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

Optimal. Leaf size=225 \[ \frac {8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac {488 d^4 \cos (a+b x)}{27 b^5}-\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}-\frac {160 d^3 (c+d x) \sin (a+b x)}{9 b^4}+\frac {80 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}+\frac {4 d^2 (c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{9 b^3}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {8 d (c+d x)^3 \sin (a+b x)}{3 b^2}-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b} \]

[Out]

-488/27*d^4*cos(b*x+a)/b^5+80/9*d^2*(d*x+c)^2*cos(b*x+a)/b^3-2/3*(d*x+c)^4*cos(b*x+a)/b+8/81*d^4*cos(b*x+a)^3/
b^5-160/9*d^3*(d*x+c)*sin(b*x+a)/b^4+8/3*d*(d*x+c)^3*sin(b*x+a)/b^2+4/9*d^2*(d*x+c)^2*cos(b*x+a)*sin(b*x+a)^2/
b^3-1/3*(d*x+c)^4*cos(b*x+a)*sin(b*x+a)^2/b-8/27*d^3*(d*x+c)*sin(b*x+a)^3/b^4+4/9*d*(d*x+c)^3*sin(b*x+a)^3/b^2

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Rubi [A]  time = 0.25, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3311, 3296, 2638, 2633} \[ -\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}-\frac {160 d^3 (c+d x) \sin (a+b x)}{9 b^4}+\frac {80 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}+\frac {4 d^2 (c+d x)^2 \sin ^2(a+b x) \cos (a+b x)}{9 b^3}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {8 d (c+d x)^3 \sin (a+b x)}{3 b^2}+\frac {8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac {488 d^4 \cos (a+b x)}{27 b^5}-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}-\frac {(c+d x)^4 \sin ^2(a+b x) \cos (a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Sin[a + b*x]^3,x]

[Out]

(-488*d^4*Cos[a + b*x])/(27*b^5) + (80*d^2*(c + d*x)^2*Cos[a + b*x])/(9*b^3) - (2*(c + d*x)^4*Cos[a + b*x])/(3
*b) + (8*d^4*Cos[a + b*x]^3)/(81*b^5) - (160*d^3*(c + d*x)*Sin[a + b*x])/(9*b^4) + (8*d*(c + d*x)^3*Sin[a + b*
x])/(3*b^2) + (4*d^2*(c + d*x)^2*Cos[a + b*x]*Sin[a + b*x]^2)/(9*b^3) - ((c + d*x)^4*Cos[a + b*x]*Sin[a + b*x]
^2)/(3*b) - (8*d^3*(c + d*x)*Sin[a + b*x]^3)/(27*b^4) + (4*d*(c + d*x)^3*Sin[a + b*x]^3)/(9*b^2)

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2638

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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]

