3.147 \(\int (c+d x)^3 \cos ^3(a+b x) \sin ^2(a+b x) \, dx\)

Optimal. Leaf size=259 \[ -\frac {3 d^3 \cos (a+b x)}{4 b^4}+\frac {d^3 \cos (3 a+3 b x)}{216 b^4}+\frac {3 d^3 \cos (5 a+5 b x)}{5000 b^4}-\frac {3 d^2 (c+d x) \sin (a+b x)}{4 b^3}+\frac {d^2 (c+d x) \sin (3 a+3 b x)}{72 b^3}+\frac {3 d^2 (c+d x) \sin (5 a+5 b x)}{1000 b^3}+\frac {3 d (c+d x)^2 \cos (a+b x)}{8 b^2}-\frac {d (c+d x)^2 \cos (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \cos (5 a+5 b x)}{400 b^2}+\frac {(c+d x)^3 \sin (a+b x)}{8 b}-\frac {(c+d x)^3 \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^3 \sin (5 a+5 b x)}{80 b} \]

[Out]

-3/4*d^3*cos(b*x+a)/b^4+3/8*d*(d*x+c)^2*cos(b*x+a)/b^2+1/216*d^3*cos(3*b*x+3*a)/b^4-1/48*d*(d*x+c)^2*cos(3*b*x
+3*a)/b^2+3/5000*d^3*cos(5*b*x+5*a)/b^4-3/400*d*(d*x+c)^2*cos(5*b*x+5*a)/b^2-3/4*d^2*(d*x+c)*sin(b*x+a)/b^3+1/
8*(d*x+c)^3*sin(b*x+a)/b+1/72*d^2*(d*x+c)*sin(3*b*x+3*a)/b^3-1/48*(d*x+c)^3*sin(3*b*x+3*a)/b+3/1000*d^2*(d*x+c
)*sin(5*b*x+5*a)/b^3-1/80*(d*x+c)^3*sin(5*b*x+5*a)/b

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Rubi [A]  time = 0.27, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4406, 3296, 2638} \[ -\frac {3 d^2 (c+d x) \sin (a+b x)}{4 b^3}+\frac {d^2 (c+d x) \sin (3 a+3 b x)}{72 b^3}+\frac {3 d^2 (c+d x) \sin (5 a+5 b x)}{1000 b^3}+\frac {3 d (c+d x)^2 \cos (a+b x)}{8 b^2}-\frac {d (c+d x)^2 \cos (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \cos (5 a+5 b x)}{400 b^2}-\frac {3 d^3 \cos (a+b x)}{4 b^4}+\frac {d^3 \cos (3 a+3 b x)}{216 b^4}+\frac {3 d^3 \cos (5 a+5 b x)}{5000 b^4}+\frac {(c+d x)^3 \sin (a+b x)}{8 b}-\frac {(c+d x)^3 \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^3 \sin (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^2,x]

[Out]

(-3*d^3*Cos[a + b*x])/(4*b^4) + (3*d*(c + d*x)^2*Cos[a + b*x])/(8*b^2) + (d^3*Cos[3*a + 3*b*x])/(216*b^4) - (d
*(c + d*x)^2*Cos[3*a + 3*b*x])/(48*b^2) + (3*d^3*Cos[5*a + 5*b*x])/(5000*b^4) - (3*d*(c + d*x)^2*Cos[5*a + 5*b
*x])/(400*b^2) - (3*d^2*(c + d*x)*Sin[a + b*x])/(4*b^3) + ((c + d*x)^3*Sin[a + b*x])/(8*b) + (d^2*(c + d*x)*Si
n[3*a + 3*b*x])/(72*b^3) - ((c + d*x)^3*Sin[3*a + 3*b*x])/(48*b) + (3*d^2*(c + d*x)*Sin[5*a + 5*b*x])/(1000*b^
3) - ((c + d*x)^3*Sin[5*a + 5*b*x])/(80*b)

