### 3.131 $$\int \sinh (a+b x) \sinh (c+b x) \, dx$$

Optimal. Leaf size=27 $\frac{\sinh (a+2 b x+c)}{4 b}-\frac{1}{2} x \cosh (a-c)$

[Out]

-(x*Cosh[a - c])/2 + Sinh[a + c + 2*b*x]/(4*b)

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Rubi [A]  time = 0.0266107, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {5613, 2637} $\frac{\sinh (a+2 b x+c)}{4 b}-\frac{1}{2} x \cosh (a-c)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sinh[a + b*x]*Sinh[c + b*x],x]

[Out]

-(x*Cosh[a - c])/2 + Sinh[a + c + 2*b*x]/(4*b)

Rule 5613

Int[Sinh[v_]^(p_.)*Sinh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Sinh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], x]))

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sinh (a+b x) \sinh (c+b x) \, dx &=\int \left (-\frac{1}{2} \cosh (a-c)+\frac{1}{2} \cosh (a+c+2 b x)\right ) \, dx\\ &=-\frac{1}{2} x \cosh (a-c)+\frac{1}{2} \int \cosh (a+c+2 b x) \, dx\\ &=-\frac{1}{2} x \cosh (a-c)+\frac{\sinh (a+c+2 b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.033118, size = 26, normalized size = 0.96 $\frac{\sinh (a+2 b x+c)-2 b x \cosh (a-c)}{4 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sinh[a + b*x]*Sinh[c + b*x],x]

[Out]

(-2*b*x*Cosh[a - c] + Sinh[a + c + 2*b*x])/(4*b)

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Maple [A]  time = 0.014, size = 24, normalized size = 0.9 \begin{align*} -{\frac{x\cosh \left ( a-c \right ) }{2}}+{\frac{\sinh \left ( 2\,bx+a+c \right ) }{4\,b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)*sinh(b*x+c),x)

[Out]

-1/2*x*cosh(a-c)+1/4*sinh(2*b*x+a+c)/b

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Maxima [B]  time = 1.1784, size = 78, normalized size = 2.89 \begin{align*} -\frac{{\left (b x + a\right )}{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{4 \, b} + \frac{e^{\left (2 \, b x + a + c\right )}}{8 \, b} - \frac{e^{\left (-2 \, b x - a - c\right )}}{8 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)*sinh(b*x+c),x, algorithm="maxima")

[Out]

-1/4*(b*x + a)*(e^(2*a) + e^(2*c))*e^(-a - c)/b + 1/8*e^(2*b*x + a + c)/b - 1/8*e^(-2*b*x - a - c)/b

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Fricas [B]  time = 1.9213, size = 232, normalized size = 8.59 \begin{align*} -\frac{2 \, b x \cosh \left (-a + c\right ) - 2 \, \cosh \left (b x + c\right ) \cosh \left (-a + c\right ) \sinh \left (b x + c\right ) + \cosh \left (b x + c\right )^{2} \sinh \left (-a + c\right ) + \sinh \left (b x + c\right )^{2} \sinh \left (-a + c\right )}{4 \,{\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)*sinh(b*x+c),x, algorithm="fricas")

[Out]

-1/4*(2*b*x*cosh(-a + c) - 2*cosh(b*x + c)*cosh(-a + c)*sinh(b*x + c) + cosh(b*x + c)^2*sinh(-a + c) + sinh(b*
x + c)^2*sinh(-a + c))/(b*cosh(-a + c)^2 - b*sinh(-a + c)^2)

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Sympy [A]  time = 1.09259, size = 58, normalized size = 2.15 \begin{align*} \begin{cases} \frac{x \sinh{\left (a + b x \right )} \sinh{\left (b x + c \right )}}{2} - \frac{x \cosh{\left (a + b x \right )} \cosh{\left (b x + c \right )}}{2} + \frac{\sinh{\left (a + b x \right )} \cosh{\left (b x + c \right )}}{2 b} & \text{for}\: b \neq 0 \\x \sinh{\left (a \right )} \sinh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)*sinh(b*x+c),x)

[Out]

Piecewise((x*sinh(a + b*x)*sinh(b*x + c)/2 - x*cosh(a + b*x)*cosh(b*x + c)/2 + sinh(a + b*x)*cosh(b*x + c)/(2*
b), Ne(b, 0)), (x*sinh(a)*sinh(c), True))

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Giac [B]  time = 1.18586, size = 96, normalized size = 3.56 \begin{align*} -\frac{2 \, b x{\left (e^{\left (2 \, a\right )} + e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} -{\left (e^{\left (2 \, b x + 2 \, a\right )} + e^{\left (2 \, b x + 2 \, c\right )} - 1\right )} e^{\left (-2 \, b x - a - c\right )} - e^{\left (2 \, b x + a + c\right )}}{8 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)*sinh(b*x+c),x, algorithm="giac")

[Out]

-1/8*(2*b*x*(e^(2*a) + e^(2*c))*e^(-a - c) - (e^(2*b*x + 2*a) + e^(2*b*x + 2*c) - 1)*e^(-2*b*x - a - c) - e^(2
*b*x + a + c))/b