3.34 \(\int y \sinh (y) \, dy\)

Optimal. Leaf size=9 \[ y \cosh (y)-\sinh (y) \]

[Out]

y*Cosh[y] - Sinh[y]

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Rubi [A]  time = 0.0191523, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ y \cosh (y)-\sinh (y) \]

Antiderivative was successfully verified.

[In]  Int[y*Sinh[y],y]

[Out]

y*Cosh[y] - Sinh[y]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ y \cosh{\left (y \right )} - \int \cosh{\left (y \right )}\, dy \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(y*sinh(y),y)

[Out]

y*cosh(y) - Integral(cosh(y), y)

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Mathematica [A]  time = 0.00513669, size = 9, normalized size = 1. \[ y \cosh (y)-\sinh (y) \]

Antiderivative was successfully verified.

[In]  Integrate[y*Sinh[y],y]

[Out]

y*Cosh[y] - Sinh[y]

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Maple [A]  time = 0.003, size = 10, normalized size = 1.1 \[ y\cosh \left ( y \right ) -\sinh \left ( y \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(y*sinh(y),y)

[Out]

y*cosh(y)-sinh(y)

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Maxima [A]  time = 1.35613, size = 46, normalized size = 5.11 \[ \frac{1}{2} \, y^{2} \sinh \left (y\right ) + \frac{1}{4} \,{\left (y^{2} + 2 \, y + 2\right )} e^{\left (-y\right )} - \frac{1}{4} \,{\left (y^{2} - 2 \, y + 2\right )} e^{y} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(y*sinh(y),y, algorithm="maxima")

[Out]

1/2*y^2*sinh(y) + 1/4*(y^2 + 2*y + 2)*e^(-y) - 1/4*(y^2 - 2*y + 2)*e^y

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Fricas [A]  time = 0.210783, size = 12, normalized size = 1.33 \[ y \cosh \left (y\right ) - \sinh \left (y\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(y*sinh(y),y, algorithm="fricas")

[Out]

y*cosh(y) - sinh(y)

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Sympy [A]  time = 0.177784, size = 7, normalized size = 0.78 \[ y \cosh{\left (y \right )} - \sinh{\left (y \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(y*sinh(y),y)

[Out]

y*cosh(y) - sinh(y)

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GIAC/XCAS [A]  time = 0.198044, size = 23, normalized size = 2.56 \[ \frac{1}{2} \,{\left (y + 1\right )} e^{\left (-y\right )} + \frac{1}{2} \,{\left (y - 1\right )} e^{y} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(y*sinh(y),y, algorithm="giac")

[Out]

1/2*(y + 1)*e^(-y) + 1/2*(y - 1)*e^y