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)^3 \cos ^3(a+b x) \sin ^2(a+b x) \, dx &=\int \left (\frac {1}{8} (c+d x)^3 \cos (a+b x)-\frac {1}{16} (c+d x)^3 \cos (3 a+3 b x)-\frac {1}{16} (c+d x)^3 \cos (5 a+5 b x)\right ) \, dx\\ &=-\left (\frac {1}{16} \int (c+d x)^3 \cos (3 a+3 b x) \, dx\right )-\frac {1}{16} \int (c+d x)^3 \cos (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^3 \cos (a+b x) \, dx\\ &=\frac {(c+d x)^3 \sin (a+b x)}{8 b}-\frac {(c+d x)^3 \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^3 \sin (5 a+5 b x)}{80 b}+\frac {(3 d) \int (c+d x)^2 \sin (5 a+5 b x) \, dx}{80 b}+\frac {d \int (c+d x)^2 \sin (3 a+3 b x) \, dx}{16 b}-\frac {(3 d) \int (c+d x)^2 \sin (a+b x) \, dx}{8 b}\\ &=\frac {3 d (c+d x)^2 \cos (a+b x)}{8 b^2}-\frac {d (c+d x)^2 \cos (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \cos (5 a+5 b x)}{400 b^2}+\frac {(c+d x)^3 \sin (a+b x)}{8 b}-\frac {(c+d x)^3 \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^3 \sin (5 a+5 b x)}{80 b}+\frac {\left (3 d^2\right ) \int (c+d x) \cos (5 a+5 b x) \, dx}{200 b^2}+\frac {d^2 \int (c+d x) \cos (3 a+3 b x) \, dx}{24 b^2}-\frac {\left (3 d^2\right ) \int (c+d x) \cos (a+b x) \, dx}{4 b^2}\\ &=\frac {3 d (c+d x)^2 \cos (a+b x)}{8 b^2}-\frac {d (c+d x)^2 \cos (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \cos (5 a+5 b x)}{400 b^2}-\frac {3 d^2 (c+d x) \sin (a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin (a+b x)}{8 b}+\frac {d^2 (c+d x) \sin (3 a+3 b x)}{72 b^3}-\frac {(c+d x)^3 \sin (3 a+3 b x)}{48 b}+\frac {3 d^2 (c+d x) \sin (5 a+5 b x)}{1000 b^3}-\frac {(c+d x)^3 \sin (5 a+5 b x)}{80 b}-\frac {\left (3 d^3\right ) \int \sin (5 a+5 b x) \, dx}{1000 b^3}-\frac {d^3 \int \sin (3 a+3 b x) \, dx}{72 b^3}+\frac {\left (3 d^3\right ) \int \sin (a+b x) \, dx}{4 b^3}\\ &=-\frac {3 d^3 \cos (a+b x)}{4 b^4}+\frac {3 d (c+d x)^2 \cos (a+b x)}{8 b^2}+\frac {d^3 \cos (3 a+3 b x)}{216 b^4}-\frac {d (c+d x)^2 \cos (3 a+3 b x)}{48 b^2}+\frac {3 d^3 \cos (5 a+5 b x)}{5000 b^4}-\frac {3 d (c+d x)^2 \cos (5 a+5 b x)}{400 b^2}-\frac {3 d^2 (c+d x) \sin (a+b x)}{4 b^3}+\frac {(c+d x)^3 \sin (a+b x)}{8 b}+\frac {d^2 (c+d x) \sin (3 a+3 b x)}{72 b^3}-\frac {(c+d x)^3 \sin (3 a+3 b x)}{48 b}+\frac {3 d^2 (c+d x) \sin (5 a+5 b x)}{1000 b^3}-\frac {(c+d x)^3 \sin (5 a+5 b x)}{80 b}\\ \end {align*}

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Mathematica [A]  time = 2.18, size = 195, normalized size = 0.75 \[ -\frac {30 b (c+d x) \sin (a+b x) \left (8 \cos (2 (a+b x)) \left (75 b^2 (c+d x)^2-38 d^2\right )+9 \cos (4 (a+b x)) \left (25 b^2 (c+d x)^2-6 d^2\right )-825 b^2 c^2-1650 b^2 c d x-825 b^2 d^2 x^2+6598 d^2\right )-101250 d \cos (a+b x) \left (b^2 (c+d x)^2-2 d^2\right )+625 d \cos (3 (a+b x)) \left (9 b^2 (c+d x)^2-2 d^2\right )+81 d \cos (5 (a+b x)) \left (25 b^2 (c+d x)^2-2 d^2\right )}{270000 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x]^2,x]

