\(\int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx\) [23]

Optimal result

Integrand size = 22, antiderivative size = 260 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {45 c d^3 x}{64 b^3}+\frac {45 d^4 x^2}{128 b^3}-\frac {3 (c+d x)^4}{32 b}-\frac {45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}+\frac {45 d^4 \sin ^2(a+b x)}{128 b^5}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}-\frac {3 d^3 (c+d x) \cos (a+b x) \sin ^3(a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos (a+b x) \sin ^3(a+b x)}{4 b^2}+\frac {3 d^4 \sin ^4(a+b x)}{128 b^5}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b} \]

[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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4489, 3392, 32, 3391} \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {3 d^4 \sin ^4(a+b x)}{128 b^5}+\frac {45 d^4 \sin ^2(a+b x)}{128 b^5}-\frac {3 d^3 (c+d x) \sin ^3(a+b x) \cos (a+b x)}{32 b^4}-\frac {45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}+\frac {d (c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b^2}+\frac {3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}+\frac {45 c d^3 x}{64 b^3}+\frac {45 d^4 x^2}{128 b^3}-\frac {3 (c+d x)^4}{32 b} \]

[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 3391

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 3392

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 4489

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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 2.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.61 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {-64 \left (3 d^4-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \cos (2 (a+b x))+\left (3 d^4-24 b^2 d^2 (c+d x)^2+32 b^4 (c+d x)^4\right ) \cos (4 (a+b x))-8 b d (c+d x) \left (-16 \left (-3 d^2+2 b^2 (c+d x)^2\right )+\left (-3 d^2+8 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (2 (a+b x))}{1024 b^5} \]

[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)

Maple [A] (verified)

Time = 1.99 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.72

method	result	size
parallelrisch	\(\frac {\left (-128 b^{4} \left (d x +c \right )^{4}+384 d^{2} \left (d x +c \right )^{2} b^{2}-192 d^{4}\right ) \cos \left (2 x b +2 a \right )+\left (32 b^{4} \left (d x +c \right )^{4}-24 d^{2} \left (d x +c \right )^{2} b^{2}+3 d^{4}\right ) \cos \left (4 x b +4 a \right )+256 b d \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) \sin \left (2 x b +2 a \right )-32 b d \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) \left (d x +c \right ) \sin \left (4 x b +4 a \right )+96 b^{4} c^{4}-360 b^{2} c^{2} d^{2}+189 d^{4}}{1024 b^{5}}\)	\(187\)
risch	\(\frac {\left (32 d^{4} x^{4} b^{4}+128 b^{4} c \,d^{3} x^{3}+192 b^{4} c^{2} d^{2} x^{2}+128 b^{4} c^{3} d x +32 b^{4} c^{4}-24 b^{2} d^{4} x^{2}-48 b^{2} c \,d^{3} x -24 b^{2} c^{2} d^{2}+3 d^{4}\right ) \cos \left (4 x b +4 a \right )}{1024 b^{5}}-\frac {d \left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (4 x b +4 a \right )}{256 b^{4}}-\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+8 b^{4} c^{3} d x +2 b^{4} c^{4}-6 b^{2} d^{4} x^{2}-12 b^{2} c \,d^{3} x -6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \cos \left (2 x b +2 a \right )}{16 b^{5}}+\frac {d \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (2 x b +2 a \right )}{8 b^{4}}\)	\(354\)
derivativedivides	\(\text {Expression too large to display}\)	\(1137\)
default	\(\text {Expression too large to display}\)	\(1137\)

[In]

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

[Out]

1/1024*((-128*b^4*(d*x+c)^4+384*d^2*(d*x+c)^2*b^2-192*d^4)*cos(2*b*x+2*a)+(32*b^4*(d*x+c)^4-24*d^2*(d*x+c)^2*b
^2+3*d^4)*cos(4*b*x+4*a)+256*b*d*((d*x+c)^2*b^2-3/2*d^2)*(d*x+c)*sin(2*b*x+2*a)-32*b*d*((d*x+c)^2*b^2-3/8*d^2)
*(d*x+c)*sin(4*b*x+4*a)+96*b^4*c^4-360*b^2*c^2*d^2+189*d^4)/b^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.67 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {20 \, b^{4} d^{4} x^{4} + 80 \, b^{4} c d^{3} x^{3} + {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 3 \, {\left (40 \, b^{4} c^{2} d^{2} - 17 \, b^{2} d^{4}\right )} x^{2} - {\left (64 \, b^{4} d^{4} x^{4} + 256 \, b^{4} c d^{3} x^{3} + 64 \, b^{4} c^{4} - 120 \, b^{2} c^{2} d^{2} + 51 \, d^{4} + 24 \, {\left (16 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 16 \, {\left (16 \, b^{4} c^{3} d - 15 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (40 \, b^{4} c^{3} d - 51 \, b^{2} c d^{3}\right )} x - 2 \, {\left (2 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - {\left (40 \, b^{3} d^{4} x^{3} + 120 \, b^{3} c d^{3} x^{2} + 40 \, b^{3} c^{3} d - 51 \, b c d^{3} + 3 \, {\left (40 \, b^{3} c^{2} d^{2} - 17 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{5}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 935 vs. \(2 (262) = 524\).

Time = 0.88 (sec) , antiderivative size = 935, normalized size of antiderivative = 3.60 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\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}{\left (a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 967 vs. \(2 (236) = 472\).

Time = 0.25 (sec) , antiderivative size = 967, normalized size of antiderivative = 3.72 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.39 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 192 \, b^{4} c^{2} d^{2} x^{2} + 128 \, b^{4} c^{3} d x + 32 \, b^{4} c^{4} - 24 \, b^{2} d^{4} x^{2} - 48 \, b^{2} c d^{3} x - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{5}} - \frac {{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 24 \, b^{3} c^{2} d^{2} x + 8 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} + \frac {{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \]

[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

Mupad [B] (verification not implemented)

Time = 24.22 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.22 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=-\frac {192\,d^4\,\cos \left (2\,a+2\,b\,x\right )-3\,d^4\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )-32\,b^4\,c^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )+32\,b^3\,c^3\,d\,\sin \left (4\,a+4\,b\,x\right )-384\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^2\,c^2\,d^2\,\cos \left (4\,a+4\,b\,x\right )-384\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^2\,d^4\,x^2\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )-32\,b^4\,d^4\,x^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )+32\,b^3\,d^4\,x^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )-12\,b\,c\,d^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )-12\,b\,d^4\,x\,\sin \left (4\,a+4\,b\,x\right )+768\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-192\,b^4\,c^2\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )-768\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+512\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+48\,b^2\,c\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )-128\,b^4\,c^3\,d\,x\,\cos \left (4\,a+4\,b\,x\right )+512\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-128\,b^4\,c\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-768\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-768\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+96\,b^3\,c^2\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )+96\,b^3\,c\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{1024\,b^5} \]

[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)