Optimal. Leaf size=164 \[ \frac{3 x^2 \text{Chi}(b x) \cosh (b x)}{2 b^2}-\frac{3 \text{Chi}(2 b x)}{2 b^4}-\frac{3 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{3 \text{Chi}(b x) \cosh (b x)}{b^4}-\frac{x^2}{4 b^2}+\frac{x^2 \sinh ^2(b x)}{4 b^2}-\frac{3 \log (x)}{2 b^4}+\frac{13 \sinh ^2(b x)}{8 b^4}+\frac{3 \cosh ^2(b x)}{8 b^4}-\frac{x \sinh (b x) \cosh (b x)}{b^3}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.235408, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {6537, 6543, 12, 5372, 3310, 30, 6549, 2564, 6547, 3312, 3301} \[ \frac{3 x^2 \text{Chi}(b x) \cosh (b x)}{2 b^2}-\frac{3 \text{Chi}(2 b x)}{2 b^4}-\frac{3 x \text{Chi}(b x) \sinh (b x)}{b^3}+\frac{3 \text{Chi}(b x) \cosh (b x)}{b^4}-\frac{x^2}{4 b^2}+\frac{x^2 \sinh ^2(b x)}{4 b^2}-\frac{3 \log (x)}{2 b^4}+\frac{13 \sinh ^2(b x)}{8 b^4}+\frac{3 \cosh ^2(b x)}{8 b^4}-\frac{x \sinh (b x) \cosh (b x)}{b^3}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6537
Rule 6543
Rule 12
Rule 5372
Rule 3310
Rule 30
Rule 6549
Rule 2564
Rule 6547
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int x^3 \text{Chi}(b x)^2 \, dx &=\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{1}{2} \int x^3 \cosh (b x) \text{Chi}(b x) \, dx\\ &=\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}+\frac{1}{2} \int \frac{x^2 \cosh (b x) \sinh (b x)}{b} \, dx+\frac{3 \int x^2 \text{Chi}(b x) \sinh (b x) \, dx}{2 b}\\ &=\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}-\frac{3 \int x \cosh (b x) \text{Chi}(b x) \, dx}{b^2}+\frac{\int x^2 \cosh (b x) \sinh (b x) \, dx}{2 b}-\frac{3 \int \frac{x \cosh ^2(b x)}{b} \, dx}{2 b}\\ &=\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{3 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}+\frac{x^2 \sinh ^2(b x)}{4 b^2}+\frac{3 \int \text{Chi}(b x) \sinh (b x) \, dx}{b^3}-\frac{\int x \sinh ^2(b x) \, dx}{2 b^2}-\frac{3 \int x \cosh ^2(b x) \, dx}{2 b^2}+\frac{3 \int \frac{\cosh (b x) \sinh (b x)}{b} \, dx}{b^2}\\ &=\frac{3 \cosh ^2(b x)}{8 b^4}+\frac{3 \cosh (b x) \text{Chi}(b x)}{b^4}+\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{x \cosh (b x) \sinh (b x)}{b^3}-\frac{3 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}+\frac{\sinh ^2(b x)}{8 b^4}+\frac{x^2 \sinh ^2(b x)}{4 b^2}-\frac{3 \int \frac{\cosh ^2(b x)}{b x} \, dx}{b^3}+\frac{3 \int \cosh (b x) \sinh (b x) \, dx}{b^3}+\frac{\int x \, dx}{4 b^2}-\frac{3 \int x \, dx}{4 b^2}\\ &=-\frac{x^2}{4 b^2}+\frac{3 \cosh ^2(b x)}{8 b^4}+\frac{3 \cosh (b x) \text{Chi}(b x)}{b^4}+\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{x \cosh (b x) \sinh (b x)}{b^3}-\frac{3 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}+\frac{\sinh ^2(b x)}{8 b^4}+\frac{x^2 \sinh ^2(b x)}{4 b^2}-\frac{3 \int \frac{\cosh ^2(b x)}{x} \, dx}{b^4}-\frac{3 \operatorname{Subst}(\int x \, dx,x,i \sinh (b x))}{b^4}\\ &=-\frac{x^2}{4 b^2}+\frac{3 \cosh ^2(b x)}{8 b^4}+\frac{3 \cosh (b x) \text{Chi}(b x)}{b^4}+\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{x \cosh (b x) \sinh (b x)}{b^3}-\frac{3 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}+\frac{13 \sinh ^2(b x)}{8 b^4}+\frac{x^2 \sinh ^2(b x)}{4 b^2}-\frac{3 \int \left (\frac{1}{2 x}+\frac{\cosh (2 b x)}{2 x}\right ) \, dx}{b^4}\\ &=-\frac{x^2}{4 b^2}+\frac{3 \cosh ^2(b x)}{8 b^4}+\frac{3 \cosh (b x) \text{Chi}(b x)}{b^4}+\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Chi}(b x)^2-\frac{3 \log (x)}{2 b^4}-\frac{x \cosh (b x) \sinh (b x)}{b^3}-\frac{3 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}+\frac{13 \sinh ^2(b x)}{8 b^4}+\frac{x^2 \sinh ^2(b x)}{4 b^2}-\frac{3 \int \frac{\cosh (2 b x)}{x} \, dx}{2 b^4}\\ &=-\frac{x^2}{4 b^2}+\frac{3 \cosh ^2(b x)}{8 b^4}+\frac{3 \cosh (b x) \text{Chi}(b x)}{b^4}+\frac{3 x^2 \cosh (b x) \text{Chi}(b x)}{2 b^2}+\frac{1}{4} x^4 \text{Chi}(b x)^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{3 x \text{Chi}(b x) \sinh (b x)}{b^3}-\frac{x^3 \text{Chi}(b x) \sinh (b x)}{2 b}+\frac{13 \sinh ^2(b x)}{8 b^4}+\frac{x^2 \sinh ^2(b x)}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.100262, size = 107, normalized size = 0.65 \[ \frac{2 b^4 x^4 \text{Chi}(b x)^2-4 \text{Chi}(b x) \left (b x \left (b^2 x^2+6\right ) \sinh (b x)-3 \left (b^2 x^2+2\right ) \cosh (b x)\right )-3 b^2 x^2+b^2 x^2 \cosh (2 b x)-12 \text{Chi}(2 b x)-4 b x \sinh (2 b x)+8 \cosh (2 b x)-12 \log (x)}{8 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 138, normalized size = 0.8 \begin{align*}{\frac{{x}^{4} \left ({\it Chi} \left ( bx \right ) \right ) ^{2}}{4}}-{\frac{{x}^{3}{\it Chi} \left ( bx \right ) \sinh \left ( bx \right ) }{2\,b}}+{\frac{3\,{x}^{2}{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{2\,{b}^{2}}}-3\,{\frac{x{\it Chi} \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{3}}}+3\,{\frac{{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{{b}^{4}}}+{\frac{{x}^{2} \left ( \cosh \left ( bx \right ) \right ) ^{2}}{4\,{b}^{2}}}-{\frac{x\cosh \left ( bx \right ) \sinh \left ( bx \right ) }{{b}^{3}}}-{\frac{{x}^{2}}{2\,{b}^{2}}}+2\,{\frac{ \left ( \cosh \left ( bx \right ) \right ) ^{2}}{{b}^{4}}}-{\frac{3\,\ln \left ( bx \right ) }{2\,{b}^{4}}}-{\frac{3\,{\it Chi} \left ( 2\,bx \right ) }{2\,{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3}{\rm Chi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{Chi}\left (b x\right )^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{Chi}^{2}\left (b x\right )\, dx \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 x^{3}{\rm Chi}\left (b x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]