[Out]

-1/270000*(-101250*d*(-2*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + 625*d*(-2*d^2 + 9*b^2*(c + d*x)^2)*Cos[3*(a + b
*x)] + 81*d*(-2*d^2 + 25*b^2*(c + d*x)^2)*Cos[5*(a + b*x)] + 30*b*(c + d*x)*(-825*b^2*c^2 + 6598*d^2 - 1650*b^
2*c*d*x - 825*b^2*d^2*x^2 + 8*(-38*d^2 + 75*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] + 9*(-6*d^2 + 25*b^2*(c + d*x)^2
)*Cos[4*(a + b*x)])*Sin[a + b*x])/b^4

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fricas [A]  time = 0.77, size = 342, normalized size = 1.32 \[ -\frac {81 \, {\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{5} - 5 \, {\left (225 \, b^{2} d^{3} x^{2} + 450 \, b^{2} c d^{2} x + 225 \, b^{2} c^{2} d + 22 \, d^{3}\right )} \cos \left (b x + a\right )^{3} - 30 \, {\left (225 \, b^{2} d^{3} x^{2} + 450 \, b^{2} c d^{2} x + 225 \, b^{2} c^{2} d - 428 \, d^{3}\right )} \cos \left (b x + a\right ) - 15 \, {\left (150 \, b^{3} d^{3} x^{3} + 450 \, b^{3} c d^{2} x^{2} + 150 \, b^{3} c^{3} - 9 \, {\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 25 \, b^{3} c^{3} - 6 \, b c d^{2} + 3 \, {\left (25 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} - 856 \, b c d^{2} + {\left (75 \, b^{3} d^{3} x^{3} + 225 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{3} + 22 \, b c d^{2} + {\left (225 \, b^{3} c^{2} d + 22 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (225 \, b^{3} c^{2} d - 428 \, b d^{3}\right )} x\right )} \sin \left (b x + a\right )}{16875 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/16875*(81*(25*b^2*d^3*x^2 + 50*b^2*c*d^2*x + 25*b^2*c^2*d - 2*d^3)*cos(b*x + a)^5 - 5*(225*b^2*d^3*x^2 + 45
0*b^2*c*d^2*x + 225*b^2*c^2*d + 22*d^3)*cos(b*x + a)^3 - 30*(225*b^2*d^3*x^2 + 450*b^2*c*d^2*x + 225*b^2*c^2*d
- 428*d^3)*cos(b*x + a) - 15*(150*b^3*d^3*x^3 + 450*b^3*c*d^2*x^2 + 150*b^3*c^3 - 9*(25*b^3*d^3*x^3 + 75*b^3*
c*d^2*x^2 + 25*b^3*c^3 - 6*b*c*d^2 + 3*(25*b^3*c^2*d - 2*b*d^3)*x)*cos(b*x + a)^4 - 856*b*c*d^2 + (75*b^3*d^3*
x^3 + 225*b^3*c*d^2*x^2 + 75*b^3*c^3 + 22*b*c*d^2 + (225*b^3*c^2*d + 22*b*d^3)*x)*cos(b*x + a)^2 + 2*(225*b^3*
c^2*d - 428*b*d^3)*x)*sin(b*x + a))/b^4

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giac [A]  time = 0.36, size = 351, normalized size = 1.36 \[ -\frac {3 \, {\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (5 \, b x + 5 \, a\right )}{10000 \, b^{4}} - \frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{432 \, b^{4}} + \frac {3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )}{8 \, b^{4}} - \frac {{\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{2} d x + 25 \, b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \sin \left (5 \, b x + 5 \, a\right )}{2000 \, b^{4}} - \frac {{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \sin \left (3 \, b x + 3 \, a\right )}{144 \, b^{4}} + \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \sin \left (b x + a\right )}{8 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="giac")

[Out]

