Optimal. Leaf size=149 \[ -\frac {3 \text {Chi}(2 b x)}{2 b^4}+\frac {3 \text {Shi}(b x) \sinh (b x)}{b^4}+\frac {3 \log (x)}{2 b^4}+\frac {2 \sinh ^2(b x)}{b^4}-\frac {3 x \text {Shi}(b x) \cosh (b x)}{b^3}-\frac {x \sinh (b x) \cosh (b x)}{b^3}+\frac {3 x^2 \text {Shi}(b x) \sinh (b x)}{2 b^2}+\frac {x^2}{2 b^2}+\frac {x^2 \sinh ^2(b x)}{4 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2-\frac {x^3 \text {Shi}(b x) \cosh (b x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.100, Rules used = {6536, 6542, 12, 5372, 3310, 30, 6548, 2564, 6546, 3312, 3301} \[ -\frac {3 \text {Chi}(2 b x)}{2 b^4}+\frac {3 x^2 \text {Shi}(b x) \sinh (b x)}{2 b^2}+\frac {3 \text {Shi}(b x) \sinh (b x)}{b^4}-\frac {3 x \text {Shi}(b x) \cosh (b x)}{b^3}+\frac {x^2}{2 b^2}+\frac {x^2 \sinh ^2(b x)}{4 b^2}+\frac {3 \log (x)}{2 b^4}+\frac {2 \sinh ^2(b x)}{b^4}-\frac {x \sinh (b x) \cosh (b x)}{b^3}+\frac {1}{4} x^4 \text {Shi}(b x)^2-\frac {x^3 \text {Shi}(b x) \cosh (b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 2564
Rule 3301
Rule 3310
Rule 3312
Rule 5372
Rule 6536
Rule 6542
Rule 6546
Rule 6548
Rubi steps
\begin {align*} \int x^3 \text {Shi}(b x)^2 \, dx &=\frac {1}{4} x^4 \text {Shi}(b x)^2-\frac {1}{2} \int x^3 \sinh (b x) \text {Shi}(b x) \, dx\\ &=-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {1}{4} x^4 \text {Shi}(b x)^2+\frac {1}{2} \int \frac {x^2 \cosh (b x) \sinh (b x)}{b} \, dx+\frac {3 \int x^2 \cosh (b x) \text {Shi}(b x) \, dx}{2 b}\\ &=-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2-\frac {3 \int x \sinh (b x) \text {Shi}(b x) \, dx}{b^2}+\frac {\int x^2 \cosh (b x) \sinh (b x) \, dx}{2 b}-\frac {3 \int \frac {x \sinh ^2(b x)}{b} \, dx}{2 b}\\ &=\frac {x^2 \sinh ^2(b x)}{4 b^2}-\frac {3 x \cosh (b x) \text {Shi}(b x)}{b^3}-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2+\frac {3 \int \cosh (b x) \text {Shi}(b x) \, dx}{b^3}-\frac {\int x \sinh ^2(b x) \, dx}{2 b^2}-\frac {3 \int x \sinh ^2(b x) \, dx}{2 b^2}+\frac {3 \int \frac {\cosh (b x) \sinh (b x)}{b} \, dx}{b^2}\\ &=-\frac {x \cosh (b x) \sinh (b x)}{b^3}+\frac {\sinh ^2(b x)}{2 b^4}+\frac {x^2 \sinh ^2(b x)}{4 b^2}-\frac {3 x \cosh (b x) \text {Shi}(b x)}{b^3}-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 \sinh (b x) \text {Shi}(b x)}{b^4}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2+\frac {3 \int \cosh (b x) \sinh (b x) \, dx}{b^3}-\frac {3 \int \frac {\sinh ^2(b x)}{b x} \, dx}{b^3}+\frac {\int x \, dx}{4 b^2}+\frac {3 \int x \, dx}{4 b^2}\\ &=\frac {x^2}{2 b^2}-\frac {x \cosh (b x) \sinh (b x)}{b^3}+\frac {\sinh ^2(b x)}{2 b^4}+\frac {x^2 \sinh ^2(b x)}{4 b^2}-\frac {3 x \cosh (b x) \text {Shi}(b x)}{b^3}-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 \sinh (b x) \text {Shi}(b x)}{b^4}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2-\frac {3 \int \frac {\sinh ^2(b x)}{x} \, dx}{b^4}-\frac {3 \operatorname {Subst}(\int x \, dx,x,i \sinh (b x))}{b^4}\\ &=\frac {x^2}{2 b^2}-\frac {x \cosh (b x) \sinh (b x)}{b^3}+\frac {2 \sinh ^2(b x)}{b^4}+\frac {x^2 \sinh ^2(b x)}{4 b^2}-\frac {3 x \cosh (b x) \text {Shi}(b x)}{b^3}-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 \sinh (b x) \text {Shi}(b x)}{b^4}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2+\frac {3 \int \left (\frac {1}{2 x}-\frac {\cosh (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=\frac {x^2}{2 b^2}+\frac {3 \log (x)}{2 b^4}-\frac {x \cosh (b x) \sinh (b x)}{b^3}+\frac {2 \sinh ^2(b x)}{b^4}+\frac {x^2 \sinh ^2(b x)}{4 b^2}-\frac {3 x \cosh (b x) \text {Shi}(b x)}{b^3}-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 \sinh (b x) \text {Shi}(b x)}{b^4}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2-\frac {3 \int \frac {\cosh (2 b x)}{x} \, dx}{2 b^4}\\ &=\frac {x^2}{2 b^2}-\frac {3 \text {Chi}(2 b x)}{2 b^4}+\frac {3 \log (x)}{2 b^4}-\frac {x \cosh (b x) \sinh (b x)}{b^3}+\frac {2 \sinh ^2(b x)}{b^4}+\frac {x^2 \sinh ^2(b x)}{4 b^2}-\frac {3 x \cosh (b x) \text {Shi}(b x)}{b^3}-\frac {x^3 \cosh (b x) \text {Shi}(b x)}{2 b}+\frac {3 \sinh (b x) \text {Shi}(b x)}{b^4}+\frac {3 x^2 \sinh (b x) \text {Shi}(b x)}{2 b^2}+\frac {1}{4} x^4 \text {Shi}(b x)^2\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 107, normalized size = 0.72 \[ \frac {2 b^4 x^4 \text {Shi}(b x)^2-4 \text {Shi}(b x) \left (b x \left (b^2 x^2+6\right ) \cosh (b x)-3 \left (b^2 x^2+2\right ) \sinh (b x)\right )+3 b^2 x^2+b^2 x^2 \cosh (2 b x)-12 \text {Chi}(2 b x)-4 b x \sinh (2 b x)+8 \cosh (2 b x)+12 \log (x)}{8 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} \operatorname {Shi}\left (b x\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm Shi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 138, normalized size = 0.93 \[ \frac {x^{4} \Shi \left (b x \right )^{2}}{4}-\frac {x^{3} \cosh \left (b x \right ) \Shi \left (b x \right )}{2 b}+\frac {3 x^{2} \Shi \left (b x \right ) \sinh \left (b x \right )}{2 b^{2}}-\frac {3 x \cosh \left (b x \right ) \Shi \left (b x \right )}{b^{3}}+\frac {3 \Shi \left (b x \right ) \sinh \left (b x \right )}{b^{4}}+\frac {x^{2} \left (\cosh ^{2}\left (b x \right )\right )}{4 b^{2}}-\frac {x \cosh \left (b x \right ) \sinh \left (b x \right )}{b^{3}}+\frac {x^{2}}{4 b^{2}}+\frac {2 \left (\cosh ^{2}\left (b x \right )\right )}{b^{4}}+\frac {3 \ln \left (b x \right )}{2 b^{4}}-\frac {3 \Chi \left (2 b x \right )}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} {\rm Shi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\mathrm {sinhint}\left (b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {Shi}^{2}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________