3.2.80 \(\int (\cos (x) \cosh (x)+\sin (x) \sinh (x)) \, dx\) [180]

Optimal. Leaf size=5 \[ \cosh (x) \sin (x) \]

[Out]

cosh(x)*sin(x)

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Rubi [F]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int (\cos (x) \cosh (x)+\sin (x) \sinh (x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Cos[x]*Cosh[x] + Sin[x]*Sinh[x],x]

[Out]

Defer[Int][Cos[x]*Cosh[x], x] + Defer[Int][Sin[x]*Sinh[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \cos (x) \cosh (x) \, dx+\int \sin (x) \sinh (x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 5, normalized size = 1.00 \begin {gather*} \cosh (x) \sin (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Cosh[x] + Sin[x]*Sinh[x],x]

[Out]

Cosh[x]*Sin[x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(15\) vs. \(2(5)=10\).
time = 0.71, size = 16, normalized size = 3.20

method	result	size
default	\(\frac {{\mathrm e}^{x} \sin \left (x \right )}{2}+\frac {{\mathrm e}^{-x} \sin \left (x \right )}{2}\)	\(16\)
parts	\(\frac {{\mathrm e}^{x} \sin \left (x \right )}{2}+\frac {{\mathrm e}^{-x} \sin \left (x \right )}{2}\)	\(16\)
risch	\(\frac {i {\mathrm e}^{\left (1-i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (-1-i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (1+i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (-1+i\right ) x}}{4}\)	\(38\)
meijerg	\(\pi ^{\frac {3}{2}} \left (\frac {{\mathrm e}^{x} \cos \left (x \right )}{4 \pi ^{\frac {3}{2}}}+\frac {{\mathrm e}^{x} \sin \left (x \right )}{4 \pi ^{\frac {3}{2}}}-\frac {{\mathrm e}^{-x} \cos \left (x \right )}{4 \pi ^{\frac {3}{2}}}+\frac {{\mathrm e}^{-x} \sin \left (x \right )}{4 \pi ^{\frac {3}{2}}}\right )+\pi ^{\frac {3}{2}} \left (-\frac {{\mathrm e}^{x} \cos \left (x \right )}{4 \pi ^{\frac {3}{2}}}+\frac {{\mathrm e}^{x} \sin \left (x \right )}{4 \pi ^{\frac {3}{2}}}+\frac {{\mathrm e}^{-x} \cos \left (x \right )}{4 \pi ^{\frac {3}{2}}}+\frac {{\mathrm e}^{-x} \sin \left (x \right )}{4 \pi ^{\frac {3}{2}}}\right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cosh(x)+sin(x)*sinh(x),x,method=_RETURNVERBOSE)

[Out]

1/2*exp(x)*sin(x)+1/2*exp(-x)*sin(x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (5) = 10\).
time = 0.38, size = 52, normalized size = 10.40 \begin {gather*} \frac {1}{4} \, {\left ({\left (e^{\left (2 \, x\right )} - 1\right )} \cos \left (x\right ) + {\left (e^{\left (2 \, x\right )} + 1\right )} \sin \left (x\right )\right )} e^{\left (-x\right )} - \frac {1}{4} \, {\left ({\left (e^{\left (2 \, x\right )} - 1\right )} \cos \left (x\right ) - {\left (e^{\left (2 \, x\right )} + 1\right )} \sin \left (x\right )\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cosh(x)+sin(x)*sinh(x),x, algorithm="maxima")

[Out]

1/4*((e^(2*x) - 1)*cos(x) + (e^(2*x) + 1)*sin(x))*e^(-x) - 1/4*((e^(2*x) - 1)*cos(x) - (e^(2*x) + 1)*sin(x))*e
^(-x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (5) = 10\).
time = 0.59, size = 34, normalized size = 6.80 \begin {gather*} \frac {2 \, \cosh \left (x\right ) \sin \left (x\right ) \sinh \left (x\right ) + \sin \left (x\right ) \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cosh(x)+sin(x)*sinh(x),x, algorithm="fricas")

[Out]

1/2*(2*cosh(x)*sin(x)*sinh(x) + sin(x)*sinh(x)^2 + (cosh(x)^2 + 1)*sin(x))/(cosh(x) + sinh(x))

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Sympy [A]
time = 0.17, size = 5, normalized size = 1.00 \begin {gather*} \sin {\left (x \right )} \cosh {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cosh(x)+sin(x)*sinh(x),x)

[Out]

sin(x)*cosh(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (5) = 10\).
time = 0.46, size = 45, normalized size = 9.00 \begin {gather*} \frac {1}{4} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} - \frac {1}{4} \, {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{\left (-x\right )} + \frac {1}{4} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{x} - \frac {1}{4} \, {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*cosh(x)+sin(x)*sinh(x),x, algorithm="giac")

[Out]

1/4*(cos(x) + sin(x))*e^(-x) - 1/4*(cos(x) - sin(x))*e^(-x) + 1/4*(cos(x) + sin(x))*e^x - 1/4*(cos(x) - sin(x)
)*e^x

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Mupad [B]
time = 0.06, size = 5, normalized size = 1.00 \begin {gather*} \mathrm {cosh}\left (x\right )\,\sin \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*cosh(x) + sin(x)*sinh(x),x)

[Out]

cosh(x)*sin(x)

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Chatgpt [F] Failed to verify
time = 1.00, size = 13, normalized size = 2.60 \begin {gather*} \frac {\sinh \left (2 x \right )}{4}+\frac {\cosh \left (2 x \right )}{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(cos(x)*cosh(x)+sin(x)*sinh(x),x)

[Out]

1/4*sinh(2*x)+1/2*cosh(2*x)

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