Optimal. Leaf size=91 \[ -\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5448, 5428,
3378, 3384, 3379, 3382} \begin {gather*} -\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5428
Rule 5448
Rubi steps
\begin {align*} \int \frac {\sinh ^3\left (a+b x^2\right )}{x^3} \, dx &=\int \left (-\frac {3 \sinh \left (a+b x^2\right )}{4 x^3}+\frac {\sinh \left (3 a+3 b x^2\right )}{4 x^3}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh \left (3 a+3 b x^2\right )}{x^3} \, dx-\frac {3}{4} \int \frac {\sinh \left (a+b x^2\right )}{x^3} \, dx\\ &=\frac {1}{8} \text {Subst}\left (\int \frac {\sinh (3 a+3 b x)}{x^2} \, dx,x,x^2\right )-\frac {3}{8} \text {Subst}\left (\int \frac {\sinh (a+b x)}{x^2} \, dx,x,x^2\right )\\ &=\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\cosh (a+b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\cosh (3 a+3 b x)}{x} \, dx,x,x^2\right )\\ &=\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {1}{8} (3 b \cosh (a)) \text {Subst}\left (\int \frac {\cosh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \cosh (3 a)) \text {Subst}\left (\int \frac {\cosh (3 b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b \sinh (a)) \text {Subst}\left (\int \frac {\sinh (b x)}{x} \, dx,x,x^2\right )+\frac {1}{8} (3 b \sinh (3 a)) \text {Subst}\left (\int \frac {\sinh (3 b x)}{x} \, dx,x,x^2\right )\\ &=-\frac {3}{8} b \cosh (a) \text {Chi}\left (b x^2\right )+\frac {3}{8} b \cosh (3 a) \text {Chi}\left (3 b x^2\right )+\frac {3 \sinh \left (a+b x^2\right )}{8 x^2}-\frac {\sinh \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \sinh (a) \text {Shi}\left (b x^2\right )+\frac {3}{8} b \sinh (3 a) \text {Shi}\left (3 b x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 90, normalized size = 0.99 \begin {gather*} -\frac {3 b x^2 \cosh (a) \text {Chi}\left (b x^2\right )-3 b x^2 \cosh (3 a) \text {Chi}\left (3 b x^2\right )-3 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )+3 b x^2 \sinh (a) \text {Shi}\left (b x^2\right )-3 b x^2 \sinh (3 a) \text {Shi}\left (3 b x^2\right )}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.06, size = 120, normalized size = 1.32
method | result | size |
risch | \(\frac {{\mathrm e}^{-3 a} {\mathrm e}^{-3 x^{2} b}}{16 x^{2}}-\frac {3 \,{\mathrm e}^{-3 a} b \expIntegral \left (1, 3 x^{2} b \right )}{16}-\frac {3 \,{\mathrm e}^{-a} {\mathrm e}^{-x^{2} b}}{16 x^{2}}+\frac {3 \,{\mathrm e}^{-a} b \expIntegral \left (1, x^{2} b \right )}{16}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{x^{2} b}}{16 x^{2}}+\frac {3 \,{\mathrm e}^{a} b \expIntegral \left (1, -x^{2} b \right )}{16}-\frac {{\mathrm e}^{3 a} {\mathrm e}^{3 x^{2} b}}{16 x^{2}}-\frac {3 \,{\mathrm e}^{3 a} b \expIntegral \left (1, -3 x^{2} b \right )}{16}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.34, size = 58, normalized size = 0.64 \begin {gather*} \frac {3}{16} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x^{2}\right ) - \frac {3}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x^{2}\right ) - \frac {3}{16} \, b e^{a} \Gamma \left (-1, -b x^{2}\right ) + \frac {3}{16} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.50, size = 160, normalized size = 1.76 \begin {gather*} -\frac {2 \, \sinh \left (b x^{2} + a\right )^{3} - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) + b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \cosh \left (3 \, a\right ) + 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) + b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \cosh \left (a\right ) + 6 \, {\left (\cosh \left (b x^{2} + a\right )^{2} - 1\right )} \sinh \left (b x^{2} + a\right ) - 3 \, {\left (b x^{2} {\rm Ei}\left (3 \, b x^{2}\right ) - b x^{2} {\rm Ei}\left (-3 \, b x^{2}\right )\right )} \sinh \left (3 \, a\right ) + 3 \, {\left (b x^{2} {\rm Ei}\left (b x^{2}\right ) - b x^{2} {\rm Ei}\left (-b x^{2}\right )\right )} \sinh \left (a\right )}{16 \, 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 {\sinh ^{3}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 223 vs.
\(2 (80) = 160\).
time = 0.45, size = 223, normalized size = 2.45 \begin {gather*} \frac {3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, a b^{2} {\rm Ei}\left (3 \, b x^{2}\right ) e^{\left (3 \, a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, a b^{2} {\rm Ei}\left (-b x^{2}\right ) e^{\left (-a\right )} + 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \, a b^{2} {\rm Ei}\left (-3 \, b x^{2}\right ) e^{\left (-3 \, a\right )} - 3 \, {\left (b x^{2} + a\right )} b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} + 3 \, a b^{2} {\rm Ei}\left (b x^{2}\right ) e^{a} - b^{2} e^{\left (3 \, b x^{2} + 3 \, a\right )} + 3 \, b^{2} e^{\left (b x^{2} + a\right )} - 3 \, b^{2} e^{\left (-b x^{2} - a\right )} + b^{2} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{16 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {sinh}\left (b\,x^2+a\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________