### 3.193 $$\int \sinh (x) \sinh (3 x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{8} \sinh (4 x)-\frac{1}{4} \sinh (2 x)$

[Out]

-Sinh[2*x]/4 + Sinh[4*x]/8

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Rubi [A]  time = 0.0094791, antiderivative size = 17, normalized size of antiderivative = 1., 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 = {4282} $\frac{1}{8} \sinh (4 x)-\frac{1}{4} \sinh (2 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sinh[x]*Sinh[3*x],x]

[Out]

-Sinh[2*x]/4 + Sinh[4*x]/8

Rule 4282

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 (3 x) \, dx &=-\frac{1}{4} \sinh (2 x)+\frac{1}{8} \sinh (4 x)\\ \end{align*}

Mathematica [A]  time = 0.0064357, size = 17, normalized size = 1. $\frac{1}{8} \sinh (4 x)-\frac{1}{4} \sinh (2 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sinh[x]*Sinh[3*x],x]

[Out]

-Sinh[2*x]/4 + Sinh[4*x]/8

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Maple [A]  time = 0.019, size = 14, normalized size = 0.8 \begin{align*} -{\frac{\sinh \left ( 2\,x \right ) }{4}}+{\frac{\sinh \left ( 4\,x \right ) }{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)*sinh(3*x),x)

[Out]

-1/4*sinh(2*x)+1/8*sinh(4*x)

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Maxima [A]  time = 1.05222, size = 36, normalized size = 2.12 \begin{align*} -\frac{1}{16} \,{\left (2 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} + \frac{1}{8} \, e^{\left (-2 \, x\right )} - \frac{1}{16} \, e^{\left (-4 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(3*x),x, algorithm="maxima")

[Out]

-1/16*(2*e^(-2*x) - 1)*e^(4*x) + 1/8*e^(-2*x) - 1/16*e^(-4*x)

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Fricas [A]  time = 2.11509, size = 80, normalized size = 4.71 \begin{align*} \frac{1}{2} \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \frac{1}{2} \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(3*x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.632957, size = 20, normalized size = 1.18 \begin{align*} \frac{3 \sinh{\left (x \right )} \cosh{\left (3 x \right )}}{8} - \frac{\sinh{\left (3 x \right )} \cosh{\left (x \right )}}{8} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(3*x),x)

[Out]

3*sinh(x)*cosh(3*x)/8 - sinh(3*x)*cosh(x)/8

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Giac [A]  time = 1.21103, size = 36, normalized size = 2.12 \begin{align*} \frac{1}{16} \,{\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} + \frac{1}{16} \, e^{\left (4 \, x\right )} - \frac{1}{8} \, e^{\left (2 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)*sinh(3*x),x, algorithm="giac")

[Out]

1/16*(2*e^(2*x) - 1)*e^(-4*x) + 1/16*e^(4*x) - 1/8*e^(2*x)