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

Optimal. Leaf size=235 \[ \frac {a^2 \sin (c+d x)}{d^2}-\frac {a^2 x \cos (c+d x)}{d}-\frac {48 a b \cos (c+d x)}{d^5}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {24 a b x^2 \cos (c+d x)}{d^3}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {5040 b^2 \sin (c+d x)}{d^8}+\frac {5040 b^2 x \cos (c+d x)}{d^7}+\frac {2520 b^2 x^2 \sin (c+d x)}{d^6}-\frac {840 b^2 x^3 \cos (c+d x)}{d^5}-\frac {210 b^2 x^4 \sin (c+d x)}{d^4}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}-\frac {b^2 x^7 \cos (c+d x)}{d} \]

[Out]

-48*a*b*cos(d*x+c)/d^5+5040*b^2*x*cos(d*x+c)/d^7-a^2*x*cos(d*x+c)/d+24*a*b*x^2*cos(d*x+c)/d^3-840*b^2*x^3*cos(
d*x+c)/d^5-2*a*b*x^4*cos(d*x+c)/d+42*b^2*x^5*cos(d*x+c)/d^3-b^2*x^7*cos(d*x+c)/d-5040*b^2*sin(d*x+c)/d^8+a^2*s
in(d*x+c)/d^2-48*a*b*x*sin(d*x+c)/d^4+2520*b^2*x^2*sin(d*x+c)/d^6+8*a*b*x^3*sin(d*x+c)/d^2-210*b^2*x^4*sin(d*x
+c)/d^4+7*b^2*x^6*sin(d*x+c)/d^2

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Rubi [A]  time = 0.33, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3339, 3296, 2637, 2638} \[ \frac {a^2 \sin (c+d x)}{d^2}-\frac {a^2 x \cos (c+d x)}{d}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {48 a b x \sin (c+d x)}{d^4}-\frac {48 a b \cos (c+d x)}{d^5}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}-\frac {210 b^2 x^4 \sin (c+d x)}{d^4}+\frac {2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}-\frac {840 b^2 x^3 \cos (c+d x)}{d^5}-\frac {5040 b^2 \sin (c+d x)}{d^8}+\frac {5040 b^2 x \cos (c+d x)}{d^7}-\frac {b^2 x^7 \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^3)^2*Sin[c + d*x],x]

[Out]

(-48*a*b*Cos[c + d*x])/d^5 + (5040*b^2*x*Cos[c + d*x])/d^7 - (a^2*x*Cos[c + d*x])/d + (24*a*b*x^2*Cos[c + d*x]
)/d^3 - (840*b^2*x^3*Cos[c + d*x])/d^5 - (2*a*b*x^4*Cos[c + d*x])/d + (42*b^2*x^5*Cos[c + d*x])/d^3 - (b^2*x^7
*Cos[c + d*x])/d - (5040*b^2*Sin[c + d*x])/d^8 + (a^2*Sin[c + d*x])/d^2 - (48*a*b*x*Sin[c + d*x])/d^4 + (2520*
b^2*x^2*Sin[c + d*x])/d^6 + (8*a*b*x^3*Sin[c + d*x])/d^2 - (210*b^2*x^4*Sin[c + d*x])/d^4 + (7*b^2*x^6*Sin[c +
d*x])/d^2