-3/10000*(25*b^2*d^3*x^2 + 50*b^2*c*d^2*x + 25*b^2*c^2*d - 2*d^3)*cos(5*b*x + 5*a)/b^4 - 1/432*(9*b^2*d^3*x^2
+ 18*b^2*c*d^2*x + 9*b^2*c^2*d - 2*d^3)*cos(3*b*x + 3*a)/b^4 + 3/8*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d -
2*d^3)*cos(b*x + a)/b^4 - 1/2000*(25*b^3*d^3*x^3 + 75*b^3*c*d^2*x^2 + 75*b^3*c^2*d*x + 25*b^3*c^3 - 6*b*d^3*x
- 6*b*c*d^2)*sin(5*b*x + 5*a)/b^4 - 1/144*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 9*b^3*c^2*d*x + 3*b^3*c^3 - 2*b*d
^3*x - 2*b*c*d^2)*sin(3*b*x + 3*a)/b^4 + 1/8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3 - 6*b*d^
3*x - 6*b*c*d^2)*sin(b*x + a)/b^4

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maple [B]  time = 0.02, size = 1016, normalized size = 3.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^2,x)

[Out]

1/b*(1/b^3*d^3*(1/3*(b*x+a)^3*(2+cos(b*x+a)^2)*sin(b*x+a)+2/5*(b*x+a)^2*cos(b*x+a)-856/1125*cos(b*x+a)-4/5*(b*
x+a)*sin(b*x+a)+1/15*(b*x+a)^2*cos(b*x+a)^3-2/45*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+22/3375*cos(b*x+a)^3-1/5*
(b*x+a)^3*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-3/25*(b*x+a)^2*cos(b*x+a)^5+6/125*(b*x+a)*(8/3+cos(b*
x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)+6/625*cos(b*x+a)^5)-3/b^3*a*d^3*(1/3*(b*x+a)^2*(2+cos(b*x+a)^2)*sin(b*x+a)
-4/15*sin(b*x+a)+4/15*(b*x+a)*cos(b*x+a)+2/45*(b*x+a)*cos(b*x+a)^3-2/135*(2+cos(b*x+a)^2)*sin(b*x+a)-1/5*(b*x+
a)^2*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-2/25*(b*x+a)*cos(b*x+a)^5+2/125*(8/3+cos(b*x+a)^4+4/3*cos(
b*x+a)^2)*sin(b*x+a))+3/b^2*c*d^2*(1/3*(b*x+a)^2*(2+cos(b*x+a)^2)*sin(b*x+a)-4/15*sin(b*x+a)+4/15*(b*x+a)*cos(
b*x+a)+2/45*(b*x+a)*cos(b*x+a)^3-2/135*(2+cos(b*x+a)^2)*sin(b*x+a)-1/5*(b*x+a)^2*(8/3+cos(b*x+a)^4+4/3*cos(b*x
+a)^2)*sin(b*x+a)-2/25*(b*x+a)*cos(b*x+a)^5+2/125*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a))+3/b^3*a^2*d^
3*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1/45*cos(b*x+a)^3+2/15*cos(b*x+a)-1/5*(b*x+a)*(8/3+cos(b*x+a)^4+4/3
*cos(b*x+a)^2)*sin(b*x+a)-1/25*cos(b*x+a)^5)-6/b^2*a*c*d^2*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1/45*cos(b
*x+a)^3+2/15*cos(b*x+a)-1/5*(b*x+a)*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-1/25*cos(b*x+a)^5)+3/b*c^2*
d*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1/45*cos(b*x+a)^3+2/15*cos(b*x+a)-1/5*(b*x+a)*(8/3+cos(b*x+a)^4+4/3
*cos(b*x+a)^2)*sin(b*x+a)-1/25*cos(b*x+a)^5)-1/b^3*a^3*d^3*(-1/5*cos(b*x+a)^4*sin(b*x+a)+1/15*(2+cos(b*x+a)^2)
*sin(b*x+a))+3/b^2*a^2*c*d^2*(-1/5*cos(b*x+a)^4*sin(b*x+a)+1/15*(2+cos(b*x+a)^2)*sin(b*x+a))-3/b*a*c^2*d*(-1/5
*cos(b*x+a)^4*sin(b*x+a)+1/15*(2+cos(b*x+a)^2)*sin(b*x+a))+c^3*(-1/5*cos(b*x+a)^4*sin(b*x+a)+1/15*(2+cos(b*x+a
)^2)*sin(b*x+a)))

