Optimal. Leaf size=431 \[ \frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 b^2}+\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}+\frac{\sqrt{-a} d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{\sqrt{-a} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}+\frac{\sqrt{-a} d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}+\frac{\sqrt{-a} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\sin (c+d x)}{2 b^2} \]
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Rubi [A] time = 0.661348, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {3343, 3345, 3303, 3299, 3302, 3346, 2637, 3334} \[ \frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 b^2}+\frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}+\frac{\sqrt{-a} d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{\sqrt{-a} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}+\frac{\sqrt{-a} d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}+\frac{\sqrt{-a} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{4 b^{5/2}}-\frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\sin (c+d x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 3343
Rule 3345
Rule 3303
Rule 3299
Rule 3302
Rule 3346
Rule 2637
Rule 3334
Rubi steps
\begin{align*} \int \frac{x^3 \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x \sin (c+d x)}{a+b x^2} \, dx}{b}+\frac{d \int \frac{x^2 \cos (c+d x)}{a+b x^2} \, dx}{2 b}\\ &=-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}+\frac{\int \left (-\frac{\sin (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sin (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{b}+\frac{d \int \left (\frac{\cos (c+d x)}{b}-\frac{a \cos (c+d x)}{b \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac{\int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 b^{3/2}}+\frac{\int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 b^{3/2}}+\frac{d \int \cos (c+d x) \, dx}{2 b^2}-\frac{(a d) \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{2 b^2}\\ &=\frac{\sin (c+d x)}{2 b^2}-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac{(a d) \int \left (\frac{\sqrt{-a} \cos (c+d x)}{2 a \left (\sqrt{-a}-\sqrt{b} x\right )}+\frac{\sqrt{-a} \cos (c+d x)}{2 a \left (\sqrt{-a}+\sqrt{b} x\right )}\right ) \, dx}{2 b^2}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 b^{3/2}}+\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 b^{3/2}}+\frac{\sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{\sqrt{-a}+\sqrt{b} x} \, dx}{2 b^{3/2}}-\frac{\sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \int \frac{\cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{\sqrt{-a}-\sqrt{b} x} \, dx}{2 b^{3/2}}\\ &=\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{\sin (c+d x)}{2 b^2}-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 b^2}-\frac{\left (\sqrt{-a} d\right ) \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{4 b^2}-\frac{\left (\sqrt{-a} d\right ) \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{4 b^2}\\ &=\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{\sin (c+d x)}{2 b^2}-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 b^2}-\frac{\left (\sqrt{-a} 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}{4 b^2}-\frac{\left (\sqrt{-a} 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}{4 b^2}+\frac{\left (\sqrt{-a} 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}{4 b^2}-\frac{\left (\sqrt{-a} 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}{4 b^2}\\ &=\frac{\sqrt{-a} d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}-\frac{\sqrt{-a} d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}+\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right ) \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{\text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right ) \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{2 b^2}+\frac{\sin (c+d x)}{2 b^2}-\frac{x^2 \sin (c+d x)}{2 b \left (a+b x^2\right )}-\frac{\cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{2 b^2}+\frac{\sqrt{-a} d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{4 b^{5/2}}+\frac{\cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{2 b^2}+\frac{\sqrt{-a} d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{4 b^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.852512, size = 583, normalized size = 1.35 \[ \frac{i a^{3/2} d \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i a^{3/2} d \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+2 b^{3/2} x^2 \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-2 b^{3/2} x^2 \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+\left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (2 \sqrt{b} \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )-i \sqrt{a} d \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )+\left (a+b x^2\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right ) \left (2 \sqrt{b} \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )+i \sqrt{a} d \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right )\right )+i \sqrt{a} b d x^2 \sin \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i \sqrt{a} b d x^2 \sin \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+2 a \sqrt{b} \cos \left (c-\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (d \left (x+\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )-2 a \sqrt{b} \cos \left (c+\frac{i \sqrt{a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{i \sqrt{a} d}{\sqrt{b}}-d x\right )+2 a \sqrt{b} \sin (c+d x)}{4 b^{5/2} \left (a+b x^2\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.076, size = 2563, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.88757, size = 641, normalized size = 1.49 \begin{align*} \frac{{\left (-4 i \, b x^{2} + 2 \,{\left (-i \, b x^{2} - i \, a\right )} \sqrt{\frac{a d^{2}}{b}} - 4 i \, a\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 (-4 i \, b x^{2} + 2 \,{\left (i \, b x^{2} + i \, a\right )} \sqrt{\frac{a d^{2}}{b}} - 4 i \, a\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 (4 i \, b x^{2} + 2 \,{\left (i \, b x^{2} + i \, a\right )} \sqrt{\frac{a d^{2}}{b}} + 4 i \, a\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 (4 i \, b x^{2} + 2 \,{\left (-i \, b x^{2} - i \, a\right )} \sqrt{\frac{a d^{2}}{b}} + 4 i \, a\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 \, a \sin \left (d x + c\right )}{16 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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