3.2.92 \(\int \sinh (x) \sinh (2 x) \, dx\) [192]

Optimal. Leaf size=8 \[ \frac {2 \sinh ^3(x)}{3} \]

[Out]

2/3*sinh(x)^3

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.88, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4367} \begin {gather*} \frac {1}{6} \sinh (3 x)-\frac {\sinh (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x]*Sinh[2*x],x]

[Out]

-1/2*Sinh[x] + Sinh[3*x]/6

Rule 4367

Int[sin[(a_.) + (b_.)*(x_)]*sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin {align*} \int \sinh (x) \sinh (2 x) \, dx &=-\frac {\sinh (x)}{2}+\frac {1}{6} \sinh (3 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.88 \begin {gather*} -\frac {\sinh (x)}{2}+\frac {1}{6} \sinh (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]*Sinh[2*x],x]

[Out]

-1/2*Sinh[x] + Sinh[3*x]/6

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Maple [A]
time = 0.53, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {2 \left (\sinh ^{3}\left (x \right )\right )}{3}\)	\(7\)
default	\(\frac {2 \left (\sinh ^{3}\left (x \right )\right )}{3}\)	\(7\)
risch	\(\frac {{\mathrm e}^{3 x}}{12}-\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{-x}}{4}-\frac {{\mathrm e}^{-3 x}}{12}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)*sinh(2*x),x,method=_RETURNVERBOSE)

[Out]

2/3*sinh(x)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (6) = 12\).
time = 0.26, size = 27, normalized size = 3.38 \begin {gather*} -\frac {1}{12} \, {\left (3 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} + \frac {1}{4} \, e^{\left (-x\right )} - \frac {1}{12} \, e^{\left (-3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(2*x),x, algorithm="maxima")

[Out]

-1/12*(3*e^(-2*x) - 1)*e^(3*x) + 1/4*e^(-x) - 1/12*e^(-3*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (6) = 12\).
time = 0.40, size = 17, normalized size = 2.12 \begin {gather*} \frac {1}{6} \, \sinh \left (x\right )^{3} + \frac {1}{2} \, {\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(2*x),x, algorithm="fricas")

[Out]

1/6*sinh(x)^3 + 1/2*(cosh(x)^2 - 1)*sinh(x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (7) = 14\).
time = 0.13, size = 20, normalized size = 2.50 \begin {gather*} \frac {2 \sinh {\left (x \right )} \cosh {\left (2 x \right )}}{3} - \frac {\sinh {\left (2 x \right )} \cosh {\left (x \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(2*x),x)

[Out]

2*sinh(x)*cosh(2*x)/3 - sinh(2*x)*cosh(x)/3

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (6) = 12\).
time = 0.40, size = 25, normalized size = 3.12 \begin {gather*} \frac {1}{12} \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{12} \, e^{\left (3 \, x\right )} - \frac {1}{4} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(2*x),x, algorithm="giac")

[Out]

1/12*(3*e^(2*x) - 1)*e^(-3*x) + 1/12*e^(3*x) - 1/4*e^x

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Mupad [B]
time = 0.07, size = 6, normalized size = 0.75 \begin {gather*} \frac {2\,{\mathrm {sinh}\left (x\right )}^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(2*x)*sinh(x),x)

[Out]

(2*sinh(x)^3)/3

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