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 3339

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

Rubi steps

\begin {align*} \int x \left (a+b x^3\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 x \sin (c+d x)+2 a b x^4 \sin (c+d x)+b^2 x^7 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x \sin (c+d x) \, dx+(2 a b) \int x^4 \sin (c+d x) \, dx+b^2 \int x^7 \sin (c+d x) \, dx\\ &=-\frac {a^2 x \cos (c+d x)}{d}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^7 \cos (c+d x)}{d}+\frac {a^2 \int \cos (c+d x) \, dx}{d}+\frac {(8 a b) \int x^3 \cos (c+d x) \, dx}{d}+\frac {\left (7 b^2\right ) \int x^6 \cos (c+d x) \, dx}{d}\\ &=-\frac {a^2 x \cos (c+d x)}{d}-\frac {2 a b x^4 \cos (c+d x)}{d}-\frac {b^2 x^7 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}-\frac {(24 a b) \int x^2 \sin (c+d x) \, dx}{d^2}-\frac {\left (42 b^2\right ) \int x^5 \sin (c+d x) \, dx}{d^2}\\ &=-\frac {a^2 x \cos (c+d x)}{d}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}-\frac {b^2 x^7 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}+\frac {8 a b x^3 \sin (c+d x)}{d^2}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}-\frac {(48 a b) \int x \cos (c+d x) \, dx}{d^3}-\frac {\left (210 b^2\right ) \int x^4 \cos (c+d x) \, dx}{d^3}\\ &=-\frac {a^2 x \cos (c+d x)}{d}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}-\frac {b^2 x^7 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {210 b^2 x^4 \sin (c+d x)}{d^4}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}+\frac {(48 a b) \int \sin (c+d x) \, dx}{d^4}+\frac {\left (840 b^2\right ) \int x^3 \sin (c+d x) \, dx}{d^4}\\ &=-\frac {48 a b \cos (c+d x)}{d^5}-\frac {a^2 x \cos (c+d x)}{d}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {840 b^2 x^3 \cos (c+d x)}{d^5}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}-\frac {b^2 x^7 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {210 b^2 x^4 \sin (c+d x)}{d^4}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}+\frac {\left (2520 b^2\right ) \int x^2 \cos (c+d x) \, dx}{d^5}\\ &=-\frac {48 a b \cos (c+d x)}{d^5}-\frac {a^2 x \cos (c+d x)}{d}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {840 b^2 x^3 \cos (c+d x)}{d^5}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}-\frac {b^2 x^7 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {210 b^2 x^4 \sin (c+d x)}{d^4}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}-\frac {\left (5040 b^2\right ) \int x \sin (c+d x) \, dx}{d^6}\\ &=-\frac {48 a b \cos (c+d x)}{d^5}+\frac {5040 b^2 x \cos (c+d x)}{d^7}-\frac {a^2 x \cos (c+d x)}{d}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {840 b^2 x^3 \cos (c+d x)}{d^5}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}-\frac {b^2 x^7 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {210 b^2 x^4 \sin (c+d x)}{d^4}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}-\frac {\left (5040 b^2\right ) \int \cos (c+d x) \, dx}{d^7}\\ &=-\frac {48 a b \cos (c+d x)}{d^5}+\frac {5040 b^2 x \cos (c+d x)}{d^7}-\frac {a^2 x \cos (c+d x)}{d}+\frac {24 a b x^2 \cos (c+d x)}{d^3}-\frac {840 b^2 x^3 \cos (c+d x)}{d^5}-\frac {2 a b x^4 \cos (c+d x)}{d}+\frac {42 b^2 x^5 \cos (c+d x)}{d^3}-\frac {b^2 x^7 \cos (c+d x)}{d}-\frac {5040 b^2 \sin (c+d x)}{d^8}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {48 a b x \sin (c+d x)}{d^4}+\frac {2520 b^2 x^2 \sin (c+d x)}{d^6}+\frac {8 a b x^3 \sin (c+d x)}{d^2}-\frac {210 b^2 x^4 \sin (c+d x)}{d^4}+\frac {7 b^2 x^6 \sin (c+d x)}{d^2}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 139, normalized size = 0.59 \[ \frac {\left (a^2 d^6+8 a b d^4 x \left (d^2 x^2-6\right )+7 b^2 \left (d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right )\right ) \sin (c+d x)-d \left (a^2 d^6 x+2 a b d^2 \left (d^4 x^4-12 d^2 x^2+24\right )+b^2 x \left (d^6 x^6-42 d^4 x^4+840 d^2 x^2-5040\right )\right ) \cos (c+d x)}{d^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(d*(a^2*d^6*x + 2*a*b*d^2*(24 - 12*d^2*x^2 + d^4*x^4) + b^2*x*(-5040 + 840*d^2*x^2 - 42*d^4*x^4 + d^6*x^6))*
Cos[c + d*x]) + (a^2*d^6 + 8*a*b*d^4*x*(-6 + d^2*x^2) + 7*b^2*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Sin
[c + d*x])/d^8

