3.2.3 \(\int \frac {x^2 \sin (c+d x)}{(a+b x^3)^2} \, dx\) [103]

Optimal. Leaf size=371 \[ -\frac {\sqrt [3]{-1} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}-\frac {\sin (c+d x)}{3 b \left (a+b x^3\right )}-\frac {\sqrt [3]{-1} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}-\frac {(-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}} \]

[Out]

1/9*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)-1/9*(-1)^(1/3)*d*Ci((-1)^(1/3)*a^(1/3
)*d/b^(1/3)-d*x)*cos(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)+1/9*(-1)^(2/3)*d*Ci((-1)^(2/3)*a^(1/3)*d/
b^(1/3)+d*x)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)-1/9*d*Si(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)
*d/b^(1/3))/a^(2/3)/b^(4/3)+1/9*(-1)^(1/3)*d*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c+(-1)^(1/3)*a^(1/3)*d/
b^(1/3))/a^(2/3)/b^(4/3)-1/9*(-1)^(2/3)*d*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1/3)*d/b^(1
/3))/a^(2/3)/b^(4/3)-1/3*sin(d*x+c)/b/(b*x^3+a)

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Rubi [A]
time = 0.37, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3422, 3415, 3384, 3380, 3383} \begin {gather*} -\frac {\sqrt [3]{-1} d \cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {CosIntegral}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}-\frac {\sqrt [3]{-1} d \sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac {(-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (x d+\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac {\sin (c+d x)}{3 b \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*((-1)^(1/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/
(a^(2/3)*b^(4/3)) + (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)
) + ((-1)^(2/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(
9*a^(2/3)*b^(4/3)) - Sin[c + d*x]/(3*b*(a + b*x^3)) - ((-1)^(1/3)*d*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*Si
nIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) - (d*Sin[c - (a^(1/3)*d)/b^(1/3)]*SinInte
gral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(2/3)*d*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*S
inIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3))

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

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

Rule 3415

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 3422

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[e^m*(a + b*x^n
)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))), x] - Dist[d*(e^m/(b*n*(p + 1))), Int[(a + b*x^n)^(p + 1)*Cos[c + d*x],
x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && (IntegerQ[n] || GtQ[e, 0])

Rubi steps

\begin {align*} \int \frac {x^2 \sin (c+d x)}{\left (a+b x^3\right )^2} \, dx &=-\frac {\sin (c+d x)}{3 b \left (a+b x^3\right )}+\frac {d \int \frac {\cos (c+d x)}{a+b x^3} \, dx}{3 b}\\ &=-\frac {\sin (c+d x)}{3 b \left (a+b x^3\right )}+\frac {d \int \left (-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac {\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{3 b}\\ &=-\frac {\sin (c+d x)}{3 b \left (a+b x^3\right )}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac {d \int \frac {\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}\\ &=-\frac {\sin (c+d x)}{3 b \left (a+b x^3\right )}-\frac {\left (d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac {\left (d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac {\left (d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\cos \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}+\frac {\left (d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac {\left (d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}+\frac {\left (d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {\sin \left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}\\ &=-\frac {\sqrt [3]{-1} d \cos \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac {d \cos \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac {(-1)^{2/3} d \cos \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Ci}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}-\frac {\sin (c+d x)}{3 b \left (a+b x^3\right )}-\frac {\sqrt [3]{-1} d \sin \left (c+\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}-\frac {d \sin \left (c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}-\frac {(-1)^{2/3} d \sin \left (c-\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text {Si}\left (\frac {(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.13, size = 214, normalized size = 0.58 \begin {gather*} \frac {d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]+d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {\cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-\sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))}{\text {$\#$1}^2}\&\right ]-\frac {6 b \sin (c+d x)}{a+b x^3}}{18 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*RootSum[a + b*#1^3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] - I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - I
*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] + d*RootSum[a + b*#1^
3 & , (Cos[c + d*#1]*CosIntegral[d*(x - #1)] + I*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + I*Cos[c + d*#1]*SinIn
tegral[d*(x - #1)] - Sin[c + d*#1]*SinIntegral[d*(x - #1)])/#1^2 & ] - (6*b*Sin[c + d*x])/(a + b*x^3))/(18*b^2
)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.10, size = 823, normalized size = 2.22

