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

Optimal. Leaf size=260 \[ \frac {3 d^4 \sin ^4(a+b x)}{128 b^5}+\frac {45 d^4 \sin ^2(a+b x)}{128 b^5}-\frac {3 d^3 (c+d x) \sin ^3(a+b x) \cos (a+b x)}{32 b^4}-\frac {45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}+\frac {d (c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b^2}+\frac {3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}+\frac {45 c d^3 x}{64 b^3}+\frac {45 d^4 x^2}{128 b^3}-\frac {3 (c+d x)^4}{32 b} \]

[Out]

45/64*c*d^3*x/b^3+45/128*d^4*x^2/b^3-3/32*(d*x+c)^4/b-45/64*d^3*(d*x+c)*cos(b*x+a)*sin(b*x+a)/b^4+3/8*d*(d*x+c
)^3*cos(b*x+a)*sin(b*x+a)/b^2+45/128*d^4*sin(b*x+a)^2/b^5-9/16*d^2*(d*x+c)^2*sin(b*x+a)^2/b^3-3/32*d^3*(d*x+c)
*cos(b*x+a)*sin(b*x+a)^3/b^4+1/4*d*(d*x+c)^3*cos(b*x+a)*sin(b*x+a)^3/b^2+3/128*d^4*sin(b*x+a)^4/b^5-3/16*d^2*(
d*x+c)^2*sin(b*x+a)^4/b^3+1/4*(d*x+c)^4*sin(b*x+a)^4/b

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Rubi [A]  time = 0.24, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4404, 3311, 32, 3310} \[ -\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}-\frac {3 d^3 (c+d x) \sin ^3(a+b x) \cos (a+b x)}{32 b^4}-\frac {45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}+\frac {d (c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b^2}+\frac {3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}+\frac {3 d^4 \sin ^4(a+b x)}{128 b^5}+\frac {45 d^4 \sin ^2(a+b x)}{128 b^5}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}+\frac {45 c d^3 x}{64 b^3}+\frac {45 d^4 x^2}{128 b^3}-\frac {3 (c+d x)^4}{32 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(45*c*d^3*x)/(64*b^3) + (45*d^4*x^2)/(128*b^3) - (3*(c + d*x)^4)/(32*b) - (45*d^3*(c + d*x)*Cos[a + b*x]*Sin[a
+ b*x])/(64*b^4) + (3*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^2) + (45*d^4*Sin[a + b*x]^2)/(128*b^5) -
(9*d^2*(c + d*x)^2*Sin[a + b*x]^2)/(16*b^3) - (3*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^3)/(32*b^4) + (d*(c +
d*x)^3*Cos[a + b*x]*Sin[a + b*x]^3)/(4*b^2) + (3*d^4*Sin[a + b*x]^4)/(128*b^5) - (3*d^2*(c + d*x)^2*Sin[a + b
*x]^4)/(16*b^3) + ((c + d*x)^4*Sin[a + b*x]^4)/(4*b)

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 4404

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c +
d*x)^m*Sin[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n +
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rubi steps

\begin {align*} \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx &=\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \int (c+d x)^3 \sin ^4(a+b x) \, dx}{b}\\ &=\frac {d (c+d x)^3 \cos (a+b x) \sin ^3(a+b x)}{4 b^2}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {(3 d) \int (c+d x)^3 \sin ^2(a+b x) \, dx}{4 b}+\frac {\left (3 d^3\right ) \int (c+d x) \sin ^4(a+b x) \, dx}{8 b^3}\\ &=\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}-\frac {3 d^3 (c+d x) \cos (a+b x) \sin ^3(a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos (a+b x) \sin ^3(a+b x)}{4 b^2}+\frac {3 d^4 \sin ^4(a+b x)}{128 b^5}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {(3 d) \int (c+d x)^3 \, dx}{8 b}+\frac {\left (9 d^3\right ) \int (c+d x) \sin ^2(a+b x) \, dx}{32 b^3}+\frac {\left (9 d^3\right ) \int (c+d x) \sin ^2(a+b x) \, dx}{8 b^3}\\ &=-\frac {3 (c+d x)^4}{32 b}-\frac {45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}+\frac {45 d^4 \sin ^2(a+b x)}{128 b^5}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}-\frac {3 d^3 (c+d x) \cos (a+b x) \sin ^3(a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos (a+b x) \sin ^3(a+b x)}{4 b^2}+\frac {3 d^4 \sin ^4(a+b x)}{128 b^5}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}+\frac {\left (9 d^3\right ) \int (c+d x) \, dx}{64 b^3}+\frac {\left (9 d^3\right ) \int (c+d x) \, dx}{16 b^3}\\ &=\frac {45 c d^3 x}{64 b^3}+\frac {45 d^4 x^2}{128 b^3}-\frac {3 (c+d x)^4}{32 b}-\frac {45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}+\frac {45 d^4 \sin ^2(a+b x)}{128 b^5}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}-\frac {3 d^3 (c+d x) \cos (a+b x) \sin ^3(a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos (a+b x) \sin ^3(a+b x)}{4 b^2}+\frac {3 d^4 \sin ^4(a+b x)}{128 b^5}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}\\ \end {align*}

