### 3.192 $$\int \sinh (x) \sinh (2 x) \, dx$$

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

[Out]

(2*Sinh[x]^3)/3

________________________________________________________________________________________

Rubi [A]  time = 0.0090377, 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 = {4282} $\frac{1}{6} \sinh (3 x)-\frac{\sinh (x)}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sinh[x]/2 + Sinh[3*x]/6

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

Mathematica [A]  time = 0.0052166, size = 15, normalized size = 1.88 $\frac{1}{6} \sinh (3 x)-\frac{\sinh (x)}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Sinh[x]/2 + Sinh[3*x]/6

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 7, normalized size = 0.9 \begin{align*}{\frac{2\, \left ( \sinh \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sinh(x)^3

________________________________________________________________________________________

Maxima [B]  time = 1.04915, size = 36, normalized size = 4.5 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Fricas [B]  time = 2.03822, size = 61, normalized size = 7.62 \begin{align*} \frac{1}{6} \, \sinh \left (x\right )^{3} + \frac{1}{2} \,{\left (\cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) \end{align*}

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)

________________________________________________________________________________________

Sympy [B]  time = 0.643073, size = 20, normalized size = 2.5 \begin{align*} \frac{2 \sinh{\left (x \right )} \cosh{\left (2 x \right )}}{3} - \frac{\sinh{\left (2 x \right )} \cosh{\left (x \right )}}{3} \end{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.25114, size = 34, normalized size = 4.25 \begin{align*} \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{align*}

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