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

Optimal. Leaf size=233 \[ -\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b} \]

[Out]

-9/128*d^4*cos(2*b*x+2*a)/b^5+9/64*d^2*(d*x+c)^2*cos(2*b*x+2*a)/b^3-3/64*(d*x+c)^4*cos(2*b*x+2*a)/b+1/10368*d^
4*cos(6*b*x+6*a)/b^5-1/576*d^2*(d*x+c)^2*cos(6*b*x+6*a)/b^3+1/192*(d*x+c)^4*cos(6*b*x+6*a)/b-9/64*d^3*(d*x+c)*
sin(2*b*x+2*a)/b^4+3/32*d*(d*x+c)^3*sin(2*b*x+2*a)/b^2+1/1728*d^3*(d*x+c)*sin(6*b*x+6*a)/b^4-1/288*d*(d*x+c)^3
*sin(6*b*x+6*a)/b^2

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Rubi [A]  time = 0.27, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4406, 3296, 2638} \[ -\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}-\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*d^4*Cos[2*a + 2*b*x])/(128*b^5) + (9*d^2*(c + d*x)^2*Cos[2*a + 2*b*x])/(64*b^3) - (3*(c + d*x)^4*Cos[2*a +
2*b*x])/(64*b) + (d^4*Cos[6*a + 6*b*x])/(10368*b^5) - (d^2*(c + d*x)^2*Cos[6*a + 6*b*x])/(576*b^3) + ((c + d*
x)^4*Cos[6*a + 6*b*x])/(192*b) - (9*d^3*(c + d*x)*Sin[2*a + 2*b*x])/(64*b^4) + (3*d*(c + d*x)^3*Sin[2*a + 2*b*
x])/(32*b^2) + (d^3*(c + d*x)*Sin[6*a + 6*b*x])/(1728*b^4) - (d*(c + d*x)^3*Sin[6*a + 6*b*x])/(288*b^2)

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 4406

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

Rubi steps

\begin {align*} \int (c+d x)^4 \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac {3}{32} (c+d x)^4 \sin (2 a+2 b x)-\frac {1}{32} (c+d x)^4 \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac {1}{32} \int (c+d x)^4 \sin (6 a+6 b x) \, dx\right )+\frac {3}{32} \int (c+d x)^4 \sin (2 a+2 b x) \, dx\\ &=-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {d \int (c+d x)^3 \cos (6 a+6 b x) \, dx}{48 b}+\frac {(3 d) \int (c+d x)^3 \cos (2 a+2 b x) \, dx}{16 b}\\ &=-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}+\frac {d^2 \int (c+d x)^2 \sin (6 a+6 b x) \, dx}{96 b^2}-\frac {\left (9 d^2\right ) \int (c+d x)^2 \sin (2 a+2 b x) \, dx}{32 b^2}\\ &=\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}+\frac {d^3 \int (c+d x) \cos (6 a+6 b x) \, dx}{288 b^3}-\frac {\left (9 d^3\right ) \int (c+d x) \cos (2 a+2 b x) \, dx}{32 b^3}\\ &=\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}-\frac {d^4 \int \sin (6 a+6 b x) \, dx}{1728 b^4}+\frac {\left (9 d^4\right ) \int \sin (2 a+2 b x) \, dx}{64 b^4}\\ &=-\frac {9 d^4 \cos (2 a+2 b x)}{128 b^5}+\frac {9 d^2 (c+d x)^2 \cos (2 a+2 b x)}{64 b^3}-\frac {3 (c+d x)^4 \cos (2 a+2 b x)}{64 b}+\frac {d^4 \cos (6 a+6 b x)}{10368 b^5}-\frac {d^2 (c+d x)^2 \cos (6 a+6 b x)}{576 b^3}+\frac {(c+d x)^4 \cos (6 a+6 b x)}{192 b}-\frac {9 d^3 (c+d x) \sin (2 a+2 b x)}{64 b^4}+\frac {3 d (c+d x)^3 \sin (2 a+2 b x)}{32 b^2}+\frac {d^3 (c+d x) \sin (6 a+6 b x)}{1728 b^4}-\frac {d (c+d x)^3 \sin (6 a+6 b x)}{288 b^2}\\ \end {align*}