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(823\)
default	\(\text {Expression too large to display}\)	\(823\)
risch	\(-\frac {\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-3 i b \textit {\_R1} \,c^{2}+2 \textit {\_R1}^{2} b c +a \,d^{3}-b \,c^{3}-2 b \textit {\_R1} c \right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}}{18 a \,b^{2}}-\frac {c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} +c -2 i\right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{18 a b}-\frac {c \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-i c \textit {\_R1} +\textit {\_R1}^{2}-i c -\textit {\_R1} \right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{9 a b}-\frac {\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-3 i b \textit {\_R1} \,c^{2}+2 \textit {\_R1}^{2} b c +a \,d^{3}-b \,c^{3}+2 b \textit {\_R1} c \right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}}{18 a \,b^{2}}-\frac {c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} +c +2 i\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{18 a b}-\frac {c \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-i c \textit {\_R1} +\textit {\_R1}^{2}+i c +\textit {\_R1} \right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{9 a b}+\frac {i \left (\frac {i \left (-3 i b \left (i d x +i c \right ) c^{2}+2 \left (i d x +i c \right )^{2} b c +a \,d^{3}-b \,c^{3}\right ) d^{3}}{3 \left (i b \left (i d x +i c \right )^{3}-3 i b \left (i d x +i c \right ) c^{2}+3 \left (i d x +i c \right )^{2} b c +a \,d^{3}-b \,c^{3}\right ) b a}-\frac {i c^{2} d^{3} \left (c +i \left (i d x +i c \right )\right )}{3 \left (-3 \left (i d x +i c \right )^{2} b c -a \,d^{3}+b \,c^{3}-i b \left (i d x +i c \right )^{3}+3 i b \left (i d x +i c \right ) c^{2}\right ) a}+\frac {2 c \left (i d x +i c \right ) d^{3} \left (c +i \left (i d x +i c \right )\right )}{3 \left (-3 \left (i d x +i c \right )^{2} b c -a \,d^{3}+b \,c^{3}-i b \left (i d x +i c \right )^{3}+3 i b \left (i d x +i c \right ) c^{2}\right ) a}\right ) \sin \left (d x +c \right )}{d^{3}}\)	\(926\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(d*x+c)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/d^3*(d^6*c^2*(sin(d*x+c)*(1/3/a/d^3*(d*x+c)-1/3*c/a/d^3)/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x
+c)^3)+2/9/a/d^3/b*sum(1/(_R1^2-2*_R1*c+c^2)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_R1=RootOf(_Z^3
*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/9/a/d^3/b*sum(1/(-_RR1+c)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*c
os(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3)))+sin(d*x+c)*(-2/3*d^3*c/a*(d*x+c)^2+2/3*c^2*d
^3/a*(d*x+c))/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^3)-2/9*d^3*c/a/b*sum((c+_R1)/(_R1^2-2*_R1
*c+c^2)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3)
)-2/9*d^3*c/a/b*sum(_RR1/(-_RR1+c)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=RootOf(_Z^3*b-3*_
Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+sin(d*x+c)*(2/3*d^3*c/a*(d*x+c)^2-c^2*d^3/a*(d*x+c)-1/3*d^3*(a*d^3-b*c^3)/a/b
)/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^3)+2/9*d^3*c/a/b*sum(_R1/(_R1^2-2*_R1*c+c^2)*(-Si(-d*
x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/9*d^3/a/b^2*
sum((-2*_RR1^2*b*c+3*_RR1*b*c^2+a*d^3-b*c^3)/(_RR1^2-2*_RR1*c+c^2)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*c
os(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((cos(c)^2 + sin(c)^2)*d*x^2*cos(d*x + c) + 4*(cos(c)^2 + sin(c)^2)*x*sin(d*x + c) + ((d*x^2*cos(c) - 4*x
*sin(c))*cos(d*x + c)^2 + (d*x^2*cos(c) - 4*x*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + 2*(((b^2*cos(c)^2 + b^2
*sin(c)^2)*d^2*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^2*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^2)*cos(d*x + c)
^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^2*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^2*x^3 + (a^2*cos(c)^2 + a^2*si
n(c)^2)*d^2)*sin(d*x + c)^2)*integrate(-(3*a*d*x*cos(d*x + c) - 2*(5*b*x^3 - a)*sin(d*x + c))/(b^3*d^2*x^9 + 3
*a*b^2*d^2*x^6 + 3*a^2*b*d^2*x^3 + a^3*d^2), x) + 2*(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^2*x^6 + 2*(a*b*cos(c)^2
+ a*b*sin(c)^2)*d^2*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d
^2*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^2*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^2)*sin(d*x + c)^2)*integrat
e(-(3*a*d*x*cos(d*x + c) - 2*(5*b*x^3 - a)*sin(d*x + c))/((b^3*d^2*x^9 + 3*a*b^2*d^2*x^6 + 3*a^2*b*d^2*x^3 + a
^3*d^2)*cos(d*x + c)^2 + (b^3*d^2*x^9 + 3*a*b^2*d^2*x^6 + 3*a^2*b*d^2*x^3 + a^3*d^2)*sin(d*x + c)^2), x) + ((d
*x^2*sin(c) + 4*x*cos(c))*cos(d*x + c)^2 + (d*x^2*sin(c) + 4*x*cos(c))*sin(d*x + c)^2)*sin(d*x + 2*c))/(((b^2*
cos(c)^2 + b^2*sin(c)^2)*d^2*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^2*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^2
)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^2*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^2*x^3 + (a^2*cos
(c)^2 + a^2*sin(c)^2)*d^2)*sin(d*x + c)^2)