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fricas [A]  time = 0.56, size = 161, normalized size = 0.69 \[ -\frac {{\left (b^{2} d^{7} x^{7} + 2 \, a b d^{7} x^{4} - 42 \, b^{2} d^{5} x^{5} - 24 \, a b d^{5} x^{2} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{3} + {\left (a^{2} d^{7} - 5040 \, b^{2} d\right )} x\right )} \cos \left (d x + c\right ) - {\left (7 \, b^{2} d^{6} x^{6} + 8 \, a b d^{6} x^{3} - 210 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} - 5040 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2*d^7*x^7 + 2*a*b*d^7*x^4 - 42*b^2*d^5*x^5 - 24*a*b*d^5*x^2 + 840*b^2*d^3*x^3 + 48*a*b*d^3 + (a^2*d^7 - 5
040*b^2*d)*x)*cos(d*x + c) - (7*b^2*d^6*x^6 + 8*a*b*d^6*x^3 - 210*b^2*d^4*x^4 + a^2*d^6 - 48*a*b*d^4*x + 2520*
b^2*d^2*x^2 - 5040*b^2)*sin(d*x + c))/d^8

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giac [A]  time = 0.53, size = 161, normalized size = 0.69 \[ -\frac {{\left (b^{2} d^{7} x^{7} + 2 \, a b d^{7} x^{4} - 42 \, b^{2} d^{5} x^{5} + a^{2} d^{7} x - 24 \, a b d^{5} x^{2} + 840 \, b^{2} d^{3} x^{3} + 48 \, a b d^{3} - 5040 \, b^{2} d x\right )} \cos \left (d x + c\right )}{d^{8}} + \frac {{\left (7 \, b^{2} d^{6} x^{6} + 8 \, a b d^{6} x^{3} - 210 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 48 \, a b d^{4} x + 2520 \, b^{2} d^{2} x^{2} - 5040 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*d^7*x^7 + 2*a*b*d^7*x^4 - 42*b^2*d^5*x^5 + a^2*d^7*x - 24*a*b*d^5*x^2 + 840*b^2*d^3*x^3 + 48*a*b*d^3 - 5
040*b^2*d*x)*cos(d*x + c)/d^8 + (7*b^2*d^6*x^6 + 8*a*b*d^6*x^3 - 210*b^2*d^4*x^4 + a^2*d^6 - 48*a*b*d^4*x + 25
20*b^2*d^2*x^2 - 5040*b^2)*sin(d*x + c)/d^8

