Optimal. Leaf size=185 \[ \frac {a^2 \sin (c+d x)}{d^2}-\frac {a^2 x \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {12 a b x \cos (c+d x)}{d^3}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {120 b^2 \sin (c+d x)}{d^6}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {b^2 x^5 \cos (c+d x)}{d} \]
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Rubi [A] time = 0.24, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3339, 3296, 2637} \[ \frac {a^2 \sin (c+d x)}{d^2}-\frac {a^2 x \cos (c+d x)}{d}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}+\frac {120 b^2 \sin (c+d x)}{d^6}-\frac {120 b^2 x \cos (c+d x)}{d^5}-\frac {b^2 x^5 \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3339
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 x \sin (c+d x)+2 a b x^3 \sin (c+d x)+b^2 x^5 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x \sin (c+d x) \, dx+(2 a b) \int x^3 \sin (c+d x) \, dx+b^2 \int x^5 \sin (c+d x) \, dx\\ &=-\frac {a^2 x \cos (c+d x)}{d}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+\frac {a^2 \int \cos (c+d x) \, dx}{d}+\frac {(6 a b) \int x^2 \cos (c+d x) \, dx}{d}+\frac {\left (5 b^2\right ) \int x^4 \cos (c+d x) \, dx}{d}\\ &=-\frac {a^2 x \cos (c+d x)}{d}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}+\frac {6 a b x^2 \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {(12 a b) \int x \sin (c+d x) \, dx}{d^2}-\frac {\left (20 b^2\right ) \int x^3 \sin (c+d x) \, dx}{d^2}\\ &=\frac {12 a b x \cos (c+d x)}{d^3}-\frac {a^2 x \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+\frac {a^2 \sin (c+d x)}{d^2}+\frac {6 a b x^2 \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}-\frac {(12 a b) \int \cos (c+d x) \, dx}{d^3}-\frac {\left (60 b^2\right ) \int x^2 \cos (c+d x) \, dx}{d^3}\\ &=\frac {12 a b x \cos (c+d x)}{d^3}-\frac {a^2 x \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {6 a b x^2 \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+\frac {\left (120 b^2\right ) \int x \sin (c+d x) \, dx}{d^4}\\ &=-\frac {120 b^2 x \cos (c+d x)}{d^5}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {a^2 x \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {6 a b x^2 \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}+\frac {\left (120 b^2\right ) \int \cos (c+d x) \, dx}{d^5}\\ &=-\frac {120 b^2 x \cos (c+d x)}{d^5}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {a^2 x \cos (c+d x)}{d}+\frac {20 b^2 x^3 \cos (c+d x)}{d^3}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^5 \cos (c+d x)}{d}+\frac {120 b^2 \sin (c+d x)}{d^6}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {a^2 \sin (c+d x)}{d^2}-\frac {60 b^2 x^2 \sin (c+d x)}{d^4}+\frac {6 a b x^2 \sin (c+d x)}{d^2}+\frac {5 b^2 x^4 \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 113, normalized size = 0.61 \[ \frac {\left (a^2 d^4+6 a b d^2 \left (d^2 x^2-2\right )+5 b^2 \left (d^4 x^4-12 d^2 x^2+24\right )\right ) \sin (c+d x)-d x \left (a^2 d^4+2 a b d^2 \left (d^2 x^2-6\right )+b^2 \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \cos (c+d x)}{d^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 126, normalized size = 0.68 \[ -\frac {{\left (b^{2} d^{5} x^{5} + 2 \, {\left (a b d^{5} - 10 \, b^{2} d^{3}\right )} x^{3} + {\left (a^{2} d^{5} - 12 \, a b d^{3} + 120 \, b^{2} d\right )} x\right )} \cos \left (d x + c\right ) - {\left (5 \, b^{2} d^{4} x^{4} + a^{2} d^{4} - 12 \, a b d^{2} + 6 \, {\left (a b d^{4} - 10 \, b^{2} d^{2}\right )} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 129, normalized size = 0.70 \[ -\frac {{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} + a^{2} d^{5} x - 20 \, b^{2} d^{3} x^{3} - 12 \, a b d^{3} x + 120 \, b^{2} d x\right )} \cos \left (d x + c\right )}{d^{6}} + \frac {{\left (5 \, b^{2} d^{4} x^{4} + 6 \, a b d^{4} x^{2} + a^{2} d^{4} - 60 \, b^{2} d^{2} x^{2} - 12 \, a b d^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 514, normalized size = 2.78 \[ \frac {\frac {b^{2} \left (-\left (d x +c \right )^{5} \cos \left (d x +c \right )+5 \left (d x +c \right )^{4} \sin \left (d x +c \right )+20 \left (d x +c \right )^{3} \cos \left (d x +c \right )-60 \left (d x +c \right )^{2} \sin \left (d x +c \right )+120 \sin \left (d x +c \right )-120 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{4}}-\frac {5 b^{2} c \left (-\left (d x +c \right )^{4} \cos \left (d x +c \right )+4 \left (d x +c \right )^{3} \sin \left (d x +c \right )+12 \left (d x +c \right )^{2} \cos \left (d x +c \right )-24 \cos \left (d x +c \right )-24 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}+\frac {2 a b \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}+\frac {10 b^{2} c^{2} \left (-\left (d x +c \right )^{3} \cos \left (d x +c \right )+3 \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )+6 \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{4}}-\frac {6 a b c \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{2}}-\frac {10 b^{2} c^{3} \left (-\left (d x +c \right )^{2} \cos \left (d x +c \right )+2 \cos \left (d x +c \right )+2 \left (d x +c \right ) \sin \left (d x +c \right )\right )}{d^{4}}+a^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )+\frac {6 a b \,c^{2} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{2}}+\frac {5 b^{2} c^{4} \left (\sin \left (d x +c \right )-\left (d x +c \right ) \cos \left (d x +c \right )\right )}{d^{4}}+a^{2} c \cos \left (d x +c \right )+\frac {2 a b \,c^{3} \cos \left (d x +c \right )}{d^{2}}+\frac {b^{2} c^{5} \cos \left (d x +c \right )}{d^{4}}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 438, normalized size = 2.37 \[ \frac {a^{2} c \cos \left (d x + c\right ) + \frac {b^{2} c^{5} \cos \left (d x + c\right )}{d^{4}} + \frac {2 \, a b c^{3} \cos \left (d x + c\right )}{d^{2}} - {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a^{2} - \frac {5 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{4}}{d^{4}} - \frac {6 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b c^{2}}{d^{2}} + \frac {10 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} b^{2} c^{3}}{d^{4}} + \frac {6 \, {\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \, {\left (d x + c\right )} \sin \left (d x + c\right )\right )} a b c}{d^{2}} - \frac {10 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b^{2} c^{2}}{d^{4}} - \frac {2 \, {\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} a b}{d^{2}} + \frac {5 \, {\left ({\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \, {\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b^{2} c}{d^{4}} - \frac {{\left ({\left ({\left (d x + c\right )}^{5} - 20 \, {\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \, {\left ({\left (d x + c\right )}^{4} - 12 \, {\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b^{2}}{d^{4}}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.93, size = 151, normalized size = 0.82 \[ \frac {\sin \left (c+d\,x\right )\,\left (a^2\,d^4-12\,a\,b\,d^2+120\,b^2\right )}{d^6}-\frac {b^2\,x^5\,\cos \left (c+d\,x\right )}{d}+\frac {5\,b^2\,x^4\,\sin \left (c+d\,x\right )}{d^2}-\frac {x\,\cos \left (c+d\,x\right )\,\left (a^2\,d^4-12\,a\,b\,d^2+120\,b^2\right )}{d^5}+\frac {2\,x^3\,\cos \left (c+d\,x\right )\,\left (10\,b^2-a\,b\,d^2\right )}{d^3}-\frac {6\,x^2\,\sin \left (c+d\,x\right )\,\left (10\,b^2-a\,b\,d^2\right )}{d^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.55, size = 226, normalized size = 1.22 \[ \begin {cases} - \frac {a^{2} x \cos {\left (c + d x \right )}}{d} + \frac {a^{2} \sin {\left (c + d x \right )}}{d^{2}} - \frac {2 a b x^{3} \cos {\left (c + d x \right )}}{d} + \frac {6 a b x^{2} \sin {\left (c + d x \right )}}{d^{2}} + \frac {12 a b x \cos {\left (c + d x \right )}}{d^{3}} - \frac {12 a b \sin {\left (c + d x \right )}}{d^{4}} - \frac {b^{2} x^{5} \cos {\left (c + d x \right )}}{d} + \frac {5 b^{2} x^{4} \sin {\left (c + d x \right )}}{d^{2}} + \frac {20 b^{2} x^{3} \cos {\left (c + d x \right )}}{d^{3}} - \frac {60 b^{2} x^{2} \sin {\left (c + d x \right )}}{d^{4}} - \frac {120 b^{2} x \cos {\left (c + d x \right )}}{d^{5}} + \frac {120 b^{2} \sin {\left (c + d x \right )}}{d^{6}} & \text {for}\: d \neq 0 \\\left (\frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6}\right ) \sin {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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