Optimal. Leaf size=227 \[ \frac {-2 a^2-b^2}{4 x^2}+\frac {b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2}-\frac {a b d e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{2 \sqrt [3]{-i d x^3}}-\frac {a b d e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{2 \sqrt [3]{i d x^3}}+\frac {i b^2 d e^{2 i c} x \Gamma \left (\frac {1}{3},-2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac {i b^2 d e^{-2 i c} x \Gamma \left (\frac {1}{3},2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac {a b \sin \left (c+d x^3\right )}{x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 225, normalized size of antiderivative = 0.99, number of steps
used = 11, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3484, 6, 3469,
3436, 2239, 3468, 3437} \begin {gather*} -\frac {a b e^{i c} d x \text {Gamma}\left (\frac {1}{3},-i d x^3\right )}{2 \sqrt [3]{-i d x^3}}-\frac {a b e^{-i c} d x \text {Gamma}\left (\frac {1}{3},i d x^3\right )}{2 \sqrt [3]{i d x^3}}+\frac {i b^2 e^{2 i c} d x \text {Gamma}\left (\frac {1}{3},-2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac {i b^2 e^{-2 i c} d x \text {Gamma}\left (\frac {1}{3},2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac {2 a^2+b^2}{4 x^2}-\frac {a b \sin \left (c+d x^3\right )}{x^2}+\frac {b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 2239
Rule 3436
Rule 3437
Rule 3468
Rule 3469
Rule 3484
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin \left (c+d x^3\right )\right )^2}{x^3} \, dx &=\int \left (\frac {a^2}{x^3}+\frac {b^2}{2 x^3}-\frac {b^2 \cos \left (2 c+2 d x^3\right )}{2 x^3}+\frac {2 a b \sin \left (c+d x^3\right )}{x^3}\right ) \, dx\\ &=\int \left (\frac {a^2+\frac {b^2}{2}}{x^3}-\frac {b^2 \cos \left (2 c+2 d x^3\right )}{2 x^3}+\frac {2 a b \sin \left (c+d x^3\right )}{x^3}\right ) \, dx\\ &=-\frac {2 a^2+b^2}{4 x^2}+(2 a b) \int \frac {\sin \left (c+d x^3\right )}{x^3} \, dx-\frac {1}{2} b^2 \int \frac {\cos \left (2 c+2 d x^3\right )}{x^3} \, dx\\ &=-\frac {2 a^2+b^2}{4 x^2}+\frac {b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2}-\frac {a b \sin \left (c+d x^3\right )}{x^2}+(3 a b d) \int \cos \left (c+d x^3\right ) \, dx+\frac {1}{2} \left (3 b^2 d\right ) \int \sin \left (2 c+2 d x^3\right ) \, dx\\ &=-\frac {2 a^2+b^2}{4 x^2}+\frac {b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2}-\frac {a b \sin \left (c+d x^3\right )}{x^2}+\frac {1}{2} (3 a b d) \int e^{-i c-i d x^3} \, dx+\frac {1}{2} (3 a b d) \int e^{i c+i d x^3} \, dx+\frac {1}{4} \left (3 i b^2 d\right ) \int e^{-2 i c-2 i d x^3} \, dx-\frac {1}{4} \left (3 i b^2 d\right ) \int e^{2 i c+2 i d x^3} \, dx\\ &=-\frac {2 a^2+b^2}{4 x^2}+\frac {b^2 \cos \left (2 c+2 d x^3\right )}{4 x^2}-\frac {a b d e^{i c} x \Gamma \left (\frac {1}{3},-i d x^3\right )}{2 \sqrt [3]{-i d x^3}}-\frac {a b d e^{-i c} x \Gamma \left (\frac {1}{3},i d x^3\right )}{2 \sqrt [3]{i d x^3}}+\frac {i b^2 d e^{2 i c} x \Gamma \left (\frac {1}{3},-2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{-i d x^3}}-\frac {i b^2 d e^{-2 i c} x \Gamma \left (\frac {1}{3},2 i d x^3\right )}{4 \sqrt [3]{2} \sqrt [3]{i d x^3}}-\frac {a b \sin \left (c+d x^3\right )}{x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.35, size = 332, normalized size = 1.46 \begin {gather*} \frac {-4 a^2 \sqrt [3]{d^2 x^6}-2 b^2 \sqrt [3]{d^2 x^6}+2 b^2 \sqrt [3]{d^2 x^6} \cos \left (2 \left (c+d x^3\right )\right )+2^{2/3} b^2 \left (i d x^3\right )^{4/3} \cos (2 c) \Gamma \left (\frac {1}{3},-2 i d x^3\right )+2^{2/3} b^2 \left (-i d x^3\right )^{4/3} \cos (2 c) \Gamma \left (\frac {1}{3},2 i d x^3\right )-4 i a b \left (-i d x^3\right )^{4/3} \Gamma \left (\frac {1}{3},i d x^3\right ) (\cos (c)-i \sin (c))+4 i a b \left (i d x^3\right )^{4/3} \Gamma \left (\frac {1}{3},-i d x^3\right ) (\cos (c)+i \sin (c))+i 2^{2/3} b^2 \left (i d x^3\right )^{4/3} \Gamma \left (\frac {1}{3},-2 i d x^3\right ) \sin (2 c)-i 2^{2/3} b^2 \left (-i d x^3\right )^{4/3} \Gamma \left (\frac {1}{3},2 i d x^3\right ) \sin (2 c)-8 a b \sqrt [3]{d^2 x^6} \sin \left (c+d x^3\right )}{8 x^2 \sqrt [3]{d^2 x^6}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sin \left (d \,x^{3}+c \right )\right )^{2}}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.35, size = 188, normalized size = 0.83 \begin {gather*} \frac {\left (d x^{3}\right )^{\frac {2}{3}} {\left ({\left ({\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {2}{3}, i \, d x^{3}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {2}{3}, -i \, d x^{3}\right )\right )} \cos \left (c\right ) - {\left ({\left (i \, \sqrt {3} + 1\right )} \Gamma \left (-\frac {2}{3}, i \, d x^{3}\right ) + {\left (-i \, \sqrt {3} + 1\right )} \Gamma \left (-\frac {2}{3}, -i \, d x^{3}\right )\right )} \sin \left (c\right )\right )} a b}{6 \, x^{2}} - \frac {{\left (2^{\frac {2}{3}} \left (d x^{3}\right )^{\frac {2}{3}} {\left ({\left ({\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {2}{3}, 2 i \, d x^{3}\right ) + {\left (i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {2}{3}, -2 i \, d x^{3}\right )\right )} \cos \left (2 \, c\right ) - {\left ({\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {2}{3}, 2 i \, d x^{3}\right ) + {\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {2}{3}, -2 i \, d x^{3}\right )\right )} \sin \left (2 \, c\right )\right )} + 6\right )} b^{2}}{24 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.12, size = 139, normalized size = 0.61 \begin {gather*} -\frac {b^{2} \left (2 i \, d\right )^{\frac {2}{3}} x^{2} e^{\left (-2 i \, c\right )} \Gamma \left (\frac {1}{3}, 2 i \, d x^{3}\right ) - 4 i \, a b \left (i \, d\right )^{\frac {2}{3}} x^{2} e^{\left (-i \, c\right )} \Gamma \left (\frac {1}{3}, i \, d x^{3}\right ) + 4 i \, a b \left (-i \, d\right )^{\frac {2}{3}} x^{2} e^{\left (i \, c\right )} \Gamma \left (\frac {1}{3}, -i \, d x^{3}\right ) + b^{2} \left (-2 i \, d\right )^{\frac {2}{3}} x^{2} e^{\left (2 i \, c\right )} \Gamma \left (\frac {1}{3}, -2 i \, d x^{3}\right ) - 4 \, b^{2} \cos \left (d x^{3} + c\right )^{2} + 8 \, a b \sin \left (d x^{3} + c\right ) + 4 \, a^{2} + 4 \, b^{2}}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sin {\left (c + d x^{3} \right )}\right )^{2}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\sin \left (d\,x^3+c\right )\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________