Integrand size = 6, antiderivative size = 31 \[ \int \text {Shi}(b x)^2 \, dx=-\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b} \]
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Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6669, 6675, 12, 5556, 3379} \[ \int \text {Shi}(b x)^2 \, dx=x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b}-\frac {2 \text {Shi}(b x) \cosh (b x)}{b} \]
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Rule 12
Rule 3379
Rule 5556
Rule 6669
Rule 6675
Rubi steps \begin{align*} \text {integral}& = x \text {Shi}(b x)^2-2 \int \sinh (b x) \text {Shi}(b x) \, dx \\ & = -\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+2 \int \frac {\cosh (b x) \sinh (b x)}{b x} \, dx \\ & = -\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {2 \int \frac {\cosh (b x) \sinh (b x)}{x} \, dx}{b} \\ & = -\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {2 \int \frac {\sinh (2 b x)}{2 x} \, dx}{b} \\ & = -\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {\int \frac {\sinh (2 b x)}{x} \, dx}{b} \\ & = -\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \text {Shi}(b x)^2 \, dx=-\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b} \]
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Time = 0.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (b x \right )^{2} b x -2 \cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )+\operatorname {Shi}\left (2 b x \right )}{b}\) | \(30\) |
default | \(\frac {\operatorname {Shi}\left (b x \right )^{2} b x -2 \cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )+\operatorname {Shi}\left (2 b x \right )}{b}\) | \(30\) |
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\[ \int \text {Shi}(b x)^2 \, dx=\int { {\rm Shi}\left (b x\right )^{2} \,d x } \]
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\[ \int \text {Shi}(b x)^2 \, dx=\int \operatorname {Shi}^{2}{\left (b x \right )}\, dx \]
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\[ \int \text {Shi}(b x)^2 \, dx=\int { {\rm Shi}\left (b x\right )^{2} \,d x } \]
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\[ \int \text {Shi}(b x)^2 \, dx=\int { {\rm Shi}\left (b x\right )^{2} \,d x } \]
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Timed out. \[ \int \text {Shi}(b x)^2 \, dx=\int {\mathrm {sinhint}\left (b\,x\right )}^2 \,d x \]
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