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Mathematica [A]  time = 1.70, size = 158, normalized size = 0.61 \[ \frac {-8 b d (c+d x) \sin (2 (a+b x)) \left (\cos (2 (a+b x)) \left (8 b^2 (c+d x)^2-3 d^2\right )-16 \left (2 b^2 (c+d x)^2-3 d^2\right )\right )-64 \cos (2 (a+b x)) \left (2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right )+\cos (4 (a+b x)) \left (32 b^4 (c+d x)^4-24 b^2 d^2 (c+d x)^2+3 d^4\right )}{1024 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-64*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + (3*d^4 - 24*b^2*d^2*(c + d*x)^2 +
32*b^4*(c + d*x)^4)*Cos[4*(a + b*x)] - 8*b*d*(c + d*x)*(-16*(-3*d^2 + 2*b^2*(c + d*x)^2) + (-3*d^2 + 8*b^2*(c
+ d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b*x)])/(1024*b^5)

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fricas [A]  time = 0.59, size = 434, normalized size = 1.67 \[ \frac {20 \, b^{4} d^{4} x^{4} + 80 \, b^{4} c d^{3} x^{3} + {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 3 \, {\left (40 \, b^{4} c^{2} d^{2} - 17 \, b^{2} d^{4}\right )} x^{2} - {\left (64 \, b^{4} d^{4} x^{4} + 256 \, b^{4} c d^{3} x^{3} + 64 \, b^{4} c^{4} - 120 \, b^{2} c^{2} d^{2} + 51 \, d^{4} + 24 \, {\left (16 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 16 \, {\left (16 \, b^{4} c^{3} d - 15 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (40 \, b^{4} c^{3} d - 51 \, b^{2} c d^{3}\right )} x - 2 \, {\left (2 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - {\left (40 \, b^{3} d^{4} x^{3} + 120 \, b^{3} c d^{3} x^{2} + 40 \, b^{3} c^{3} d - 51 \, b c d^{3} + 3 \, {\left (40 \, b^{3} c^{2} d^{2} - 17 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(20*b^4*d^4*x^4 + 80*b^4*c*d^3*x^3 + (32*b^4*d^4*x^4 + 128*b^4*c*d^3*x^3 + 32*b^4*c^4 - 24*b^2*c^2*d^2 +
3*d^4 + 24*(8*b^4*c^2*d^2 - b^2*d^4)*x^2 + 16*(8*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)^4 + 3*(40*b^4*c^2*d
^2 - 17*b^2*d^4)*x^2 - (64*b^4*d^4*x^4 + 256*b^4*c*d^3*x^3 + 64*b^4*c^4 - 120*b^2*c^2*d^2 + 51*d^4 + 24*(16*b^
4*c^2*d^2 - 5*b^2*d^4)*x^2 + 16*(16*b^4*c^3*d - 15*b^2*c*d^3)*x)*cos(b*x + a)^2 + 2*(40*b^4*c^3*d - 51*b^2*c*d
^3)*x - 2*(2*(8*b^3*d^4*x^3 + 24*b^3*c*d^3*x^2 + 8*b^3*c^3*d - 3*b*c*d^3 + 3*(8*b^3*c^2*d^2 - b*d^4)*x)*cos(b*
x + a)^3 - (40*b^3*d^4*x^3 + 120*b^3*c*d^3*x^2 + 40*b^3*c^3*d - 51*b*c*d^3 + 3*(40*b^3*c^2*d^2 - 17*b*d^4)*x)*
cos(b*x + a))*sin(b*x + a))/b^5