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maple [B]  time = 0.02, size = 822, normalized size = 3.50 \[ \frac {\frac {b^{2} \left (-\left (d x +c \right )^{7} \cos \left (d x +c \right )+7 \left (d x +c \right )^{6} \sin \left (d x +c \right )+42 \left (d x +c \right )^{5} \cos \left (d x +c \right )-210 \left (d x +c \right )^{4} \sin \left (d x +c \right )-840 \left (d x +c \right )^{3} \cos \left (d x +c \right )+2520 \left (d x +c \right )^{2} \sin \left (d x +c \right )-5040 \sin \left (d x +c \right )+5040 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{6}}-\frac {7 b^{2} c \left (-\left (d x +c \right )^{6} \cos \left (d x +c \right )+6 \left (d x +c \right )^{5} \sin \left (d x +c \right )+30 \left (d x +c \right )^{4} \cos \left (d x +c \right )-120 \left (d x +c \right )^{3} \sin \left (d x +c \right )-360 \left (d x +c \right )^{2} \cos \left (d x +c \right )+720 \cos \left (d x +c \right )+720 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {21 b^{2} c^{2} \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{6}}+\frac {2 a b \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {35 b^{2} c^{3} \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}-\frac {8 a b c \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}+\frac {35 b^{2} c^{4} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{6}}+\frac {12 a b \,c^{2} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {21 b^{2} c^{5} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+a^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )-\frac {8 a b \,c^{3} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}+\frac {7 b^{2} c^{6} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{6}}+a^{2} c \cos \left (d x +c \right )-\frac {2 a b \,c^{4} \cos \left (d x +c \right )}{d^{3}}+\frac {b^{2} c^{7} \cos \left (d x +c \right )}{d^{6}}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(1/d^6*b^2*(-(d*x+c)^7*cos(d*x+c)+7*(d*x+c)^6*sin(d*x+c)+42*(d*x+c)^5*cos(d*x+c)-210*(d*x+c)^4*sin(d*x+c
)-840*(d*x+c)^3*cos(d*x+c)+2520*(d*x+c)^2*sin(d*x+c)-5040*sin(d*x+c)+5040*(d*x+c)*cos(d*x+c))-7/d^6*b^2*c*(-(d
*x+c)^6*cos(d*x+c)+6*(d*x+c)^5*sin(d*x+c)+30*(d*x+c)^4*cos(d*x+c)-120*(d*x+c)^3*sin(d*x+c)-360*(d*x+c)^2*cos(d
*x+c)+720*cos(d*x+c)+720*(d*x+c)*sin(d*x+c))+21/d^6*b^2*c^2*(-(d*x+c)^5*cos(d*x+c)+5*(d*x+c)^4*sin(d*x+c)+20*(
d*x+c)^3*cos(d*x+c)-60*(d*x+c)^2*sin(d*x+c)+120*sin(d*x+c)-120*(d*x+c)*cos(d*x+c))+2/d^3*a*b*(-(d*x+c)^4*cos(d
*x+c)+4*(d*x+c)^3*sin(d*x+c)+12*(d*x+c)^2*cos(d*x+c)-24*cos(d*x+c)-24*(d*x+c)*sin(d*x+c))-35/d^6*b^2*c^3*(-(d*
x+c)^4*cos(d*x+c)+4*(d*x+c)^3*sin(d*x+c)+12*(d*x+c)^2*cos(d*x+c)-24*cos(d*x+c)-24*(d*x+c)*sin(d*x+c))-8/d^3*a*
b*c*(-(d*x+c)^3*cos(d*x+c)+3*(d*x+c)^2*sin(d*x+c)-6*sin(d*x+c)+6*(d*x+c)*cos(d*x+c))+35/d^6*b^2*c^4*(-(d*x+c)^
3*cos(d*x+c)+3*(d*x+c)^2*sin(d*x+c)-6*sin(d*x+c)+6*(d*x+c)*cos(d*x+c))+12/d^3*a*b*c^2*(-(d*x+c)^2*cos(d*x+c)+2
*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))-21/d^6*b^2*c^5*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))+a^2
*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-8/d^3*a*b*c^3*(sin(d*x+c)-(d*x+c)*cos(d*x+c))+7/d^6*b^2*c^6*(sin(d*x+c)-(d*x+
c)*cos(d*x+c))+a^2*c*cos(d*x+c)-2/d^3*a*b*c^4*cos(d*x+c)+1/d^6*b^2*c^7*cos(d*x+c))

