Integrand size = 8, antiderivative size = 48 \[ \int \text {Shi}(a+b x)^2 \, dx=-\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {(a+b x) \text {Shi}(a+b x)^2}{b}+\frac {\text {Shi}(2 a+2 b x)}{b} \]
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Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6669, 6675, 5556, 12, 3379} \[ \int \text {Shi}(a+b x)^2 \, dx=\frac {(a+b x) \text {Shi}(a+b x)^2}{b}+\frac {\text {Shi}(2 a+2 b x)}{b}-\frac {2 \text {Shi}(a+b x) \cosh (a+b x)}{b} \]
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Rule 12
Rule 3379
Rule 5556
Rule 6669
Rule 6675
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x) \text {Shi}(a+b x)^2}{b}-2 \int \sinh (a+b x) \text {Shi}(a+b x) \, dx \\ & = -\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {(a+b x) \text {Shi}(a+b x)^2}{b}+2 \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx \\ & = -\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {(a+b x) \text {Shi}(a+b x)^2}{b}+2 \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx \\ & = -\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {(a+b x) \text {Shi}(a+b x)^2}{b}+\int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx \\ & = -\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {(a+b x) \text {Shi}(a+b x)^2}{b}+\frac {\text {Shi}(2 a+2 b x)}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85 \[ \int \text {Shi}(a+b x)^2 \, dx=\frac {-2 \cosh (a+b x) \text {Shi}(a+b x)+(a+b x) \text {Shi}(a+b x)^2+\text {Shi}(2 (a+b x))}{b} \]
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Time = 0.45 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(\frac {\operatorname {Shi}\left (b x +a \right )^{2} \left (b x +a \right )-2 \cosh \left (b x +a \right ) \operatorname {Shi}\left (b x +a \right )+\operatorname {Shi}\left (2 b x +2 a \right )}{b}\) | \(43\) |
default | \(\frac {\operatorname {Shi}\left (b x +a \right )^{2} \left (b x +a \right )-2 \cosh \left (b x +a \right ) \operatorname {Shi}\left (b x +a \right )+\operatorname {Shi}\left (2 b x +2 a \right )}{b}\) | \(43\) |
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\[ \int \text {Shi}(a+b x)^2 \, dx=\int { {\rm Shi}\left (b x + a\right )^{2} \,d x } \]
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\[ \int \text {Shi}(a+b x)^2 \, dx=\int \operatorname {Shi}^{2}{\left (a + b x \right )}\, dx \]
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\[ \int \text {Shi}(a+b x)^2 \, dx=\int { {\rm Shi}\left (b x + a\right )^{2} \,d x } \]
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\[ \int \text {Shi}(a+b x)^2 \, dx=\int { {\rm Shi}\left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int \text {Shi}(a+b x)^2 \, dx=\int {\mathrm {sinhint}\left (a+b\,x\right )}^2 \,d x \]
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