Optimal. Leaf size=512 \[ -\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d^2 \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.77, antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3341, 3334, 3297, 3303, 3299, 3302} \[ \frac {d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {d^2 \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d^2 \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {d \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 (-a)^{3/2} b^{3/2}}-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3334
Rule 3341
Rubi steps
\begin {align*} \int \frac {x \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx &=-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \frac {\cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \left (-\frac {b \cos (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}-b x\right )^2}-\frac {b \cos (c+d x)}{4 a \left (\sqrt {-a} \sqrt {b}+b x\right )^2}-\frac {b \cos (c+d x)}{2 a \left (-a b-b^2 x^2\right )}\right ) \, dx}{4 b}\\ &=-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}-b x\right )^2} \, dx}{16 a}-\frac {d \int \frac {\cos (c+d x)}{\left (\sqrt {-a} \sqrt {b}+b x\right )^2} \, dx}{16 a}-\frac {d \int \frac {\cos (c+d x)}{-a b-b^2 x^2} \, dx}{8 a}\\ &=-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d \int \left (-\frac {\sqrt {-a} \cos (c+d x)}{2 a b \left (\sqrt {-a}-\sqrt {b} x\right )}-\frac {\sqrt {-a} \cos (c+d x)}{2 a b \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{8 a}-\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b}+\frac {d^2 \int \frac {\sin (c+d x)}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b}\\ &=-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {d \int \frac {\cos (c+d x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\left (d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b}+\frac {\left (d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b}+\frac {\left (d^2 \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a} \sqrt {b}+b x} \, dx}{16 a b}-\frac {\left (d^2 \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a} \sqrt {b}-b x} \, dx}{16 a b}\\ &=-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}+\frac {\left (d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\left (d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}-\frac {\left (d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{\sqrt {-a}+\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}+\frac {\left (d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )\right ) \int \frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{\sqrt {-a}-\sqrt {b} x} \, dx}{16 (-a)^{3/2} b}\\ &=-\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {d \cos (c+d x)}{16 a b^{3/2} \left (\sqrt {-a}+\sqrt {b} x\right )}-\frac {d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}+\frac {d^2 \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 a b^2}-\frac {\sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 a b^2}-\frac {d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 (-a)^{3/2} b^{3/2}}+\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 a b^2}-\frac {d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 (-a)^{3/2} b^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.82, size = 634, normalized size = 1.24 \[ \frac {\frac {d^2 \cos (c) \left (i \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-i \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )}{b}+\frac {d^2 \sin (c) \left (\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+i \sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )}{b}+\frac {d \cos (c) \left (-i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )-\text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )\right )\right )}{\sqrt {a} \sqrt {b}}-\frac {d \sin (c) \left (\sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\sinh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+i \cosh \left (\frac {\sqrt {a} d}{\sqrt {b}}\right ) \left (\text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+\text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )\right )\right )}{\sqrt {a} \sqrt {b}}+\frac {2 \cos (d x) \left (d x \cos (c) \left (a+b x^2\right )-2 a \sin (c)\right )}{\left (a+b x^2\right )^2}-\frac {2 \sin (d x) \left (d x \sin (c) \left (a+b x^2\right )+2 a \cos (c)\right )}{\left (a+b x^2\right )^2}}{16 a b} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.68, size = 487, normalized size = 0.95 \[ -\frac {16 \, a^{2} b \sin \left (d x + c\right ) - {\left (-2 i \, a b^{2} d^{2} x^{4} - 4 i \, a^{2} b d^{2} x^{2} - 2 i \, a^{3} d^{2} + 2 \, {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (-2 i \, a b^{2} d^{2} x^{4} - 4 i \, a^{2} b d^{2} x^{2} - 2 i \, a^{3} d^{2} + 2 \, {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (2 i \, a b^{2} d^{2} x^{4} + 4 i \, a^{2} b d^{2} x^{2} + 2 i \, a^{3} d^{2} + 2 \, {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - {\left (2 i \, a b^{2} d^{2} x^{4} + 4 i \, a^{2} b d^{2} x^{2} + 2 i \, a^{3} d^{2} + 2 \, {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 8 \, {\left (a b^{2} d x^{3} + a^{2} b d x\right )} \cos \left (d x + c\right )}{64 \, {\left (a^{2} b^{4} x^{4} + 2 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1374, normalized size = 2.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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