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giac [A]  time = 2.89, size = 361, normalized size = 1.39 \[ \frac {{\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 192 \, b^{4} c^{2} d^{2} x^{2} + 128 \, b^{4} c^{3} d x + 32 \, b^{4} c^{4} - 24 \, b^{2} d^{4} x^{2} - 48 \, b^{2} c d^{3} x - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{5}} - \frac {{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 24 \, b^{3} c^{2} d^{2} x + 8 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} + \frac {{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*(32*b^4*d^4*x^4 + 128*b^4*c*d^3*x^3 + 192*b^4*c^2*d^2*x^2 + 128*b^4*c^3*d*x + 32*b^4*c^4 - 24*b^2*d^4*x
^2 - 48*b^2*c*d^3*x - 24*b^2*c^2*d^2 + 3*d^4)*cos(4*b*x + 4*a)/b^5 - 1/16*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 1
2*b^4*c^2*d^2*x^2 + 8*b^4*c^3*d*x + 2*b^4*c^4 - 6*b^2*d^4*x^2 - 12*b^2*c*d^3*x - 6*b^2*c^2*d^2 + 3*d^4)*cos(2*
b*x + 2*a)/b^5 - 1/256*(8*b^3*d^4*x^3 + 24*b^3*c*d^3*x^2 + 24*b^3*c^2*d^2*x + 8*b^3*c^3*d - 3*b*d^4*x - 3*b*c*
d^3)*sin(4*b*x + 4*a)/b^5 + 1/8*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 6*b^3*c^2*d^2*x + 2*b^3*c^3*d - 3*b*d^4*x -
3*b*c*d^3)*sin(2*b*x + 2*a)/b^5

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maple [B]  time = 0.12, size = 1143, normalized size = 4.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/b^4*d^4*(1/4*(b*x+a)^4*sin(b*x+a)^4-(b*x+a)^3*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/
8*a)-3/16*(b*x+a)^2*sin(b*x+a)^4+3/8*(b*x+a)*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+27/
128*(b*x+a)^2+3/128*sin(b*x+a)^4+45/128*sin(b*x+a)^2+9/16*(b*x+a)^2*cos(b*x+a)^2-9/8*(b*x+a)*(1/2*cos(b*x+a)*s
in(b*x+a)+1/2*b*x+1/2*a)+9/32*(b*x+a)^4)-4/b^4*a*d^4*(1/4*(b*x+a)^3*sin(b*x+a)^4-3/4*(b*x+a)^2*(-1/4*(sin(b*x+
a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-3/32*(b*x+a)*sin(b*x+a)^4-3/128*(sin(b*x+a)^3+3/2*sin(b*x+a))*c
os(b*x+a)-27/256*b*x-27/256*a+9/32*(b*x+a)*cos(b*x+a)^2-9/64*cos(b*x+a)*sin(b*x+a)+3/16*(b*x+a)^3)+4/b^3*c*d^3
*(1/4*(b*x+a)^3*sin(b*x+a)^4-3/4*(b*x+a)^2*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-3/32*
(b*x+a)*sin(b*x+a)^4-3/128*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-27/256*b*x-27/256*a+9/32*(b*x+a)*cos(b*x+a
)^2-9/64*cos(b*x+a)*sin(b*x+a)+3/16*(b*x+a)^3)+6/b^4*a^2*d^4*(1/4*(b*x+a)^2*sin(b*x+a)^4-1/2*(b*x+a)*(-1/4*(si
n(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+3/32*(b*x+a)^2-1/32*sin(b*x+a)^4-3/32*sin(b*x+a)^2)-12/b^
3*a*c*d^3*(1/4*(b*x+a)^2*sin(b*x+a)^4-1/2*(b*x+a)*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a
)+3/32*(b*x+a)^2-1/32*sin(b*x+a)^4-3/32*sin(b*x+a)^2)+6/b^2*c^2*d^2*(1/4*(b*x+a)^2*sin(b*x+a)^4-1/2*(b*x+a)*(-
1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+3/32*(b*x+a)^2-1/32*sin(b*x+a)^4-3/32*sin(b*x+a)^2
)-4/b^4*a^3*d^4*(1/4*(b*x+a)*sin(b*x+a)^4+1/16*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-3/32*b*x-3/32*a)+12/b^
3*a^2*c*d^3*(1/4*(b*x+a)*sin(b*x+a)^4+1/16*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-3/32*b*x-3/32*a)-12/b^2*a*
c^2*d^2*(1/4*(b*x+a)*sin(b*x+a)^4+1/16*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-3/32*b*x-3/32*a)+4/b*c^3*d*(1/
4*(b*x+a)*sin(b*x+a)^4+1/16*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-3/32*b*x-3/32*a)+1/4/b^4*a^4*d^4*sin(b*x+
a)^4-1/b^3*a^3*c*d^3*sin(b*x+a)^4+3/2/b^2*a^2*c^2*d^2*sin(b*x+a)^4-1/b*a*c^3*d*sin(b*x+a)^4+1/4*c^4*sin(b*x+a)
^4)