Rubi steps

\begin {align*} \int (c+d x)^4 \sin ^3(a+b x) \, dx &=-\frac {(c+d x)^4 \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {2}{3} \int (c+d x)^4 \sin (a+b x) \, dx-\frac {\left (4 d^2\right ) \int (c+d x)^2 \sin ^3(a+b x) \, dx}{3 b^2}\\ &=-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}+\frac {4 d^2 (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{9 b^3}-\frac {(c+d x)^4 \cos (a+b x) \sin ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {(8 d) \int (c+d x)^3 \cos (a+b x) \, dx}{3 b}-\frac {\left (8 d^2\right ) \int (c+d x)^2 \sin (a+b x) \, dx}{9 b^2}+\frac {\left (8 d^4\right ) \int \sin ^3(a+b x) \, dx}{27 b^4}\\ &=\frac {8 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}+\frac {8 d (c+d x)^3 \sin (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{9 b^3}-\frac {(c+d x)^4 \cos (a+b x) \sin ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {\left (8 d^2\right ) \int (c+d x)^2 \sin (a+b x) \, dx}{b^2}-\frac {\left (16 d^3\right ) \int (c+d x) \cos (a+b x) \, dx}{9 b^3}-\frac {\left (8 d^4\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{27 b^5}\\ &=-\frac {8 d^4 \cos (a+b x)}{27 b^5}+\frac {80 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}+\frac {8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac {16 d^3 (c+d x) \sin (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sin (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{9 b^3}-\frac {(c+d x)^4 \cos (a+b x) \sin ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}-\frac {\left (16 d^3\right ) \int (c+d x) \cos (a+b x) \, dx}{b^3}+\frac {\left (16 d^4\right ) \int \sin (a+b x) \, dx}{9 b^4}\\ &=-\frac {56 d^4 \cos (a+b x)}{27 b^5}+\frac {80 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}+\frac {8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac {160 d^3 (c+d x) \sin (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sin (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{9 b^3}-\frac {(c+d x)^4 \cos (a+b x) \sin ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}+\frac {\left (16 d^4\right ) \int \sin (a+b x) \, dx}{b^4}\\ &=-\frac {488 d^4 \cos (a+b x)}{27 b^5}+\frac {80 d^2 (c+d x)^2 \cos (a+b x)}{9 b^3}-\frac {2 (c+d x)^4 \cos (a+b x)}{3 b}+\frac {8 d^4 \cos ^3(a+b x)}{81 b^5}-\frac {160 d^3 (c+d x) \sin (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \sin (a+b x)}{3 b^2}+\frac {4 d^2 (c+d x)^2 \cos (a+b x) \sin ^2(a+b x)}{9 b^3}-\frac {(c+d x)^4 \cos (a+b x) \sin ^2(a+b x)}{3 b}-\frac {8 d^3 (c+d x) \sin ^3(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \sin ^3(a+b x)}{9 b^2}\\ \end {align*}

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Mathematica [A]  time = 1.06, size = 150, normalized size = 0.67 \[ \frac {-24 b d (c+d x) \sin (a+b x) \left (\cos (2 (a+b x)) \left (3 b^2 (c+d x)^2-2 d^2\right )-39 b^2 (c+d x)^2+242 d^2\right )-243 \cos (a+b x) \left (b^4 (c+d x)^4-12 b^2 d^2 (c+d x)^2+24 d^4\right )+\cos (3 (a+b x)) \left (27 b^4 (c+d x)^4-36 b^2 d^2 (c+d x)^2+8 d^4\right )}{324 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-243*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x] + (8*d^4 - 36*b^2*d^2*(c + d*x)^2 + 27*
b^4*(c + d*x)^4)*Cos[3*(a + b*x)] - 24*b*d*(c + d*x)*(242*d^2 - 39*b^2*(c + d*x)^2 + (-2*d^2 + 3*b^2*(c + d*x)
^2)*Cos[2*(a + b*x)])*Sin[a + b*x])/(324*b^5)

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fricas [A]  time = 0.85, size = 351, normalized size = 1.56 \[ \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 2 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 252 \, b^{2} c^{2} d^{2} + 488 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 14 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 14 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right ) + 12 \, {\left (21 \, b^{3} d^{4} x^{3} + 63 \, b^{3} c d^{3} x^{2} + 21 \, b^{3} c^{3} d - 122 \, b c d^{3} - {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + {\left (63 \, b^{3} c^{2} d^{2} - 122 \, b d^{4}\right )} x\right )} \sin \left (b x + a\right )}{81 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/81*((27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 - 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 - 2*b^2*d^
4)*x^2 + 36*(3*b^4*c^3*d - 2*b^2*c*d^3)*x)*cos(b*x + a)^3 - 3*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4
- 252*b^2*c^2*d^2 + 488*d^4 + 18*(9*b^4*c^2*d^2 - 14*b^2*d^4)*x^2 + 36*(3*b^4*c^3*d - 14*b^2*c*d^3)*x)*cos(b*
x + a) + 12*(21*b^3*d^4*x^3 + 63*b^3*c*d^3*x^2 + 21*b^3*c^3*d - 122*b*c*d^3 - (3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2
+ 3*b^3*c^3*d - 2*b*c*d^3 + (9*b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^2 + (63*b^3*c^2*d^2 - 122*b*d^4)*x)*sin
(b*x + a))/b^5