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maxima [B]  time = 1.05, size = 662, normalized size = 2.82 \[ \frac {a^{2} c \cos \left (d x + c\right ) + \frac {b^{2} c^{7} \cos \left (d x + c\right )}{d^{6}} - \frac {2 \, a b c^{4} \cos \left (d x + c\right )}{d^{3}} - {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a^{2} - \frac {7 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{6}}{d^{6}} + \frac {8 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b c^{3}}{d^{3}} + \frac {21 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b^{2} c^{5}}{d^{6}} - \frac {12 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a b c^{2}}{d^{3}} - \frac {35 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b^{2} c^{4}}{d^{6}} + \frac {8 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} a b c}{d^{3}} + \frac {35 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b^{2} c^{3}}{d^{6}} - \frac {2 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} a b}{d^{3}} - \frac {21 \, {\left ({\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \, {\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b^{2} c^{2}}{d^{6}} + \frac {7 \, {\left ({\left ({\left (d x + c\right )}^{6} - 30 \, {\left (d x + c\right )}^{4} + 360 \, {\left (d x + c\right )}^{2} - 720\right )} \cos \left (d x + c\right ) - 6 \, {\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \sin \left (d x + c\right )\right )} b^{2} c}{d^{6}} - \frac {{\left ({\left ({\left (d x + c\right )}^{7} - 42 \, {\left (d x + c\right )}^{5} + 840 \, {\left (d x + c\right )}^{3} - 5040 \, d x - 5040 \, c\right )} \cos \left (d x + c\right ) - 7 \, {\left ({\left (d x + c\right )}^{6} - 30 \, {\left (d x + c\right )}^{4} + 360 \, {\left (d x + c\right )}^{2} - 720\right )} \sin \left (d x + c\right )\right )} b^{2}}{d^{6}}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*c*cos(d*x + c) + b^2*c^7*cos(d*x + c)/d^6 - 2*a*b*c^4*cos(d*x + c)/d^3 - ((d*x + c)*cos(d*x + c) - sin(d*
x + c))*a^2 - 7*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b^2*c^6/d^6 + 8*((d*x + c)*cos(d*x + c) - sin(d*x + c)
)*a*b*c^3/d^3 + 21*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b^2*c^5/d^6 - 12*(((d*x + c)^2
- 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a*b*c^2/d^3 - 35*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*(
(d*x + c)^2 - 2)*sin(d*x + c))*b^2*c^4/d^6 + 8*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)
*sin(d*x + c))*a*b*c/d^3 + 35*(((d*x + c)^4 - 12*(d*x + c)^2 + 24)*cos(d*x + c) - 4*((d*x + c)^3 - 6*d*x - 6*c
)*sin(d*x + c))*b^2*c^3/d^6 - 2*(((d*x + c)^4 - 12*(d*x + c)^2 + 24)*cos(d*x + c) - 4*((d*x + c)^3 - 6*d*x - 6
*c)*sin(d*x + c))*a*b/d^3 - 21*(((d*x + c)^5 - 20*(d*x + c)^3 + 120*d*x + 120*c)*cos(d*x + c) - 5*((d*x + c)^4
- 12*(d*x + c)^2 + 24)*sin(d*x + c))*b^2*c^2/d^6 + 7*(((d*x + c)^6 - 30*(d*x + c)^4 + 360*(d*x + c)^2 - 720)*
cos(d*x + c) - 6*((d*x + c)^5 - 20*(d*x + c)^3 + 120*d*x + 120*c)*sin(d*x + c))*b^2*c/d^6 - (((d*x + c)^7 - 42
*(d*x + c)^5 + 840*(d*x + c)^3 - 5040*d*x - 5040*c)*cos(d*x + c) - 7*((d*x + c)^6 - 30*(d*x + c)^4 + 360*(d*x
+ c)^2 - 720)*sin(d*x + c))*b^2/d^6)/d^2

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mupad [B]  time = 5.09, size = 225, normalized size = 0.96 \[ \frac {42\,b^2\,x^5\,\cos \left (c+d\,x\right )+24\,a\,b\,x^2\,\cos \left (c+d\,x\right )}{d^3}-\frac {b^2\,x^7\,\cos \left (c+d\,x\right )+a^2\,x\,\cos \left (c+d\,x\right )+2\,a\,b\,x^4\,\cos \left (c+d\,x\right )}{d}-\frac {840\,b^2\,x^3\,\cos \left (c+d\,x\right )+48\,a\,b\,\cos \left (c+d\,x\right )}{d^5}+\frac {a^2\,\sin \left (c+d\,x\right )+7\,b^2\,x^6\,\sin \left (c+d\,x\right )+8\,a\,b\,x^3\,\sin \left (c+d\,x\right )}{d^2}-\frac {210\,b^2\,x^4\,\sin \left (c+d\,x\right )+48\,a\,b\,x\,\sin \left (c+d\,x\right )}{d^4}-\frac {5040\,b^2\,\sin \left (c+d\,x\right )}{d^8}+\frac {2520\,b^2\,x^2\,\sin \left (c+d\,x\right )}{d^6}+\frac {5040\,b^2\,x\,\cos \left (c+d\,x\right )}{d^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sin(c + d*x)*(a + b*x^3)^2,x)