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maxima [B]  time = 0.42, size = 967, normalized size = 3.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*(256*c^4*sin(b*x + a)^4 - 1024*a*c^3*d*sin(b*x + a)^4/b + 1536*a^2*c^2*d^2*sin(b*x + a)^4/b^2 - 1024*a^
3*c*d^3*sin(b*x + a)^4/b^3 + 256*a^4*d^4*sin(b*x + a)^4/b^4 + 32*(4*(b*x + a)*cos(4*b*x + 4*a) - 16*(b*x + a)*
cos(2*b*x + 2*a) - sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*c^3*d/b - 96*(4*(b*x + a)*cos(4*b*x + 4*a) - 16*(b*x
+ a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*a*c^2*d^2/b^2 + 96*(4*(b*x + a)*cos(4*b*x + 4*
a) - 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*a^2*c*d^3/b^3 - 32*(4*(b*x + a)*co
s(4*b*x + 4*a) - 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*a^3*d^4/b^4 + 24*((8*(
b*x + a)^2 - 1)*cos(4*b*x + 4*a) - 16*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) + 32
*(b*x + a)*sin(2*b*x + 2*a))*c^2*d^2/b^2 - 48*((8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) - 16*(2*(b*x + a)^2 - 1)*c
os(2*b*x + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) + 32*(b*x + a)*sin(2*b*x + 2*a))*a*c*d^3/b^3 + 24*((8*(b*x + a)
^2 - 1)*cos(4*b*x + 4*a) - 16*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) + 32*(b*x +
a)*sin(2*b*x + 2*a))*a^2*d^4/b^4 + 4*(4*(8*(b*x + a)^3 - 3*b*x - 3*a)*cos(4*b*x + 4*a) - 64*(2*(b*x + a)^3 - 3
*b*x - 3*a)*cos(2*b*x + 2*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a) + 96*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a
))*c*d^3/b^3 - 4*(4*(8*(b*x + a)^3 - 3*b*x - 3*a)*cos(4*b*x + 4*a) - 64*(2*(b*x + a)^3 - 3*b*x - 3*a)*cos(2*b*
x + 2*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a) + 96*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*a*d^4/b^4 + ((32*
(b*x + a)^4 - 24*(b*x + a)^2 + 3)*cos(4*b*x + 4*a) - 64*(2*(b*x + a)^4 - 6*(b*x + a)^2 + 3)*cos(2*b*x + 2*a) -
4*(8*(b*x + a)^3 - 3*b*x - 3*a)*sin(4*b*x + 4*a) + 128*(2*(b*x + a)^3 - 3*b*x - 3*a)*sin(2*b*x + 2*a))*d^4/b^
4)/b