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giac [A]  time = 0.52, size = 351, normalized size = 1.56 \[ \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \cos \left (3 \, b x + 3 \, a\right )}{324 \, b^{5}} - \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \cos \left (b x + a\right )}{4 \, b^{5}} - \frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{27 \, b^{5}} + \frac {3 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \sin \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 108*b^4*c^3*d*x + 27*b^4*c^4 - 36*b^2*d^4*x^
2 - 72*b^2*c*d^3*x - 36*b^2*c^2*d^2 + 8*d^4)*cos(3*b*x + 3*a)/b^5 - 3/4*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4
*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4 - 12*b^2*d^4*x^2 - 24*b^2*c*d^3*x - 12*b^2*c^2*d^2 + 24*d^4)*cos(b*x +
a)/b^5 - 1/27*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 9*b^3*c^2*d^2*x + 3*b^3*c^3*d - 2*b*d^4*x - 2*b*c*d^3)*sin(3*
b*x + 3*a)/b^5 + 3*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d - 6*b*d^4*x - 6*b*c*d^3)*sin(b
*x + a)/b^5

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maple [B]  time = 0.06, size = 1023, normalized size = 4.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/b^4*d^4*(-1/3*(b*x+a)^4*(2+sin(b*x+a)^2)*cos(b*x+a)+8/3*(b*x+a)^3*sin(b*x+a)+8*(b*x+a)^2*cos(b*x+a)-160
/9*cos(b*x+a)-160/9*(b*x+a)*sin(b*x+a)+4/9*(b*x+a)^3*sin(b*x+a)^3+4/9*(b*x+a)^2*(2+sin(b*x+a)^2)*cos(b*x+a)-8/
27*(b*x+a)*sin(b*x+a)^3-8/81*(2+sin(b*x+a)^2)*cos(b*x+a))-4/b^4*a*d^4*(-1/3*(b*x+a)^3*(2+sin(b*x+a)^2)*cos(b*x
+a)+2*(b*x+a)^2*sin(b*x+a)-40/9*sin(b*x+a)+4*(b*x+a)*cos(b*x+a)+1/3*(b*x+a)^2*sin(b*x+a)^3+2/9*(b*x+a)*(2+sin(
b*x+a)^2)*cos(b*x+a)-2/27*sin(b*x+a)^3)+4/b^3*c*d^3*(-1/3*(b*x+a)^3*(2+sin(b*x+a)^2)*cos(b*x+a)+2*(b*x+a)^2*si
n(b*x+a)-40/9*sin(b*x+a)+4*(b*x+a)*cos(b*x+a)+1/3*(b*x+a)^2*sin(b*x+a)^3+2/9*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+
a)-2/27*sin(b*x+a)^3)+6/b^4*a^2*d^4*(-1/3*(b*x+a)^2*(2+sin(b*x+a)^2)*cos(b*x+a)+4/3*cos(b*x+a)+4/3*(b*x+a)*sin
(b*x+a)+2/9*(b*x+a)*sin(b*x+a)^3+2/27*(2+sin(b*x+a)^2)*cos(b*x+a))-12/b^3*a*c*d^3*(-1/3*(b*x+a)^2*(2+sin(b*x+a
)^2)*cos(b*x+a)+4/3*cos(b*x+a)+4/3*(b*x+a)*sin(b*x+a)+2/9*(b*x+a)*sin(b*x+a)^3+2/27*(2+sin(b*x+a)^2)*cos(b*x+a
))+6/b^2*c^2*d^2*(-1/3*(b*x+a)^2*(2+sin(b*x+a)^2)*cos(b*x+a)+4/3*cos(b*x+a)+4/3*(b*x+a)*sin(b*x+a)+2/9*(b*x+a)
*sin(b*x+a)^3+2/27*(2+sin(b*x+a)^2)*cos(b*x+a))-4/b^4*a^3*d^4*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+1/9*si
n(b*x+a)^3+2/3*sin(b*x+a))+12/b^3*a^2*c*d^3*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+1/9*sin(b*x+a)^3+2/3*sin
(b*x+a))-12/b^2*a*c^2*d^2*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+1/9*sin(b*x+a)^3+2/3*sin(b*x+a))+4/b*c^3*d
*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+1/9*sin(b*x+a)^3+2/3*sin(b*x+a))-1/3/b^4*a^4*d^4*(2+sin(b*x+a)^2)*c
os(b*x+a)+4/3/b^3*a^3*c*d^3*(2+sin(b*x+a)^2)*cos(b*x+a)-2/b^2*a^2*c^2*d^2*(2+sin(b*x+a)^2)*cos(b*x+a)+4/3/b*a*
c^3*d*(2+sin(b*x+a)^2)*cos(b*x+a)-1/3*c^4*(2+sin(b*x+a)^2)*cos(b*x+a))