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maxima [B]  time = 0.85, size = 766, normalized size = 2.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/270000*(18000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*c^3 - 54000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*a*c^2
*d/b + 54000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*a^2*c*d^2/b^2 - 18000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)
*a^3*d^3/b^3 + 225*(45*(b*x + a)*sin(5*b*x + 5*a) + 75*(b*x + a)*sin(3*b*x + 3*a) - 450*(b*x + a)*sin(b*x + a)
+ 9*cos(5*b*x + 5*a) + 25*cos(3*b*x + 3*a) - 450*cos(b*x + a))*c^2*d/b - 450*(45*(b*x + a)*sin(5*b*x + 5*a) +
75*(b*x + a)*sin(3*b*x + 3*a) - 450*(b*x + a)*sin(b*x + a) + 9*cos(5*b*x + 5*a) + 25*cos(3*b*x + 3*a) - 450*c
os(b*x + a))*a*c*d^2/b^2 + 225*(45*(b*x + a)*sin(5*b*x + 5*a) + 75*(b*x + a)*sin(3*b*x + 3*a) - 450*(b*x + a)*
sin(b*x + a) + 9*cos(5*b*x + 5*a) + 25*cos(3*b*x + 3*a) - 450*cos(b*x + a))*a^2*d^3/b^3 + 15*(270*(b*x + a)*co
s(5*b*x + 5*a) + 750*(b*x + a)*cos(3*b*x + 3*a) - 13500*(b*x + a)*cos(b*x + a) + 27*(25*(b*x + a)^2 - 2)*sin(5
*b*x + 5*a) + 125*(9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) - 6750*((b*x + a)^2 - 2)*sin(b*x + a))*c*d^2/b^2 - 15*(
270*(b*x + a)*cos(5*b*x + 5*a) + 750*(b*x + a)*cos(3*b*x + 3*a) - 13500*(b*x + a)*cos(b*x + a) + 27*(25*(b*x +
a)^2 - 2)*sin(5*b*x + 5*a) + 125*(9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) - 6750*((b*x + a)^2 - 2)*sin(b*x + a))*
a*d^3/b^3 + (81*(25*(b*x + a)^2 - 2)*cos(5*b*x + 5*a) + 625*(9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 101250*((b*
x + a)^2 - 2)*cos(b*x + a) + 135*(25*(b*x + a)^3 - 6*b*x - 6*a)*sin(5*b*x + 5*a) + 1875*(3*(b*x + a)^3 - 2*b*x
- 2*a)*sin(3*b*x + 3*a) - 33750*((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*d^3/b^3)/b

