### 3.167 $$\int \sinh (a+b x) \sinh (c+d x) \, dx$$

Optimal. Leaf size=43 $\frac{\sinh (a+x (b+d)+c)}{2 (b+d)}-\frac{\sinh (a+x (b-d)-c)}{2 (b-d)}$

[Out]

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

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Rubi [A]  time = 0.0421003, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {5613, 2637} $\frac{\sinh (a+x (b+d)+c)}{2 (b+d)}-\frac{\sinh (a+x (b-d)-c)}{2 (b-d)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sinh[a + b*x]*Sinh[c + d*x],x]

[Out]

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

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

Mathematica [A]  time = 0.21916, size = 43, normalized size = 1. $\frac{\sinh (a+x (b+d)+c)}{2 (b+d)}-\frac{\sinh (a+x (b-d)-c)}{2 (b-d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.018, size = 40, normalized size = 0.9 \begin{align*} -{\frac{\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{2\,b-2\,d}}+{\frac{\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{2\,b+2\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.86633, size = 170, normalized size = 3.95 \begin{align*} -\frac{d \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - b \cosh \left (b x + a\right ) \sinh \left (d x + c\right )}{{\left (b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} -{\left (b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*cosh(d*x + c)*sinh(b*x + a) - b*cosh(b*x + a)*sinh(d*x + c))/((b^2 - d^2)*cosh(b*x + a)^2 - (b^2 - d^2)*si
nh(b*x + a)^2)

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Sympy [A]  time = 2.02849, size = 153, normalized size = 3.56 \begin{align*} \begin{cases} x \sinh{\left (a \right )} \sinh{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \sinh{\left (a - d x \right )} \sinh{\left (c + d x \right )}}{2} + \frac{x \cosh{\left (a - d x \right )} \cosh{\left (c + d x \right )}}{2} + \frac{\sinh{\left (a - d x \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = - d \\\frac{x \sinh{\left (a + d x \right )} \sinh{\left (c + d x \right )}}{2} - \frac{x \cosh{\left (a + d x \right )} \cosh{\left (c + d x \right )}}{2} + \frac{\sinh{\left (a + d x \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = d \\\frac{b \sinh{\left (c + d x \right )} \cosh{\left (a + b x \right )}}{b^{2} - d^{2}} - \frac{d \sinh{\left (a + b x \right )} \cosh{\left (c + d x \right )}}{b^{2} - d^{2}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(a)*sinh(c), Eq(b, 0) & Eq(d, 0)), (x*sinh(a - d*x)*sinh(c + d*x)/2 + x*cosh(a - d*x)*cosh(c
+ d*x)/2 + sinh(a - d*x)*cosh(c + d*x)/(2*d), Eq(b, -d)), (x*sinh(a + d*x)*sinh(c + d*x)/2 - x*cosh(a + d*x)*c
osh(c + d*x)/2 + sinh(a + d*x)*cosh(c + d*x)/(2*d), Eq(b, d)), (b*sinh(c + d*x)*cosh(a + b*x)/(b**2 - d**2) -
d*sinh(a + b*x)*cosh(c + d*x)/(b**2 - d**2), True))

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Giac [B]  time = 1.16566, size = 115, normalized size = 2.67 \begin{align*} \frac{e^{\left (b x + d x + a + c\right )}}{4 \,{\left (b + d\right )}} - \frac{e^{\left (b x - d x + a - c\right )}}{4 \,{\left (b - d\right )}} + \frac{e^{\left (-b x + d x - a + c\right )}}{4 \,{\left (b - d\right )}} - \frac{e^{\left (-b x - d x - a - c\right )}}{4 \,{\left (b + d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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