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maxima [B]  time = 0.41, size = 934, normalized size = 4.15 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/324*(108*(cos(b*x + a)^3 - 3*cos(b*x + a))*c^4 - 432*(cos(b*x + a)^3 - 3*cos(b*x + a))*a*c^3*d/b + 648*(cos(
b*x + a)^3 - 3*cos(b*x + a))*a^2*c^2*d^2/b^2 - 432*(cos(b*x + a)^3 - 3*cos(b*x + a))*a^3*c*d^3/b^3 + 108*(cos(
b*x + a)^3 - 3*cos(b*x + a))*a^4*d^4/b^4 + 36*(3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a) - sin(
3*b*x + 3*a) + 27*sin(b*x + a))*c^3*d/b - 108*(3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a) - sin(
3*b*x + 3*a) + 27*sin(b*x + a))*a*c^2*d^2/b^2 + 108*(3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x + a)
- sin(3*b*x + 3*a) + 27*sin(b*x + a))*a^2*c*d^3/b^3 - 36*(3*(b*x + a)*cos(3*b*x + 3*a) - 27*(b*x + a)*cos(b*x
+ a) - sin(3*b*x + 3*a) + 27*sin(b*x + a))*a^3*d^4/b^4 + 18*((9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 81*((b*x +
a)^2 - 2)*cos(b*x + a) - 6*(b*x + a)*sin(3*b*x + 3*a) + 162*(b*x + a)*sin(b*x + a))*c^2*d^2/b^2 - 36*((9*(b*x
+ a)^2 - 2)*cos(3*b*x + 3*a) - 81*((b*x + a)^2 - 2)*cos(b*x + a) - 6*(b*x + a)*sin(3*b*x + 3*a) + 162*(b*x +
a)*sin(b*x + a))*a*c*d^3/b^3 + 18*((9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 81*((b*x + a)^2 - 2)*cos(b*x + a) -
6*(b*x + a)*sin(3*b*x + 3*a) + 162*(b*x + a)*sin(b*x + a))*a^2*d^4/b^4 + 12*(3*(3*(b*x + a)^3 - 2*b*x - 2*a)*c
os(3*b*x + 3*a) - 81*((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) - (9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) + 243*((b
*x + a)^2 - 2)*sin(b*x + a))*c*d^3/b^3 - 12*(3*(3*(b*x + a)^3 - 2*b*x - 2*a)*cos(3*b*x + 3*a) - 81*((b*x + a)^
3 - 6*b*x - 6*a)*cos(b*x + a) - (9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) + 243*((b*x + a)^2 - 2)*sin(b*x + a))*a*d
^4/b^4 + ((27*(b*x + a)^4 - 36*(b*x + a)^2 + 8)*cos(3*b*x + 3*a) - 243*((b*x + a)^4 - 12*(b*x + a)^2 + 24)*cos
(b*x + a) - 12*(3*(b*x + a)^3 - 2*b*x - 2*a)*sin(3*b*x + 3*a) + 972*((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*
d^4/b^4)/b