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mupad [B]  time = 2.43, size = 516, normalized size = 1.99 \[ -\frac {\frac {3\,d^3\,\cos \left (a+b\,x\right )}{4}-\frac {d^3\,\cos \left (3\,a+3\,b\,x\right )}{216}-\frac {3\,d^3\,\cos \left (5\,a+5\,b\,x\right )}{5000}-\frac {b^3\,c^3\,\sin \left (a+b\,x\right )}{8}+\frac {b^3\,c^3\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {b^3\,c^3\,\sin \left (5\,a+5\,b\,x\right )}{80}+\frac {b^2\,c^2\,d\,\cos \left (3\,a+3\,b\,x\right )}{48}+\frac {3\,b^2\,c^2\,d\,\cos \left (5\,a+5\,b\,x\right )}{400}-\frac {3\,b^2\,d^3\,x^2\,\cos \left (a+b\,x\right )}{8}-\frac {b^3\,d^3\,x^3\,\sin \left (a+b\,x\right )}{8}+\frac {3\,b\,c\,d^2\,\sin \left (a+b\,x\right )}{4}+\frac {3\,b\,d^3\,x\,\sin \left (a+b\,x\right )}{4}+\frac {b^2\,d^3\,x^2\,\cos \left (3\,a+3\,b\,x\right )}{48}+\frac {3\,b^2\,d^3\,x^2\,\cos \left (5\,a+5\,b\,x\right )}{400}+\frac {b^3\,d^3\,x^3\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {b^3\,d^3\,x^3\,\sin \left (5\,a+5\,b\,x\right )}{80}-\frac {3\,b^2\,c^2\,d\,\cos \left (a+b\,x\right )}{8}-\frac {b\,c\,d^2\,\sin \left (3\,a+3\,b\,x\right )}{72}-\frac {3\,b\,c\,d^2\,\sin \left (5\,a+5\,b\,x\right )}{1000}-\frac {b\,d^3\,x\,\sin \left (3\,a+3\,b\,x\right )}{72}-\frac {3\,b\,d^3\,x\,\sin \left (5\,a+5\,b\,x\right )}{1000}-\frac {3\,b^2\,c\,d^2\,x\,\cos \left (a+b\,x\right )}{4}-\frac {3\,b^3\,c^2\,d\,x\,\sin \left (a+b\,x\right )}{8}+\frac {b^2\,c\,d^2\,x\,\cos \left (3\,a+3\,b\,x\right )}{24}+\frac {3\,b^2\,c\,d^2\,x\,\cos \left (5\,a+5\,b\,x\right )}{200}+\frac {b^3\,c^2\,d\,x\,\sin \left (3\,a+3\,b\,x\right )}{16}+\frac {3\,b^3\,c^2\,d\,x\,\sin \left (5\,a+5\,b\,x\right )}{80}-\frac {3\,b^3\,c\,d^2\,x^2\,\sin \left (a+b\,x\right )}{8}+\frac {b^3\,c\,d^2\,x^2\,\sin \left (3\,a+3\,b\,x\right )}{16}+\frac {3\,b^3\,c\,d^2\,x^2\,\sin \left (5\,a+5\,b\,x\right )}{80}}{b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^3*sin(a + b*x)^2*(c + d*x)^3,x)

[Out]

-((3*d^3*cos(a + b*x))/4 - (d^3*cos(3*a + 3*b*x))/216 - (3*d^3*cos(5*a + 5*b*x))/5000 - (b^3*c^3*sin(a + b*x))
/8 + (b^3*c^3*sin(3*a + 3*b*x))/48 + (b^3*c^3*sin(5*a + 5*b*x))/80 + (b^2*c^2*d*cos(3*a + 3*b*x))/48 + (3*b^2*
c^2*d*cos(5*a + 5*b*x))/400 - (3*b^2*d^3*x^2*cos(a + b*x))/8 - (b^3*d^3*x^3*sin(a + b*x))/8 + (3*b*c*d^2*sin(a
+ b*x))/4 + (3*b*d^3*x*sin(a + b*x))/4 + (b^2*d^3*x^2*cos(3*a + 3*b*x))/48 + (3*b^2*d^3*x^2*cos(5*a + 5*b*x))
/400 + (b^3*d^3*x^3*sin(3*a + 3*b*x))/48 + (b^3*d^3*x^3*sin(5*a + 5*b*x))/80 - (3*b^2*c^2*d*cos(a + b*x))/8 -
(b*c*d^2*sin(3*a + 3*b*x))/72 - (3*b*c*d^2*sin(5*a + 5*b*x))/1000 - (b*d^3*x*sin(3*a + 3*b*x))/72 - (3*b*d^3*x
*sin(5*a + 5*b*x))/1000 - (3*b^2*c*d^2*x*cos(a + b*x))/4 - (3*b^3*c^2*d*x*sin(a + b*x))/8 + (b^2*c*d^2*x*cos(3
*a + 3*b*x))/24 + (3*b^2*c*d^2*x*cos(5*a + 5*b*x))/200 + (b^3*c^2*d*x*sin(3*a + 3*b*x))/16 + (3*b^3*c^2*d*x*si
n(5*a + 5*b*x))/80 - (3*b^3*c*d^2*x^2*sin(a + b*x))/8 + (b^3*c*d^2*x^2*sin(3*a + 3*b*x))/16 + (3*b^3*c*d^2*x^2
*sin(5*a + 5*b*x))/80)/b^4