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mupad [B]  time = 1.94, size = 576, normalized size = 2.22 \[ -\frac {192\,d^4\,\cos \left (2\,a+2\,b\,x\right )-3\,d^4\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )-32\,b^4\,c^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )+32\,b^3\,c^3\,d\,\sin \left (4\,a+4\,b\,x\right )-384\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^2\,c^2\,d^2\,\cos \left (4\,a+4\,b\,x\right )-384\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^2\,d^4\,x^2\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )-32\,b^4\,d^4\,x^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )+32\,b^3\,d^4\,x^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )-12\,b\,c\,d^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )-12\,b\,d^4\,x\,\sin \left (4\,a+4\,b\,x\right )+768\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-192\,b^4\,c^2\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )-768\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+512\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+48\,b^2\,c\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )-128\,b^4\,c^3\,d\,x\,\cos \left (4\,a+4\,b\,x\right )+512\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-128\,b^4\,c\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-768\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-768\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+96\,b^3\,c^2\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )+96\,b^3\,c\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{1024\,b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(192*d^4*cos(2*a + 2*b*x) - 3*d^4*cos(4*a + 4*b*x) + 128*b^4*c^4*cos(2*a + 2*b*x) - 32*b^4*c^4*cos(4*a + 4*b*
x) - 256*b^3*c^3*d*sin(2*a + 2*b*x) + 32*b^3*c^3*d*sin(4*a + 4*b*x) - 384*b^2*c^2*d^2*cos(2*a + 2*b*x) + 24*b^
2*c^2*d^2*cos(4*a + 4*b*x) - 384*b^2*d^4*x^2*cos(2*a + 2*b*x) + 24*b^2*d^4*x^2*cos(4*a + 4*b*x) + 128*b^4*d^4*
x^4*cos(2*a + 2*b*x) - 32*b^4*d^4*x^4*cos(4*a + 4*b*x) - 256*b^3*d^4*x^3*sin(2*a + 2*b*x) + 32*b^3*d^4*x^3*sin
(4*a + 4*b*x) + 384*b*c*d^3*sin(2*a + 2*b*x) - 12*b*c*d^3*sin(4*a + 4*b*x) + 384*b*d^4*x*sin(2*a + 2*b*x) - 12
*b*d^4*x*sin(4*a + 4*b*x) + 768*b^4*c^2*d^2*x^2*cos(2*a + 2*b*x) - 192*b^4*c^2*d^2*x^2*cos(4*a + 4*b*x) - 768*
b^2*c*d^3*x*cos(2*a + 2*b*x) + 512*b^4*c^3*d*x*cos(2*a + 2*b*x) + 48*b^2*c*d^3*x*cos(4*a + 4*b*x) - 128*b^4*c^
3*d*x*cos(4*a + 4*b*x) + 512*b^4*c*d^3*x^3*cos(2*a + 2*b*x) - 128*b^4*c*d^3*x^3*cos(4*a + 4*b*x) - 768*b^3*c^2
*d^2*x*sin(2*a + 2*b*x) - 768*b^3*c*d^3*x^2*sin(2*a + 2*b*x) + 96*b^3*c^2*d^2*x*sin(4*a + 4*b*x) + 96*b^3*c*d^
3*x^2*sin(4*a + 4*b*x))/(1024*b^5)