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Fricas [C] Result contains complex when optimal does not.
time = 0.41, size = 482, normalized size = 1.30 \begin {gather*} \frac {{\left (-i \, b x^{3} + \sqrt {3} {\left (b x^{3} + a\right )} - i \, a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (i \, b x^{3} - \sqrt {3} {\left (b x^{3} + a\right )} + i \, a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} + {\left (-i \, b x^{3} - \sqrt {3} {\left (b x^{3} + a\right )} - i \, a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left (i \, b x^{3} + \sqrt {3} {\left (b x^{3} + a\right )} + i \, a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} - 2 \, {\left (i \, b x^{3} + i \, a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - 2 \, {\left (-i \, b x^{3} - i \, a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - 12 \, a \sin \left (d x + c\right )}{36 \, {\left (a b^{2} x^{3} + a^{2} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*((-I*b*x^3 + sqrt(3)*(b*x^3 + a) - I*a)*(I*a*d^3/b)^(1/3)*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt(3) -
1))*e^(1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) - I*c) + (I*b*x^3 - sqrt(3)*(b*x^3 + a) + I*a)*(-I*a*d^3/b)^(1/3
)*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) + I*c) + (-I*b
*x^3 - sqrt(3)*(b*x^3 + a) - I*a)*(I*a*d^3/b)^(1/3)*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*
(I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) - I*c) + (I*b*x^3 + sqrt(3)*(b*x^3 + a) + I*a)*(-I*a*d^3/b)^(1/3)*Ei(I*d*x
+ 1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) + 1) + I*c) - 2*(I*b*x^3 + I*a
)*(-I*a*d^3/b)^(1/3)*Ei(I*d*x + (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) - 2*(-I*b*x^3 - I*a)*(I*a*d^3
/b)^(1/3)*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)) - 12*a*sin(d*x + c))/(a*b^2*x^3 + a^2*b)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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