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sympy [A]  time = 11.35, size = 690, normalized size = 2.66 \[ \begin {cases} \frac {2 c^{3} \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac {c^{3} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 c^{2} d x \sin ^{5}{\left (a + b x \right )}}{5 b} + \frac {c^{2} d x \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac {2 c d^{2} x^{2} \sin ^{5}{\left (a + b x \right )}}{5 b} + \frac {c d^{2} x^{2} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} + \frac {2 d^{3} x^{3} \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac {d^{3} x^{3} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 c^{2} d \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{5 b^{2}} + \frac {13 c^{2} d \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{15 b^{2}} + \frac {26 c^{2} d \cos ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {4 c d^{2} x \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{5 b^{2}} + \frac {26 c d^{2} x \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{15 b^{2}} + \frac {52 c d^{2} x \cos ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {2 d^{3} x^{2} \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{5 b^{2}} + \frac {13 d^{3} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{15 b^{2}} + \frac {26 d^{3} x^{2} \cos ^{5}{\left (a + b x \right )}}{75 b^{2}} - \frac {856 c d^{2} \sin ^{5}{\left (a + b x \right )}}{1125 b^{3}} - \frac {338 c d^{2} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{225 b^{3}} - \frac {52 c d^{2} \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{75 b^{3}} - \frac {856 d^{3} x \sin ^{5}{\left (a + b x \right )}}{1125 b^{3}} - \frac {338 d^{3} x \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{225 b^{3}} - \frac {52 d^{3} x \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{75 b^{3}} - \frac {856 d^{3} \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{1125 b^{4}} - \frac {5114 d^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3375 b^{4}} - \frac {12568 d^{3} \cos ^{5}{\left (a + b x \right )}}{16875 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin ^{2}{\relax (a )} \cos ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*cos(b*x+a)**3*sin(b*x+a)**2,x)

[Out]

Piecewise((2*c**3*sin(a + b*x)**5/(15*b) + c**3*sin(a + b*x)**3*cos(a + b*x)**2/(3*b) + 2*c**2*d*x*sin(a + b*x
)**5/(5*b) + c**2*d*x*sin(a + b*x)**3*cos(a + b*x)**2/b + 2*c*d**2*x**2*sin(a + b*x)**5/(5*b) + c*d**2*x**2*si
n(a + b*x)**3*cos(a + b*x)**2/b + 2*d**3*x**3*sin(a + b*x)**5/(15*b) + d**3*x**3*sin(a + b*x)**3*cos(a + b*x)*
*2/(3*b) + 2*c**2*d*sin(a + b*x)**4*cos(a + b*x)/(5*b**2) + 13*c**2*d*sin(a + b*x)**2*cos(a + b*x)**3/(15*b**2
) + 26*c**2*d*cos(a + b*x)**5/(75*b**2) + 4*c*d**2*x*sin(a + b*x)**4*cos(a + b*x)/(5*b**2) + 26*c*d**2*x*sin(a
+ b*x)**2*cos(a + b*x)**3/(15*b**2) + 52*c*d**2*x*cos(a + b*x)**5/(75*b**2) + 2*d**3*x**2*sin(a + b*x)**4*cos
(a + b*x)/(5*b**2) + 13*d**3*x**2*sin(a + b*x)**2*cos(a + b*x)**3/(15*b**2) + 26*d**3*x**2*cos(a + b*x)**5/(75
*b**2) - 856*c*d**2*sin(a + b*x)**5/(1125*b**3) - 338*c*d**2*sin(a + b*x)**3*cos(a + b*x)**2/(225*b**3) - 52*c
*d**2*sin(a + b*x)*cos(a + b*x)**4/(75*b**3) - 856*d**3*x*sin(a + b*x)**5/(1125*b**3) - 338*d**3*x*sin(a + b*x
)**3*cos(a + b*x)**2/(225*b**3) - 52*d**3*x*sin(a + b*x)*cos(a + b*x)**4/(75*b**3) - 856*d**3*sin(a + b*x)**4*
cos(a + b*x)/(1125*b**4) - 5114*d**3*sin(a + b*x)**2*cos(a + b*x)**3/(3375*b**4) - 12568*d**3*cos(a + b*x)**5/
(16875*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4)*sin(a)**2*cos(a)**3, True))

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