Optimal. Leaf size=188 \[ \frac {720 b^2 \cos (c+d x)}{d^7}-\frac {a^2 \cos (c+d x)}{d}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {720 b^2 x \sin (c+d x)}{d^6}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2} \]
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Rubi [A]
time = 0.17, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3410, 2718,
3377, 2717} \begin {gather*} -\frac {a^2 \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 {720 b^2 \cos (c+d x)}{d^7}+\frac {720 b^2 x \sin (c+d x)}{d^6}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {b^2 x^6 \cos (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 2718
Rule 3377
Rule 3410
Rubi steps
\begin {align*} \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 \sin (c+d x)+2 a b x^3 \sin (c+d x)+b^2 x^6 \sin (c+d x)\right ) \, dx\\ &=a^2 \int \sin (c+d x) \, dx+(2 a b) \int x^3 \sin (c+d x) \, dx+b^2 \int x^6 \sin (c+d x) \, dx\\ &=-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {(6 a b) \int x^2 \cos (c+d x) \, dx}{d}+\frac {\left (6 b^2\right ) \int x^5 \cos (c+d x) \, dx}{d}\\ &=-\frac {a^2 \cos (c+d x)}{d}-\frac {2 a b x^3 \cos (c+d x)}{d}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {6 a b x^2 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {(12 a b) \int x \sin (c+d x) \, dx}{d^2}-\frac {\left (30 b^2\right ) \int x^4 \sin (c+d x) \, dx}{d^2}\\ &=-\frac {a^2 \cos (c+d x)}{d}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {b^2 x^6 \cos (c+d x)}{d}+\frac {6 a b x^2 \sin (c+d x)}{d^2}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {(12 a b) \int \cos (c+d x) \, dx}{d^3}-\frac {\left (120 b^2\right ) \int x^3 \cos (c+d x) \, dx}{d^3}\\ &=-\frac {a^2 \cos (c+d x)}{d}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}+\frac {\left (360 b^2\right ) \int x^2 \sin (c+d x) \, dx}{d^4}\\ &=-\frac {a^2 \cos (c+d x)}{d}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}+\frac {\left (720 b^2\right ) \int x \cos (c+d x) \, dx}{d^5}\\ &=-\frac {a^2 \cos (c+d x)}{d}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {720 b^2 x \sin (c+d x)}{d^6}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}-\frac {\left (720 b^2\right ) \int \sin (c+d x) \, dx}{d^6}\\ &=\frac {720 b^2 \cos (c+d x)}{d^7}-\frac {a^2 \cos (c+d x)}{d}+\frac {12 a b x \cos (c+d x)}{d^3}-\frac {360 b^2 x^2 \cos (c+d x)}{d^5}-\frac {2 a b x^3 \cos (c+d x)}{d}+\frac {30 b^2 x^4 \cos (c+d x)}{d^3}-\frac {b^2 x^6 \cos (c+d x)}{d}-\frac {12 a b \sin (c+d x)}{d^4}+\frac {720 b^2 x \sin (c+d x)}{d^6}+\frac {6 a b x^2 \sin (c+d x)}{d^2}-\frac {120 b^2 x^3 \sin (c+d x)}{d^4}+\frac {6 b^2 x^5 \sin (c+d x)}{d^2}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 112, normalized size = 0.60 \begin {gather*} \frac {-\left (\left (a^2 d^6+2 a b d^4 x \left (-6+d^2 x^2\right )+b^2 \left (-720+360 d^2 x^2-30 d^4 x^4+d^6 x^6\right )\right ) \cos (c+d x)\right )+6 b d \left (a d^2 \left (-2+d^2 x^2\right )+b x \left (120-20 d^2 x^2+d^4 x^4\right )\right ) \sin (c+d x)}{d^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs.
\(2(188)=376\).
time = 0.08, size = 599, normalized size = 3.19 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 489 vs.
\(2 (188) = 376\).
time = 0.34, size = 489, normalized size = 2.60 \begin {gather*} -\frac {a^{2} \cos \left (d x + c\right ) + \frac {b^{2} c^{6} \cos \left (d x + c\right )}{d^{6}} - \frac {2 \, a b c^{3} \cos \left (d x + c\right )}{d^{3}} - \frac {6 \, {\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{5}}{d^{6}} + \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^{3}} + \frac {15 \, {\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^{4}}{d^{6}} - \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^{3}} - \frac {20 \, {\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^{3}}{d^{6}} + \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^{3}} + \frac {15 \, {\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^{2}}{d^{6}} - \frac {6 \, {\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}{d^{6}} + \frac {{\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}}{d^{6}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 129, normalized size = 0.69 \begin {gather*} -\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} - 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right ) - 6 \, {\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.64, size = 226, normalized size = 1.20 \begin {gather*} \begin {cases} - \frac {a^{2} \cos {\left (c + d x \right )}}{d} - \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^{6} \cos {\left (c + d x \right )}}{d} + \frac {6 b^{2} x^{5} \sin {\left (c + d x \right )}}{d^{2}} + \frac {30 b^{2} x^{4} \cos {\left (c + d x \right )}}{d^{3}} - \frac {120 b^{2} x^{3} \sin {\left (c + d x \right )}}{d^{4}} - \frac {360 b^{2} x^{2} \cos {\left (c + d x \right )}}{d^{5}} + \frac {720 b^{2} x \sin {\left (c + d x \right )}}{d^{6}} + \frac {720 b^{2} \cos {\left (c + d x \right )}}{d^{7}} & \text {for}\: d \neq 0 \\\left (a^{2} x + \frac {a b x^{4}}{2} + \frac {b^{2} x^{7}}{7}\right ) \sin {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.84, size = 131, normalized size = 0.70 \begin {gather*} -\frac {{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} - 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{7}} + \frac {6 \, {\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.62, size = 184, normalized size = 0.98 \begin {gather*} \frac {\cos \left (c+d\,x\right )\,\left (720\,b^2-a^2\,d^6\right )}{d^7}-\frac {b^2\,x^6\,\cos \left (c+d\,x\right )}{d}+\frac {30\,b^2\,x^4\,\cos \left (c+d\,x\right )}{d^3}-\frac {360\,b^2\,x^2\,\cos \left (c+d\,x\right )}{d^5}+\frac {6\,b^2\,x^5\,\sin \left (c+d\,x\right )}{d^2}-\frac {120\,b^2\,x^3\,\sin \left (c+d\,x\right )}{d^4}-\frac {12\,a\,b\,\sin \left (c+d\,x\right )}{d^4}+\frac {720\,b^2\,x\,\sin \left (c+d\,x\right )}{d^6}-\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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