Optimal. Leaf size=476 \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 b^3}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}+\frac{3 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 b^3}+\frac{3 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{3 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 1.0076, antiderivative size = 476, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {3343, 3341, 3334, 3303, 3299, 3302, 3344, 3345} \[ -\frac{d^2 \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 b^3}-\frac{d^2 \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}+\frac{3 d \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{d^2 \cos \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 b^3}-\frac{d^2 \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 b^3}+\frac{3 d \sin \left (\frac{\sqrt{-a} d}{\sqrt{b}}+c\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}+\frac{3 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (x d+\frac{\sqrt{-a} d}{\sqrt{b}}\right )}{16 \sqrt{-a} b^{5/2}}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3343
Rule 3341
Rule 3334
Rule 3303
Rule 3299
Rule 3302
Rule 3344
Rule 3345
Rubi steps
\begin{align*} \int \frac{x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx &=-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}+\frac{\int \frac{x \sin (c+d x)}{\left (a+b x^2\right )^2} \, dx}{2 b}+\frac{d \int \frac{x^2 \cos (c+d x)}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac{d \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{8 b^2}+\frac{d \int \frac{\cos (c+d x)}{a+b x^2} \, dx}{4 b^2}-\frac{d^2 \int \frac{x \sin (c+d x)}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac{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}{8 b^2}+\frac{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}{4 b^2}-\frac{d^2 \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}{8 b^2}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 \sqrt{-a} b^2}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 \sqrt{-a} b^2}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{8 \sqrt{-a} b^2}-\frac{d \int \frac{\cos (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{8 \sqrt{-a} b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt{-a}-\sqrt{b} x} \, dx}{16 b^{5/2}}-\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt{-a}+\sqrt{b} x} \, dx}{16 b^{5/2}}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}-\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 \sqrt{-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}{8 \sqrt{-a} b^2}-\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} x} \, dx}{16 b^{5/2}}-\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 \sqrt{-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}{8 \sqrt{-a} b^2}-\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} x} \, dx}{16 b^{5/2}}+\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 \sqrt{-a} b^2}+\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}{8 \sqrt{-a} b^2}-\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} x} \, dx}{16 b^{5/2}}-\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 \sqrt{-a} b^2}-\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}{8 \sqrt{-a} b^2}+\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} x} \, dx}{16 b^{5/2}}\\ &=-\frac{d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac{3 d \cos \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\frac{3 d \cos \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Ci}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}-\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 b^3}-\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 b^3}-\frac{x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac{\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}+\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 b^3}+\frac{3 d \sin \left (c+\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}-d x\right )}{16 \sqrt{-a} b^{5/2}}-\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 b^3}+\frac{3 d \sin \left (c-\frac{\sqrt{-a} d}{\sqrt{b}}\right ) \text{Si}\left (\frac{\sqrt{-a} d}{\sqrt{b}}+d x\right )}{16 \sqrt{-a} b^{5/2}}\\ \end{align*}
Mathematica [C] time = 1.94547, size = 647, normalized size = 1.36 \[ \frac{\frac{d^2 \cos (c) \left (-i \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i \sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\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 (\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 )}{b}-\frac{d^2 \sin (c) \left (\cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\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{3 d \cos (c) \left (-i \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+i \cosh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\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{3 d \sin (c) \left (\sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\left (d \left (x-\frac{i \sqrt{a}}{\sqrt{b}}\right )\right )+\sinh \left (\frac{\sqrt{a} d}{\sqrt{b}}\right ) \text{CosIntegral}\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 \sin (c) \left (a+2 b x^2\right )\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 \cos (c) \left (a+2 b x^2\right )\right )}{\left (a+b x^2\right )^2}}{16 b^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.099, size = 3391, normalized size = 7.1 \begin{align*} \text{output 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 = 2.09553, size = 1079, normalized size = 2.27 \begin{align*} \frac{{\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 (3 i \, b^{3} x^{4} + 6 i \, a b^{2} x^{2} + 3 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 (-3 i \, b^{3} x^{4} - 6 i \, a b^{2} x^{2} - 3 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 (-3 i \, b^{3} x^{4} - 6 i \, a b^{2} x^{2} - 3 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 (3 i \, b^{3} x^{4} + 6 i \, a b^{2} x^{2} + 3 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 ) - 16 \,{\left (2 \, a b^{2} x^{2} + a^{2} b\right )} \sin \left (d x + c\right )}{64 \,{\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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