1/2*b*x)^3 + 48*b^3*c^2*d*x*tan(1/2*a) - 192*b^2*c*d^2*x*tan(1/2*b*x)*tan(1/2*a) + 144*b*d^3*x*tan(1/2*b*x)^2*
tan(1/2*a) - 72*b^2*c*d^2*x*tan(1/2*a)^2 + 144*b*d^3*x*tan(1/2*b*x)*tan(1/2*a)^2 + 24*b*d^3*x*tan(1/2*a)^3 + 6
*b^2*d^3*x^2 + 16*b^3*c^3*tan(1/2*b*x) - 36*b^2*c^2*d*tan(1/2*b*x)^2 + 24*b*c*d^2*tan(1/2*b*x)^3 - 3*d^3*tan(1
/2*b*x)^4 + 16*b^3*c^3*tan(1/2*a) - 96*b^2*c^2*d*tan(1/2*b*x)*tan(1/2*a) + 144*b*c*d^2*tan(1/2*b*x)^2*tan(1/2*
a) - 48*d^3*tan(1/2*b*x)^3*tan(1/2*a) - 36*b^2*c^2*d*tan(1/2*a)^2 + 144*b*c*d^2*tan(1/2*b*x)*tan(1/2*a)^2 - 10
8*d^3*tan(1/2*b*x)^2*tan(1/2*a)^2 + 24*b*c*d^2*tan(1/2*a)^3 - 48*d^3*tan(1/2*b*x)*tan(1/2*a)^3 - 3*d^3*tan(1/2
*a)^4 + 12*b^2*c*d^2*x - 24*b*d^3*x*tan(1/2*b*x) - 24*b*d^3*x*tan(1/2*a) + 6*b^2*c^2*d - 24*b*c*d^2*tan(1/2*b*
x) + 18*d^3*tan(1/2*b*x)^2 - 24*b*c*d^2*tan(1/2*a) + 48*d^3*tan(1/2*b*x)*tan(1/2*a) + 18*d^3*tan(1/2*a)^2 - 3*
d^3)/(b^4*tan(1/2*b*x)^4*tan(1/2*a)^4 + 2*b^4*tan(1/2*b*x)^4*tan(1/2*a)^2 + 2*b^4*tan(1/2*b*x)^2*tan(1/2*a)^4
+ b^4*tan(1/2*b*x)^4 + 4*b^4*tan(1/2*b*x)^2*tan(1/2*a)^2 + b^4*tan(1/2*a)^4 + 2*b^4*tan(1/2*b*x)^2 + 2*b^4*tan
(1/2*a)^2 + b^4)

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maple [B]  time = 0.04, size = 580, normalized size = 3.39 \[ -c^{3} x -\frac {d^{3} x^{4}}{4}+\frac {4 c^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {3 c^{2} d \,x^{2}}{2}-c \,d^{2} x^{3}+\frac {4 d^{3} \left (\left (b x +a \right )^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {3 \left (b x +a \right )^{2} \left (\cos ^{2}\left (b x +a \right )\right )}{4}-\frac {3 \left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )}{2}+\frac {3 \left (b x +a \right )^{2}}{8}+\frac {3 \left (\sin ^{2}\left (b x +a \right )\right )}{8}-\frac {3 \left (b x +a \right )^{4}}{8}-3 a \left (\left (b x +a \right )^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}-\frac {\left (b x +a \right )^{3}}{3}\right )+3 a^{2} \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )-a^{3} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{4}}+\frac {12 c^{2} d \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}-a \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {12 c \,d^{2} \left (\left (b x +a \right )^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )+\frac {\left (b x +a \right ) \left (\cos ^{2}\left (b x +a \right )\right )}{2}-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{4}-\frac {b x}{4}-\frac {a}{4}-\frac {\left (b x +a \right )^{3}}{3}-2 a \left (\left (b x +a \right ) \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )-\frac {\left (b x +a \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (b x +a \right )\right )}{4}\right )+a^{2} \left (\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{2}+\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^3*x-1/4*d^3*x^4+4*c^3/b*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-3/2*c^2*d*x^2-c*d^2*x^3+4*d^3/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*c
os(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))+12*c^2*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))+12*c*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*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^2*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a))

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maxima [A]  time = 0.37, size = 173, normalized size = 1.01 \[ \frac {{\left (b x + \sin \left (2 \, b x + 2 \, a\right )\right )} c^{3}}{b} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )\right )} c^{2} d}{2 \, b^{2}} + \frac {{\left (2 \, b^{3} x^{3} + 6 \, b x \cos \left (2 \, b x + 2 \, a\right ) + 3 \, {\left (2 \, b^{2} x^{2} - 1\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} c d^{2}}{2 \, b^{3}} + \frac {{\left (b^{4} x^{4} + 3 \, {\left (2 \, b^{2} x^{2} - 1\right )} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d^{3}}{4 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.18, size = 216, normalized size = 1.26 \[ c^3\,x+\frac {d^3\,x^4}{4}+\frac {3\,c^2\,d\,x^2}{2}+c\,d^2\,x^3-\frac {3\,d^3\,\cos \left (2\,a+2\,b\,x\right )}{4\,b^4}+\frac {c^3\,\sin \left (2\,a+2\,b\,x\right )}{b}+\frac {3\,c^2\,d\,\cos \left (2\,a+2\,b\,x\right )}{2\,b^2}-\frac {3\,c\,d^2\,\sin \left (2\,a+2\,b\,x\right )}{2\,b^3}-\frac {3\,d^3\,x\,\sin \left (2\,a+2\,b\,x\right )}{2\,b^3}+\frac {3\,d^3\,x^2\,\cos \left (2\,a+2\,b\,x\right )}{2\,b^2}+\frac {d^3\,x^3\,\sin \left (2\,a+2\,b\,x\right )}{b}+\frac {3\,c\,d^2\,x\,\cos \left (2\,a+2\,b\,x\right )}{b^2}+\frac {3\,c^2\,d\,x\,\sin \left (2\,a+2\,b\,x\right )}{b}+\frac {3\,c\,d^2\,x^2\,\sin \left (2\,a+2\,b\,x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)**3*csc(b*x+a)*sin(3*b*x+3*a),x)

[Out]

Timed out

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