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mupad [B]  time = 1.55, size = 533, normalized size = 2.37 \[ \frac {8\,x\,{\cos \left (a+b\,x\right )}^3\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^3}-\frac {2\,{\cos \left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4-360\,b^2\,c^2\,d^2+728\,d^4\right )}{81\,b^5}-\frac {\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (27\,b^4\,c^4-252\,b^2\,c^2\,d^2+488\,d^4\right )}{27\,b^5}-\frac {8\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^4}-\frac {2\,d^4\,x^4\,{\cos \left (a+b\,x\right )}^3}{3\,b}-\frac {4\,{\sin \left (a+b\,x\right )}^3\,\left (122\,c\,d^3-21\,b^2\,c^3\,d\right )}{27\,b^4}+\frac {28\,d^4\,x^3\,{\sin \left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,x\,{\sin \left (a+b\,x\right )}^3\,\left (122\,d^4-63\,b^2\,c^2\,d^2\right )}{27\,b^4}+\frac {4\,x^2\,{\cos \left (a+b\,x\right )}^3\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^3}+\frac {2\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{3\,b^3}-\frac {8\,c\,d^3\,x^3\,{\cos \left (a+b\,x\right )}^3}{3\,b}-\frac {d^4\,x^4\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b}+\frac {8\,d^4\,x^3\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{3\,b^2}+\frac {28\,c\,d^3\,x^2\,{\sin \left (a+b\,x\right )}^3}{3\,b^2}-\frac {8\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^4}+\frac {4\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{3\,b^3}-\frac {4\,c\,d^3\,x^3\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b}+\frac {8\,c\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^3*(c + d*x)^4,x)

[Out]

(8*x*cos(a + b*x)^3*(20*c*d^3 - 3*b^2*c^3*d))/(9*b^3) - (2*cos(a + b*x)^3*(728*d^4 + 27*b^4*c^4 - 360*b^2*c^2*
d^2))/(81*b^5) - (cos(a + b*x)*sin(a + b*x)^2*(488*d^4 + 27*b^4*c^4 - 252*b^2*c^2*d^2))/(27*b^5) - (8*cos(a +
b*x)^2*sin(a + b*x)*(20*c*d^3 - 3*b^2*c^3*d))/(9*b^4) - (2*d^4*x^4*cos(a + b*x)^3)/(3*b) - (4*sin(a + b*x)^3*(
122*c*d^3 - 21*b^2*c^3*d))/(27*b^4) + (28*d^4*x^3*sin(a + b*x)^3)/(9*b^2) - (4*x*sin(a + b*x)^3*(122*d^4 - 63*
b^2*c^2*d^2))/(27*b^4) + (4*x^2*cos(a + b*x)^3*(20*d^4 - 9*b^2*c^2*d^2))/(9*b^3) + (2*x^2*cos(a + b*x)*sin(a +
b*x)^2*(14*d^4 - 9*b^2*c^2*d^2))/(3*b^3) - (8*c*d^3*x^3*cos(a + b*x)^3)/(3*b) - (d^4*x^4*cos(a + b*x)*sin(a +
b*x)^2)/b + (8*d^4*x^3*cos(a + b*x)^2*sin(a + b*x))/(3*b^2) + (28*c*d^3*x^2*sin(a + b*x)^3)/(3*b^2) - (8*x*co
s(a + b*x)^2*sin(a + b*x)*(20*d^4 - 9*b^2*c^2*d^2))/(9*b^4) + (4*x*cos(a + b*x)*sin(a + b*x)^2*(14*c*d^3 - 3*b
^2*c^3*d))/(3*b^3) - (4*c*d^3*x^3*cos(a + b*x)*sin(a + b*x)^2)/b + (8*c*d^3*x^2*cos(a + b*x)^2*sin(a + b*x))/b
^2