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Mathematica [A]  time = 1.54, size = 153, normalized size = 0.66 \[ \frac {-12 b d (c+d x) \sin (2 (a+b x)) \left (\cos (4 (a+b x)) \left (6 b^2 (c+d x)^2-d^2\right )-78 b^2 (c+d x)^2+121 d^2\right )-243 \cos (2 (a+b x)) \left (2 b^4 (c+d x)^4-6 b^2 d^2 (c+d x)^2+3 d^4\right )+\cos (6 (a+b x)) \left (54 b^4 (c+d x)^4-18 b^2 d^2 (c+d x)^2+d^4\right )}{10368 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-243*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + (d^4 - 18*b^2*d^2*(c + d*x)^2 + 5
4*b^4*(c + d*x)^4)*Cos[6*(a + b*x)] - 12*b*d*(c + d*x)*(121*d^2 - 78*b^2*(c + d*x)^2 + (-d^2 + 6*b^2*(c + d*x)
^2)*Cos[4*(a + b*x)])*Sin[2*(a + b*x)])/(10368*b^5)

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fricas [B]  time = 0.50, size = 546, normalized size = 2.34 \[ \frac {27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 2 \, {\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 54 \, b^{4} c^{4} - 18 \, b^{2} c^{2} d^{2} + d^{4} + 18 \, {\left (18 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 36 \, {\left (6 \, b^{4} c^{3} d - b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{6} - 3 \, {\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 54 \, b^{4} c^{4} - 18 \, b^{2} c^{2} d^{2} + d^{4} + 18 \, {\left (18 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 36 \, {\left (6 \, b^{4} c^{3} d - b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 18 \, {\left (9 \, b^{2} d^{4} x^{2} + 18 \, b^{2} c d^{3} x + 9 \, b^{2} c^{2} d^{2} - 5 \, d^{4}\right )} \cos \left (b x + a\right )^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 5 \, b^{2} c d^{3}\right )} x - 12 \, {\left ({\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{3} d - b c d^{3} + {\left (18 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{5} - {\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{3} d - b c d^{3} + {\left (18 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 3 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 5 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 5 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{648 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/648*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 2*(54*b^4*d^4*x^4 + 216*b^4*c*d^3*x^3 + 54*b^4*c^4 - 18*b^2*c^2*d^
2 + d^4 + 18*(18*b^4*c^2*d^2 - b^2*d^4)*x^2 + 36*(6*b^4*c^3*d - b^2*c*d^3)*x)*cos(b*x + a)^6 - 3*(54*b^4*d^4*x
^4 + 216*b^4*c*d^3*x^3 + 54*b^4*c^4 - 18*b^2*c^2*d^2 + d^4 + 18*(18*b^4*c^2*d^2 - b^2*d^4)*x^2 + 36*(6*b^4*c^3
*d - b^2*c*d^3)*x)*cos(b*x + a)^4 + 18*(9*b^4*c^2*d^2 - 5*b^2*d^4)*x^2 + 18*(9*b^2*d^4*x^2 + 18*b^2*c*d^3*x +
9*b^2*c^2*d^2 - 5*d^4)*cos(b*x + a)^2 + 36*(3*b^4*c^3*d - 5*b^2*c*d^3)*x - 12*((6*b^3*d^4*x^3 + 18*b^3*c*d^3*x
^2 + 6*b^3*c^3*d - b*c*d^3 + (18*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)^5 - (6*b^3*d^4*x^3 + 18*b^3*c*d^3*x^2 +
6*b^3*c^3*d - b*c*d^3 + (18*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)^3 - 3*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^
3*c^3*d - 5*b*c*d^3 + (9*b^3*c^2*d^2 - 5*b*d^4)*x)*cos(b*x + a))*sin(b*x + a))/b^5

