Optimal. Leaf size=161 \[ a^2 \sin (c) \text {Ci}(d x)+a^2 \cos (c) \text {Si}(d x)+\frac {4 a b \cos (c+d x)}{d^3}+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {120 b^2 \sin (c+d x)}{d^6}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {b^2 x^5 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.26, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3339, 3303, 3299, 3302, 3296, 2638, 2637} \[ a^2 \sin (c) \text {CosIntegral}(d x)+a^2 \cos (c) \text {Si}(d x)+\frac {4 a b x \sin (c+d x)}{d^2}+\frac {4 a b \cos (c+d x)}{d^3}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}+\frac {120 b^2 \sin (c+d x)}{d^6}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {b^2 x^5 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3299
Rule 3302
Rule 3303
Rule 3339
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^2 \sin (c+d x)}{x} \, dx &=\int \left (\frac {a^2 \sin (c+d x)}{x}+2 a b x^2 \sin (c+d x)+b^2 x^5 \sin (c+d x)\right ) \, dx\\ &=a^2 \int \frac {\sin (c+d x)}{x} \, dx+(2 a b) \int x^2 \sin (c+d x) \, dx+b^2 \int x^5 \sin (c+d x) \, dx\\ &=-\frac {2 a b x^2 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+\frac {(4 a b) \int x \cos (c+d x) \, dx}{d}+\frac {\left (5 b^2\right ) \int x^4 \cos (c+d x) \, dx}{d}+\left (a^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx+\left (a^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {2 a b x^2 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \text {Ci}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)-\frac {(4 a b) \int \sin (c+d x) \, dx}{d^2}-\frac {\left (20 b^2\right ) \int x^3 \sin (c+d x) \, dx}{d^2}\\ &=\frac {4 a b \cos (c+d x)}{d^3}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \text {Ci}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)-\frac {\left (60 b^2\right ) \int x^2 \cos (c+d x) \, dx}{d^3}\\ &=\frac {4 a b \cos (c+d x)}{d^3}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \text {Ci}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)+\frac {\left (120 b^2\right ) \int x \sin (c+d x) \, dx}{d^4}\\ &=\frac {4 a b \cos (c+d x)}{d^3}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \text {Ci}(d x) \sin (c)+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)+\frac {\left (120 b^2\right ) \int \cos (c+d x) \, dx}{d^5}\\ &=\frac {4 a b \cos (c+d x)}{d^3}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {2 a b x^2 \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {b^2 x^5 \cos (c+d x)}{d}+a^2 \text {Ci}(d x) \sin (c)+\frac {120 b^2 \sin (c+d x)}{d^6}+\frac {4 a b x \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+a^2 \cos (c) \text {Si}(d x)\\ \end {align*}
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Mathematica [A] time = 0.53, size = 108, normalized size = 0.67 \[ a^2 \sin (c) \text {Ci}(d x)+a^2 \cos (c) \text {Si}(d x)+\frac {b \left (4 a d^4 x+5 b \left (d^4 x^4-12 d^2 x^2+24\right )\right ) \sin (c+d x)}{d^6}-\frac {b \left (2 a d^2 \left (d^2 x^2-2\right )+b x \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \cos (c+d x)}{d^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 145, normalized size = 0.90 \[ \frac {2 \, a^{2} d^{6} \cos \relax (c) \operatorname {Si}\left (d x\right ) - 2 \, {\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x - 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right ) + {\left (a^{2} d^{6} \operatorname {Ci}\left (d x\right ) + a^{2} d^{6} \operatorname {Ci}\left (-d x\right )\right )} \sin \relax (c)}{2 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.36, size = 921, normalized size = 5.72 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 487, normalized size = 3.02 \[ \frac {\left (c^{5}+c^{4}+c^{3}+c^{2}+c +1\right ) b^{2} \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{6}}-\frac {6 b^{2} c \left (c^{4}+c^{3}+c^{2}+c +1\right ) \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}+\frac {15 \left (c^{3}+c^{2}+c +1\right ) c^{2} b^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{6}}+\frac {2 \left (c^{2}+c +1\right ) a b \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{3}}-\frac {20 b^{2} c^{3} \left (c^{2}+c +1\right ) \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{6}}-\frac {6 c a b \left (1+c \right ) \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{3}}+\frac {15 \left (1+c \right ) b^{2} c^{4} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{6}}-\frac {6 c^{2} a b \cos \left (d x +c \right )}{d^{3}}+\frac {6 c^{5} b^{2} \cos \left (d x +c \right )}{d^{6}}+a^{2} \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 12.02, size = 147, normalized size = 0.91 \[ \frac {{\left (a^{2} {\left (-i \, {\rm Ei}\left (i \, d x\right ) + i \, {\rm Ei}\left (-i \, d x\right )\right )} \cos \relax (c) + a^{2} {\left ({\rm Ei}\left (i \, d x\right ) + {\rm Ei}\left (-i \, d x\right )\right )} \sin \relax (c)\right )} d^{6} - 2 \, {\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 4 \, a b d^{3} + 120 \, b^{2} d x\right )} \cos \left (d x + c\right ) + 2 \, {\left (5 \, b^{2} d^{4} x^{4} + 4 \, a b d^{4} x - 60 \, b^{2} d^{2} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.64, size = 211, normalized size = 1.31 \[ a^{2} \sin {\relax (c )} \operatorname {Ci}{\left (d x \right )} + a^{2} \cos {\relax (c )} \operatorname {Si}{\left (d x \right )} + 2 a b x^{2} \left (\begin {cases} - \cos {\relax (c )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 4 a b \left (\begin {cases} - \frac {x^{2} \cos {\relax (c )}}{2} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {x \sin {\left (c + d x \right )}}{d} + \frac {\cos {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \cos {\relax (c )}}{2} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) + b^{2} x^{5} \left (\begin {cases} - \cos {\relax (c )} & \text {for}\: d = 0 \\- \frac {\cos {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right ) - 5 b^{2} \left (\begin {cases} - \frac {x^{5} \cos {\relax (c )}}{5} & \text {for}\: d = 0 \\- \frac {\begin {cases} \frac {x^{4} \sin {\left (c + d x \right )}}{d} + \frac {4 x^{3} \cos {\left (c + d x \right )}}{d^{2}} - \frac {12 x^{2} \sin {\left (c + d x \right )}}{d^{3}} - \frac {24 x \cos {\left (c + d x \right )}}{d^{4}} + \frac {24 \sin {\left (c + d x \right )}}{d^{5}} & \text {for}\: d \neq 0 \\\frac {x^{5} \cos {\relax (c )}}{5} & \text {otherwise} \end {cases}}{d} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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