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sympy [A]  time = 10.99, size = 772, normalized size = 3.43 \[ \begin {cases} - \frac {c^{4} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 c^{4} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {4 c^{3} d x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {8 c^{3} d x \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {6 c^{2} d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {4 c^{2} d^{2} x^{2} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {4 c d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {8 c d^{3} x^{3} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {d^{4} x^{4} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b} - \frac {2 d^{4} x^{4} \cos ^{3}{\left (a + b x \right )}}{3 b} + \frac {28 c^{3} d \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {8 c^{3} d \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {28 c^{2} d^{2} x \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 c^{2} d^{2} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {28 c d^{3} x^{2} \sin ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {8 c d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {28 d^{4} x^{3} \sin ^{3}{\left (a + b x \right )}}{9 b^{2}} + \frac {8 d^{4} x^{3} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b^{2}} + \frac {28 c^{2} d^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {80 c^{2} d^{2} \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {56 c d^{3} x \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {160 c d^{3} x \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {28 d^{4} x^{2} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{3 b^{3}} + \frac {80 d^{4} x^{2} \cos ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {488 c d^{3} \sin ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {160 c d^{3} \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{4}} - \frac {488 d^{4} x \sin ^{3}{\left (a + b x \right )}}{27 b^{4}} - \frac {160 d^{4} x \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{9 b^{4}} - \frac {488 d^{4} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{27 b^{5}} - \frac {1456 d^{4} \cos ^{3}{\left (a + b x \right )}}{81 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 )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-c**4*sin(a + b*x)**2*cos(a + b*x)/b - 2*c**4*cos(a + b*x)**3/(3*b) - 4*c**3*d*x*sin(a + b*x)**2*co
s(a + b*x)/b - 8*c**3*d*x*cos(a + b*x)**3/(3*b) - 6*c**2*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)/b - 4*c**2*d**
2*x**2*cos(a + b*x)**3/b - 4*c*d**3*x**3*sin(a + b*x)**2*cos(a + b*x)/b - 8*c*d**3*x**3*cos(a + b*x)**3/(3*b)
- d**4*x**4*sin(a + b*x)**2*cos(a + b*x)/b - 2*d**4*x**4*cos(a + b*x)**3/(3*b) + 28*c**3*d*sin(a + b*x)**3/(9*
b**2) + 8*c**3*d*sin(a + b*x)*cos(a + b*x)**2/(3*b**2) + 28*c**2*d**2*x*sin(a + b*x)**3/(3*b**2) + 8*c**2*d**2
*x*sin(a + b*x)*cos(a + b*x)**2/b**2 + 28*c*d**3*x**2*sin(a + b*x)**3/(3*b**2) + 8*c*d**3*x**2*sin(a + b*x)*co
s(a + b*x)**2/b**2 + 28*d**4*x**3*sin(a + b*x)**3/(9*b**2) + 8*d**4*x**3*sin(a + b*x)*cos(a + b*x)**2/(3*b**2)
+ 28*c**2*d**2*sin(a + b*x)**2*cos(a + b*x)/(3*b**3) + 80*c**2*d**2*cos(a + b*x)**3/(9*b**3) + 56*c*d**3*x*si
n(a + b*x)**2*cos(a + b*x)/(3*b**3) + 160*c*d**3*x*cos(a + b*x)**3/(9*b**3) + 28*d**4*x**2*sin(a + b*x)**2*cos
(a + b*x)/(3*b**3) + 80*d**4*x**2*cos(a + b*x)**3/(9*b**3) - 488*c*d**3*sin(a + b*x)**3/(27*b**4) - 160*c*d**3
*sin(a + b*x)*cos(a + b*x)**2/(9*b**4) - 488*d**4*x*sin(a + b*x)**3/(27*b**4) - 160*d**4*x*sin(a + b*x)*cos(a
+ b*x)**2/(9*b**4) - 488*d**4*sin(a + b*x)**2*cos(a + b*x)/(27*b**5) - 1456*d**4*cos(a + b*x)**3/(81*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, True))

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