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giac [A]  time = 1.13, size = 359, normalized size = 1.54 \[ \frac {{\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 324 \, b^{4} c^{2} d^{2} x^{2} + 216 \, b^{4} c^{3} d x + 54 \, b^{4} c^{4} - 18 \, b^{2} d^{4} x^{2} - 36 \, b^{2} c d^{3} x - 18 \, b^{2} c^{2} d^{2} + d^{4}\right )} \cos \left (6 \, b x + 6 \, a\right )}{10368 \, b^{5}} - \frac {3 \, {\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{5}} - \frac {{\left (6 \, b^{3} d^{4} x^{3} + 18 \, b^{3} c d^{3} x^{2} + 18 \, b^{3} c^{2} d^{2} x + 6 \, b^{3} c^{3} d - b d^{4} x - b c d^{3}\right )} \sin \left (6 \, b x + 6 \, a\right )}{1728 \, b^{5}} + \frac {3 \, {\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{64 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10368*(54*b^4*d^4*x^4 + 216*b^4*c*d^3*x^3 + 324*b^4*c^2*d^2*x^2 + 216*b^4*c^3*d*x + 54*b^4*c^4 - 18*b^2*d^4*
x^2 - 36*b^2*c*d^3*x - 18*b^2*c^2*d^2 + d^4)*cos(6*b*x + 6*a)/b^5 - 3/128*(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 - 1/1728*(6*b^3*d^4*x^3 + 18*b^3*c*d^3*x^2 + 18*b^3*c^2*d^2*x + 6*b^3*c^3*d - b*d^4*x - b*c*d^3
)*sin(6*b*x + 6*a)/b^5 + 3/64*(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 = 0.13, size = 2061, normalized size = 8.85 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*cos(b*x+a)^3*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)-1/12*(b*x+a)^2*sin(b*x+a)^4+1/6*(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)+1/9
*(b*x+a)^2+1/216*sin(b*x+a)^4+5/36*sin(b*x+a)^2+1/4*(b*x+a)^2*cos(b*x+a)^2-1/2*(b*x+a)*(1/2*cos(b*x+a)*sin(b*x
+a)+1/2*b*x+1/2*a)+1/8*(b*x+a)^4-1/6*(b*x+a)^4*sin(b*x+a)^6+2/3*(b*x+a)^3*(-1/6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3
+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/16*a)+1/18*(b*x+a)^2*sin(b*x+a)^6-1/9*(b*x+a)*(-1/6*(sin(b*x+a)^5+5/4*
sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/16*a)-1/324*sin(b*x+a)^6)-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)-1/24*(b*x+a)*sin(b*x+a)^4-
1/96*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-1/18*b*x-1/18*a+1/8*(b*x+a)*cos(b*x+a)^2-1/16*cos(b*x+a)*sin(b*x
+a)+1/12*(b*x+a)^3-1/6*(b*x+a)^3*sin(b*x+a)^6+1/2*(b*x+a)^2*(-1/6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+
a))*cos(b*x+a)+5/16*b*x+5/16*a)+1/36*(b*x+a)*sin(b*x+a)^6+1/216*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a)
)*cos(b*x+a))+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)-1/24*(b*x+a)*sin(b*x+a)^4-1/96*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)-1/18*b*x-1/18*a+1/
8*(b*x+a)*cos(b*x+a)^2-1/16*cos(b*x+a)*sin(b*x+a)+1/12*(b*x+a)^3-1/6*(b*x+a)^3*sin(b*x+a)^6+1/2*(b*x+a)^2*(-1/
6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/16*a)+1/36*(b*x+a)*sin(b*x+a)^6+1/216*
(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a))+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*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+1/24*(b*x+a)^2-1/72*sin(b*x+a)^4-1/24*sin(b*
x+a)^2-1/6*(b*x+a)^2*sin(b*x+a)^6+1/3*(b*x+a)*(-1/6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)
+5/16*b*x+5/16*a)+1/108*sin(b*x+a)^6)-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)+1/24*(b*x+a)^2-1/72*sin(b*x+a)^4-1/24*sin(b*x+a)^2-1/6*(b*x+a)^2*
sin(b*x+a)^6+1/3*(b*x+a)*(-1/6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/16*a)+1/1
08*sin(b*x+a)^6)+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)+1/24*(b*x+a)^2-1/72*sin(b*x+a)^4-1/24*sin(b*x+a)^2-1/6*(b*x+a)^2*sin(b*x+a)^6+1/3*(b*x+
a)*(-1/6*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a)+5/16*b*x+5/16*a)+1/108*sin(b*x+a)^6)-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)-1/24*b*x-1/24*a-1/6*(b*x+a)*s
in(b*x+a)^6-1/36*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+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)-1/24*b*x-1/24*a-1/6*(b*x+a)*sin(b*x+a)^6-1/36*(sin(b*x
+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+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)-1/24*b*x-1/24*a-1/6*(b*x+a)*sin(b*x+a)^6-1/36*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15
/8*sin(b*x+a))*cos(b*x+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)-1
/24*b*x-1/24*a-1/6*(b*x+a)*sin(b*x+a)^6-1/36*(sin(b*x+a)^5+5/4*sin(b*x+a)^3+15/8*sin(b*x+a))*cos(b*x+a))+1/b^4
*a^4*d^4*(-1/6*sin(b*x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4)-4/b^3*a^3*c*d^3*(-1/6*sin(b*x+a)^2*cos(b*x+a)^4-1/
12*cos(b*x+a)^4)+6/b^2*a^2*c^2*d^2*(-1/6*sin(b*x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4)-4/b*a*c^3*d*(-1/6*sin(b*
x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4)+c^4*(-1/6*sin(b*x+a)^2*cos(b*x+a)^4-1/12*cos(b*x+a)^4))

