Optimal. Leaf size=777 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.52803, antiderivative size = 777, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {3341, 3332, 3346, 3297, 3303, 3299, 3302, 3334, 3345} \[ -\frac{d^2 \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{4/3} b^{5/3}}-\frac{(-1)^{2/3} d^2 \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{4/3} b^{5/3}}+\frac{\sqrt [3]{-1} d^2 \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{4/3} b^{5/3}}-\frac{\sqrt [3]{-1} d \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}+\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d^2 \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{4/3} b^{5/3}}-\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{4/3} b^{5/3}}+\frac{\sqrt [3]{-1} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{4/3} b^{5/3}}-\frac{\sqrt [3]{-1} d \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}-\frac{d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}-\frac{(-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{27 a^{5/3} b^{4/3}}-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{d \cos (c+d x)}{18 a b^2 x^2}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3341
Rule 3332
Rule 3346
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rule 3334
Rule 3345
Rubi steps
\begin{align*} \int \frac{x^2 \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{d \int \frac{\cos (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{d \int \frac{\cos (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x^2 \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{d \int \left (\frac{\cos (c+d x)}{a x^3}-\frac{b \cos (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac{d^2 \int \left (\frac{\sin (c+d x)}{a x^2}-\frac{b x \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{d \int \frac{\cos (c+d x)}{x^3} \, dx}{9 a b^2}+\frac{d \int \frac{\cos (c+d x)}{a+b x^3} \, dx}{9 a b}-\frac{d^2 \int \frac{\sin (c+d x)}{x^2} \, dx}{18 a b^2}+\frac{d^2 \int \frac{x \sin (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^2}-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{d^2 \sin (c+d x)}{18 a b^2 x}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{d \int \left (-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\cos (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{9 a b}+\frac{d^2 \int \frac{\sin (c+d x)}{x^2} \, dx}{18 a b^2}+\frac{d^2 \int \left (-\frac{\sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{(-1)^{2/3} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac{\sqrt [3]{-1} \sin (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{18 a b}-\frac{d^3 \int \frac{\cos (c+d x)}{x} \, dx}{18 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^2}-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{d \int \frac{\cos (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}+\frac{\left (\sqrt [3]{-1} d^2\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}-\frac{\left ((-1)^{2/3} d^2\right ) \int \frac{\sin (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}+\frac{d^3 \int \frac{\cos (c+d x)}{x} \, dx}{18 a b^2}-\frac{\left (d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{18 a b^2}+\frac{\left (d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{18 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^2}-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d^3 \cos (c) \text{Ci}(d x)}{18 a b^2}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{d^3 \sin (c) \text{Si}(d x)}{18 a b^2}+\frac{\left (d^3 \cos (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{18 a b^2}-\frac{\left (d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\left (d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}-\frac{\left (d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\left (\sqrt [3]{-1} d^2 \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}-\frac{\left (d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\left ((-1)^{2/3} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}-\frac{\left (d^3 \sin (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{18 a b^2}+\frac{\left (d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\left (d^2 \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}-\frac{\left (d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}+\frac{\left (\sqrt [3]{-1} d^2 \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}+\frac{\left (d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\left ((-1)^{2/3} d^2 \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{54 a^{4/3} b^{4/3}}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x^2}-\frac{d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}+\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}+\frac{(-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}-\frac{d^2 \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{4/3} b^{5/3}}-\frac{(-1)^{2/3} d^2 \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{4/3} b^{5/3}}+\frac{\sqrt [3]{-1} d^2 \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{54 a^{4/3} b^{5/3}}-\frac{\sin (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{(-1)^{2/3} d^2 \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{54 a^{4/3} b^{5/3}}-\frac{\sqrt [3]{-1} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{27 a^{5/3} b^{4/3}}-\frac{d^2 \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{4/3} b^{5/3}}-\frac{d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}+\frac{\sqrt [3]{-1} d^2 \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{54 a^{4/3} b^{5/3}}-\frac{(-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{27 a^{5/3} b^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.402902, size = 449, normalized size = 0.58 \[ \frac{i d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-2 i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+2 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-i d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+2 i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-2 i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+\frac{6 b \cos (d x) \left (d x \cos (c) \left (a+b x^3\right )-3 a \sin (c)\right )}{\left (a+b x^3\right )^2}-\frac{6 b \sin (d x) \left (d x \sin (c) \left (a+b x^3\right )+3 a \cos (c)\right )}{\left (a+b x^3\right )^2}}{108 a b^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.083, size = 1394, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.72148, size = 2221, normalized size = 2.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]