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sympy [A]  time = 13.31, size = 935, normalized size = 3.60 \[ \begin {cases} \frac {c^{4} \sin ^{4}{\left (a + b x \right )}}{4 b} + \frac {5 c^{3} d x \sin ^{4}{\left (a + b x \right )}}{8 b} - \frac {3 c^{3} d x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c^{3} d x \cos ^{4}{\left (a + b x \right )}}{8 b} + \frac {15 c^{2} d^{2} x^{2} \sin ^{4}{\left (a + b x \right )}}{16 b} - \frac {9 c^{2} d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8 b} - \frac {9 c^{2} d^{2} x^{2} \cos ^{4}{\left (a + b x \right )}}{16 b} + \frac {5 c d^{3} x^{3} \sin ^{4}{\left (a + b x \right )}}{8 b} - \frac {3 c d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4 b} - \frac {3 c d^{3} x^{3} \cos ^{4}{\left (a + b x \right )}}{8 b} + \frac {5 d^{4} x^{4} \sin ^{4}{\left (a + b x \right )}}{32 b} - \frac {3 d^{4} x^{4} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{16 b} - \frac {3 d^{4} x^{4} \cos ^{4}{\left (a + b x \right )}}{32 b} + \frac {5 c^{3} d \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {3 c^{3} d \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} + \frac {15 c^{2} d^{2} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {9 c^{2} d^{2} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} + \frac {15 c d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {9 c d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} + \frac {5 d^{4} x^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{8 b^{2}} + \frac {3 d^{4} x^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b^{2}} - \frac {15 c^{2} d^{2} \sin ^{4}{\left (a + b x \right )}}{32 b^{3}} + \frac {9 c^{2} d^{2} \cos ^{4}{\left (a + b x \right )}}{32 b^{3}} - \frac {51 c d^{3} x \sin ^{4}{\left (a + b x \right )}}{64 b^{3}} + \frac {9 c d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32 b^{3}} + \frac {45 c d^{3} x \cos ^{4}{\left (a + b x \right )}}{64 b^{3}} - \frac {51 d^{4} x^{2} \sin ^{4}{\left (a + b x \right )}}{128 b^{3}} + \frac {9 d^{4} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{64 b^{3}} + \frac {45 d^{4} x^{2} \cos ^{4}{\left (a + b x \right )}}{128 b^{3}} - \frac {51 c d^{3} \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{4}} - \frac {45 c d^{3} \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{4}} - \frac {51 d^{4} x \sin ^{3}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{64 b^{4}} - \frac {45 d^{4} x \sin {\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{64 b^{4}} + \frac {51 d^{4} \sin ^{4}{\left (a + b x \right )}}{256 b^{5}} - \frac {45 d^{4} \cos ^{4}{\left (a + b x \right )}}{256 b^{5}} & \text {for}\: b \neq 0 \\\left (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}\right ) \sin ^{3}{\relax (a )} \cos {\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**4*sin(a + b*x)**4/(4*b) + 5*c**3*d*x*sin(a + b*x)**4/(8*b) - 3*c**3*d*x*sin(a + b*x)**2*cos(a +
b*x)**2/(4*b) - 3*c**3*d*x*cos(a + b*x)**4/(8*b) + 15*c**2*d**2*x**2*sin(a + b*x)**4/(16*b) - 9*c**2*d**2*x**2
*sin(a + b*x)**2*cos(a + b*x)**2/(8*b) - 9*c**2*d**2*x**2*cos(a + b*x)**4/(16*b) + 5*c*d**3*x**3*sin(a + b*x)*
*4/(8*b) - 3*c*d**3*x**3*sin(a + b*x)**2*cos(a + b*x)**2/(4*b) - 3*c*d**3*x**3*cos(a + b*x)**4/(8*b) + 5*d**4*
x**4*sin(a + b*x)**4/(32*b) - 3*d**4*x**4*sin(a + b*x)**2*cos(a + b*x)**2/(16*b) - 3*d**4*x**4*cos(a + b*x)**4
/(32*b) + 5*c**3*d*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 3*c**3*d*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 15
*c**2*d**2*x*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 9*c**2*d**2*x*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 15*
c*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 9*c*d**3*x**2*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 5*d*
*4*x**3*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 3*d**4*x**3*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) - 15*c**2*d*
*2*sin(a + b*x)**4/(32*b**3) + 9*c**2*d**2*cos(a + b*x)**4/(32*b**3) - 51*c*d**3*x*sin(a + b*x)**4/(64*b**3) +
9*c*d**3*x*sin(a + b*x)**2*cos(a + b*x)**2/(32*b**3) + 45*c*d**3*x*cos(a + b*x)**4/(64*b**3) - 51*d**4*x**2*s
in(a + b*x)**4/(128*b**3) + 9*d**4*x**2*sin(a + b*x)**2*cos(a + b*x)**2/(64*b**3) + 45*d**4*x**2*cos(a + b*x)*
*4/(128*b**3) - 51*c*d**3*sin(a + b*x)**3*cos(a + b*x)/(64*b**4) - 45*c*d**3*sin(a + b*x)*cos(a + b*x)**3/(64*
b**4) - 51*d**4*x*sin(a + b*x)**3*cos(a + b*x)/(64*b**4) - 45*d**4*x*sin(a + b*x)*cos(a + b*x)**3/(64*b**4) +
51*d**4*sin(a + b*x)**4/(256*b**5) - 45*d**4*cos(a + b*x)**4/(256*b**5), Ne(b, 0)), ((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)*sin(a)**3*cos(a), True))

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