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maxima [B]  time = 0.39, size = 1033, normalized size = 4.43 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10368*(864*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*c^4 - 3456*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a*c^3*d/b
+ 5184*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^2*c^2*d^2/b^2 - 3456*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^3
*c*d^3/b^3 + 864*(2*sin(b*x + a)^6 - 3*sin(b*x + a)^4)*a^4*d^4/b^4 - 36*(6*(b*x + a)*cos(6*b*x + 6*a) - 54*(b*
x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b*x + 2*a))*c^3*d/b + 108*(6*(b*x + a)*cos(6*b*x + 6*a)
- 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b*x + 2*a))*a*c^2*d^2/b^2 - 108*(6*(b*x + a)*cos
(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b*x + 2*a))*a^2*c*d^3/b^3 + 36*(6*
(b*x + a)*cos(6*b*x + 6*a) - 54*(b*x + a)*cos(2*b*x + 2*a) - sin(6*b*x + 6*a) + 27*sin(2*b*x + 2*a))*a^3*d^4/b
^4 - 18*((18*(b*x + a)^2 - 1)*cos(6*b*x + 6*a) - 81*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 6*(b*x + a)*sin(6*b
*x + 6*a) + 162*(b*x + a)*sin(2*b*x + 2*a))*c^2*d^2/b^2 + 36*((18*(b*x + a)^2 - 1)*cos(6*b*x + 6*a) - 81*(2*(b
*x + a)^2 - 1)*cos(2*b*x + 2*a) - 6*(b*x + a)*sin(6*b*x + 6*a) + 162*(b*x + a)*sin(2*b*x + 2*a))*a*c*d^3/b^3 -
18*((18*(b*x + a)^2 - 1)*cos(6*b*x + 6*a) - 81*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 6*(b*x + a)*sin(6*b*x +
6*a) + 162*(b*x + a)*sin(2*b*x + 2*a))*a^2*d^4/b^4 - 6*(6*(6*(b*x + a)^3 - b*x - a)*cos(6*b*x + 6*a) - 162*(2
*(b*x + a)^3 - 3*b*x - 3*a)*cos(2*b*x + 2*a) - (18*(b*x + a)^2 - 1)*sin(6*b*x + 6*a) + 243*(2*(b*x + a)^2 - 1)
*sin(2*b*x + 2*a))*c*d^3/b^3 + 6*(6*(6*(b*x + a)^3 - b*x - a)*cos(6*b*x + 6*a) - 162*(2*(b*x + a)^3 - 3*b*x -
3*a)*cos(2*b*x + 2*a) - (18*(b*x + a)^2 - 1)*sin(6*b*x + 6*a) + 243*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*a*d^
4/b^4 - ((54*(b*x + a)^4 - 18*(b*x + a)^2 + 1)*cos(6*b*x + 6*a) - 243*(2*(b*x + a)^4 - 6*(b*x + a)^2 + 3)*cos(
2*b*x + 2*a) - 6*(6*(b*x + a)^3 - b*x - a)*sin(6*b*x + 6*a) + 486*(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 = 2.58, size = 576, normalized size = 2.47 \[ -\frac {729\,d^4\,\cos \left (2\,a+2\,b\,x\right )-d^4\,\cos \left (6\,a+6\,b\,x\right )+486\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )-54\,b^4\,c^4\,\cos \left (6\,a+6\,b\,x\right )-972\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )+36\,b^3\,c^3\,d\,\sin \left (6\,a+6\,b\,x\right )-1458\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,c^2\,d^2\,\cos \left (6\,a+6\,b\,x\right )-1458\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )+486\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )+18\,b^2\,d^4\,x^2\,\cos \left (6\,a+6\,b\,x\right )-54\,b^4\,d^4\,x^4\,\cos \left (6\,a+6\,b\,x\right )-972\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )+36\,b^3\,d^4\,x^3\,\sin \left (6\,a+6\,b\,x\right )+1458\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,c\,d^3\,\sin \left (6\,a+6\,b\,x\right )+1458\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )-6\,b\,d^4\,x\,\sin \left (6\,a+6\,b\,x\right )+2916\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-324\,b^4\,c^2\,d^2\,x^2\,\cos \left (6\,a+6\,b\,x\right )-2916\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+1944\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+36\,b^2\,c\,d^3\,x\,\cos \left (6\,a+6\,b\,x\right )-216\,b^4\,c^3\,d\,x\,\cos \left (6\,a+6\,b\,x\right )+1944\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-216\,b^4\,c\,d^3\,x^3\,\cos \left (6\,a+6\,b\,x\right )-2916\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-2916\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+108\,b^3\,c^2\,d^2\,x\,\sin \left (6\,a+6\,b\,x\right )+108\,b^3\,c\,d^3\,x^2\,\sin \left (6\,a+6\,b\,x\right )}{10368\,b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(729*d^4*cos(2*a + 2*b*x) - d^4*cos(6*a + 6*b*x) + 486*b^4*c^4*cos(2*a + 2*b*x) - 54*b^4*c^4*cos(6*a + 6*b*x)
- 972*b^3*c^3*d*sin(2*a + 2*b*x) + 36*b^3*c^3*d*sin(6*a + 6*b*x) - 1458*b^2*c^2*d^2*cos(2*a + 2*b*x) + 18*b^2
*c^2*d^2*cos(6*a + 6*b*x) - 1458*b^2*d^4*x^2*cos(2*a + 2*b*x) + 486*b^4*d^4*x^4*cos(2*a + 2*b*x) + 18*b^2*d^4*
x^2*cos(6*a + 6*b*x) - 54*b^4*d^4*x^4*cos(6*a + 6*b*x) - 972*b^3*d^4*x^3*sin(2*a + 2*b*x) + 36*b^3*d^4*x^3*sin
(6*a + 6*b*x) + 1458*b*c*d^3*sin(2*a + 2*b*x) - 6*b*c*d^3*sin(6*a + 6*b*x) + 1458*b*d^4*x*sin(2*a + 2*b*x) - 6
*b*d^4*x*sin(6*a + 6*b*x) + 2916*b^4*c^2*d^2*x^2*cos(2*a + 2*b*x) - 324*b^4*c^2*d^2*x^2*cos(6*a + 6*b*x) - 291
6*b^2*c*d^3*x*cos(2*a + 2*b*x) + 1944*b^4*c^3*d*x*cos(2*a + 2*b*x) + 36*b^2*c*d^3*x*cos(6*a + 6*b*x) - 216*b^4
*c^3*d*x*cos(6*a + 6*b*x) + 1944*b^4*c*d^3*x^3*cos(2*a + 2*b*x) - 216*b^4*c*d^3*x^3*cos(6*a + 6*b*x) - 2916*b^
3*c^2*d^2*x*sin(2*a + 2*b*x) - 2916*b^3*c*d^3*x^2*sin(2*a + 2*b*x) + 108*b^3*c^2*d^2*x*sin(6*a + 6*b*x) + 108*
b^3*c*d^3*x^2*sin(6*a + 6*b*x))/(10368*b^5)

