Optimal. Leaf size=80 \[ \frac {b c \sinh \left (\frac {2 b}{d}\right ) \text {Chi}\left (\frac {2 b c}{d (c+d x)}\right )}{d^2}-\frac {b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {b x}{c+d x}\right )}{d} \]
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Rubi [A] time = 0.15, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {5607, 3313, 12, 3303, 3298, 3301} \[ \frac {b c \sinh \left (\frac {2 b}{d}\right ) \text {Chi}\left (\frac {2 b c}{d (c+d x)}\right )}{d^2}-\frac {b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {b x}{c+d x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 3313
Rule 5607
Rubi steps
\begin {align*} \int \sinh ^2\left (\frac {b x}{c+d x}\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sinh ^2\left (\frac {b}{d}-\frac {b c x}{d}\right )}{x^2} \, dx,x,\frac {1}{c+d x}\right )}{d}\\ &=\frac {(c+d x) \sinh ^2\left (\frac {b x}{c+d x}\right )}{d}-\frac {(2 i b c) \operatorname {Subst}\left (\int \frac {i \sinh \left (\frac {2 b}{d}-\frac {2 b c x}{d}\right )}{2 x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^2\left (\frac {b x}{c+d x}\right )}{d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 b}{d}-\frac {2 b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {(c+d x) \sinh ^2\left (\frac {b x}{c+d x}\right )}{d}-\frac {\left (b c \cosh \left (\frac {2 b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}+\frac {\left (b c \sinh \left (\frac {2 b}{d}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 b c x}{d}\right )}{x} \, dx,x,\frac {1}{c+d x}\right )}{d^2}\\ &=\frac {b c \text {Chi}\left (\frac {2 b c}{d (c+d x)}\right ) \sinh \left (\frac {2 b}{d}\right )}{d^2}+\frac {(c+d x) \sinh ^2\left (\frac {b x}{c+d x}\right )}{d}-\frac {b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c}{d (c+d x)}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 85, normalized size = 1.06 \[ \frac {2 b c \sinh \left (\frac {2 b}{d}\right ) \text {Chi}\left (\frac {2 b c}{d (c+d x)}\right )-2 b c \cosh \left (\frac {2 b}{d}\right ) \text {Shi}\left (\frac {2 b c}{d (c+d x)}\right )+d \left ((c+d x) \cosh \left (\frac {2 b x}{c+d x}\right )-d x\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 277, normalized size = 3.46 \[ -\frac {d^{2} x - {\left (d^{2} x + c d\right )} \cosh \left (\frac {b x}{d x + c}\right )^{2} + {\left (b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {2 \, b}{d}\right ) - d^{2} x - c d\right )} \sinh \left (\frac {b x}{d x + c}\right )^{2} - {\left (b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (\frac {2 \, b c}{d^{2} x + c d}\right )\right )} \cosh \left (\frac {2 \, b}{d}\right ) - {\left (b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \cosh \left (\frac {b x}{d x + c}\right )^{2} - b c {\rm Ei}\left (-\frac {2 \, b c}{d^{2} x + c d}\right ) \sinh \left (\frac {b x}{d x + c}\right )^{2} + b c {\rm Ei}\left (\frac {2 \, b c}{d^{2} x + c d}\right )\right )} \sinh \left (\frac {2 \, b}{d}\right )}{2 \, {\left (d^{2} \cosh \left (\frac {b x}{d x + c}\right )^{2} - d^{2} \sinh \left (\frac {b x}{d x + c}\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh \left (\frac {b x}{d x + c}\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 120, normalized size = 1.50 \[ -\frac {x}{2}+\frac {{\mathrm e}^{-\frac {2 b x}{d x +c}} \left (d x +c \right )}{4 d}+\frac {c b \,{\mathrm e}^{-\frac {2 b}{d}} \Ei \left (1, -\frac {2 b c}{d \left (d x +c \right )}\right )}{2 d^{2}}+\frac {{\mathrm e}^{\frac {2 b x}{d x +c}} x}{4}+\frac {c \,{\mathrm e}^{\frac {2 b x}{d x +c}}}{4 d}-\frac {c b \,{\mathrm e}^{\frac {2 b}{d}} \Ei \left (1, \frac {2 b c}{d \left (d x +c \right )}\right )}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b c \int \frac {x e^{\left (\frac {2 \, b c}{d^{2} x + c d}\right )}}{d^{2} x^{2} e^{\left (\frac {2 \, b}{d}\right )} + 2 \, c d x e^{\left (\frac {2 \, b}{d}\right )} + c^{2} e^{\left (\frac {2 \, b}{d}\right )}}\,{d x} - \frac {1}{2} \, b c \int \frac {x e^{\left (-\frac {2 \, b c}{d^{2} x + c d} + \frac {2 \, b}{d}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\,{d x} + \frac {1}{4} \, {\left (x e^{\left (\frac {2 \, b c}{d^{2} x + c d}\right )} + x e^{\left (-\frac {2 \, b c}{d^{2} x + c d} + \frac {4 \, b}{d}\right )}\right )} e^{\left (-\frac {2 \, b}{d}\right )} - \frac {1}{2} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {sinh}\left (\frac {b\,x}{c+d\,x}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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