[Out]

(42*b^2*x^5*cos(c + d*x) + 24*a*b*x^2*cos(c + d*x))/d^3 - (b^2*x^7*cos(c + d*x) + a^2*x*cos(c + d*x) + 2*a*b*x
^4*cos(c + d*x))/d - (840*b^2*x^3*cos(c + d*x) + 48*a*b*cos(c + d*x))/d^5 + (a^2*sin(c + d*x) + 7*b^2*x^6*sin(
c + d*x) + 8*a*b*x^3*sin(c + d*x))/d^2 - (210*b^2*x^4*sin(c + d*x) + 48*a*b*x*sin(c + d*x))/d^4 - (5040*b^2*si
n(c + d*x))/d^8 + (2520*b^2*x^2*sin(c + d*x))/d^6 + (5040*b^2*x*cos(c + d*x))/d^7

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sympy [A]  time = 11.93, size = 284, normalized size = 1.21 \[ \begin {cases} - \frac {a^{2} x \cos {\left (c + d x \right )}}{d} + \frac {a^{2} \sin {\left (c + d x \right )}}{d^{2}} - \frac {2 a b x^{4} \cos {\left (c + d x \right )}}{d} + \frac {8 a b x^{3} \sin {\left (c + d x \right )}}{d^{2}} + \frac {24 a b x^{2} \cos {\left (c + d x \right )}}{d^{3}} - \frac {48 a b x \sin {\left (c + d x \right )}}{d^{4}} - \frac {48 a b \cos {\left (c + d x \right )}}{d^{5}} - \frac {b^{2} x^{7} \cos {\left (c + d x \right )}}{d} + \frac {7 b^{2} x^{6} \sin {\left (c + d x \right )}}{d^{2}} + \frac {42 b^{2} x^{5} \cos {\left (c + d x \right )}}{d^{3}} - \frac {210 b^{2} x^{4} \sin {\left (c + d x \right )}}{d^{4}} - \frac {840 b^{2} x^{3} \cos {\left (c + d x \right )}}{d^{5}} + \frac {2520 b^{2} x^{2} \sin {\left (c + d x \right )}}{d^{6}} + \frac {5040 b^{2} x \cos {\left (c + d x \right )}}{d^{7}} - \frac {5040 b^{2} \sin {\left (c + d x \right )}}{d^{8}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{2}}{2} + \frac {2 a b x^{5}}{5} + \frac {b^{2} x^{8}}{8}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*x*cos(c + d*x)/d + a**2*sin(c + d*x)/d**2 - 2*a*b*x**4*cos(c + d*x)/d + 8*a*b*x**3*sin(c + d*
x)/d**2 + 24*a*b*x**2*cos(c + d*x)/d**3 - 48*a*b*x*sin(c + d*x)/d**4 - 48*a*b*cos(c + d*x)/d**5 - b**2*x**7*co
s(c + d*x)/d + 7*b**2*x**6*sin(c + d*x)/d**2 + 42*b**2*x**5*cos(c + d*x)/d**3 - 210*b**2*x**4*sin(c + d*x)/d**
4 - 840*b**2*x**3*cos(c + d*x)/d**5 + 2520*b**2*x**2*sin(c + d*x)/d**6 + 5040*b**2*x*cos(c + d*x)/d**7 - 5040*
b**2*sin(c + d*x)/d**8, Ne(d, 0)), ((a**2*x**2/2 + 2*a*b*x**5/5 + b**2*x**8/8)*sin(c), True))

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