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sympy [A]  time = 31.30, size = 1334, normalized size = 5.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**4*sin(a + b*x)**6/(12*b) + c**4*sin(a + b*x)**4*cos(a + b*x)**2/(4*b) + c**3*d*x*sin(a + b*x)**6
/(6*b) + c**3*d*x*sin(a + b*x)**4*cos(a + b*x)**2/(2*b) - c**3*d*x*sin(a + b*x)**2*cos(a + b*x)**4/(2*b) - c**
3*d*x*cos(a + b*x)**6/(6*b) + c**2*d**2*x**2*sin(a + b*x)**6/(4*b) + 3*c**2*d**2*x**2*sin(a + b*x)**4*cos(a +
b*x)**2/(4*b) - 3*c**2*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)**4/(4*b) - c**2*d**2*x**2*cos(a + b*x)**6/(4*b)
+ c*d**3*x**3*sin(a + b*x)**6/(6*b) + c*d**3*x**3*sin(a + b*x)**4*cos(a + b*x)**2/(2*b) - c*d**3*x**3*sin(a +
b*x)**2*cos(a + b*x)**4/(2*b) - c*d**3*x**3*cos(a + b*x)**6/(6*b) + d**4*x**4*sin(a + b*x)**6/(24*b) + d**4*x*
*4*sin(a + b*x)**4*cos(a + b*x)**2/(8*b) - d**4*x**4*sin(a + b*x)**2*cos(a + b*x)**4/(8*b) - d**4*x**4*cos(a +
b*x)**6/(24*b) + c**3*d*sin(a + b*x)**5*cos(a + b*x)/(6*b**2) + 4*c**3*d*sin(a + b*x)**3*cos(a + b*x)**3/(9*b
**2) + c**3*d*sin(a + b*x)*cos(a + b*x)**5/(6*b**2) + c**2*d**2*x*sin(a + b*x)**5*cos(a + b*x)/(2*b**2) + 4*c*
*2*d**2*x*sin(a + b*x)**3*cos(a + b*x)**3/(3*b**2) + c**2*d**2*x*sin(a + b*x)*cos(a + b*x)**5/(2*b**2) + c*d**
3*x**2*sin(a + b*x)**5*cos(a + b*x)/(2*b**2) + 4*c*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)**3/(3*b**2) + c*d**3
*x**2*sin(a + b*x)*cos(a + b*x)**5/(2*b**2) + d**4*x**3*sin(a + b*x)**5*cos(a + b*x)/(6*b**2) + 4*d**4*x**3*si
n(a + b*x)**3*cos(a + b*x)**3/(9*b**2) + d**4*x**3*sin(a + b*x)*cos(a + b*x)**5/(6*b**2) - 7*c**2*d**2*sin(a +
b*x)**6/(36*b**3) - c**2*d**2*sin(a + b*x)**4*cos(a + b*x)**2/(3*b**3) + c**2*d**2*cos(a + b*x)**6/(12*b**3)
- 5*c*d**3*x*sin(a + b*x)**6/(18*b**3) - c*d**3*x*sin(a + b*x)**4*cos(a + b*x)**2/(3*b**3) + c*d**3*x*sin(a +
b*x)**2*cos(a + b*x)**4/(3*b**3) + 5*c*d**3*x*cos(a + b*x)**6/(18*b**3) - 5*d**4*x**2*sin(a + b*x)**6/(36*b**3
) - d**4*x**2*sin(a + b*x)**4*cos(a + b*x)**2/(6*b**3) + d**4*x**2*sin(a + b*x)**2*cos(a + b*x)**4/(6*b**3) +
5*d**4*x**2*cos(a + b*x)**6/(36*b**3) - 5*c*d**3*sin(a + b*x)**5*cos(a + b*x)/(18*b**4) - 31*c*d**3*sin(a + b*
x)**3*cos(a + b*x)**3/(54*b**4) - 5*c*d**3*sin(a + b*x)*cos(a + b*x)**5/(18*b**4) - 5*d**4*x*sin(a + b*x)**5*c
os(a + b*x)/(18*b**4) - 31*d**4*x*sin(a + b*x)**3*cos(a + b*x)**3/(54*b**4) - 5*d**4*x*sin(a + b*x)*cos(a + b*
x)**5/(18*b**4) + 61*d**4*sin(a + b*x)**6/(648*b**5) + 31*d**4*sin(a + b*x)**4*cos(a + b*x)**2/(216*b**5) - 5*
d**4*cos(a + b*x)**6/(108*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)**3, True))

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