### 3.182 $$\int \cosh (c+d x) \sinh ^2(a+b x) \, dx$$

Optimal. Leaf size=68 $\frac{\sinh (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac{\sinh (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac{\sinh (c+d x)}{2 d}$

[Out]

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

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Rubi [A]  time = 0.0536795, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.133, Rules used = {5618, 2637} $\frac{\sinh (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac{\sinh (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac{\sinh (c+d x)}{2 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[c + d*x]*Sinh[a + b*x]^2,x]

[Out]

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

Rule 5618

Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Cosh[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 \cosh (c+d x) \sinh ^2(a+b x) \, dx &=\int \left (\frac{1}{4} \cosh (2 a-c+(2 b-d) x)-\frac{1}{2} \cosh (c+d x)+\frac{1}{4} \cosh (2 a+c+(2 b+d) x)\right ) \, dx\\ &=\frac{1}{4} \int \cosh (2 a-c+(2 b-d) x) \, dx+\frac{1}{4} \int \cosh (2 a+c+(2 b+d) x) \, dx-\frac{1}{2} \int \cosh (c+d x) \, dx\\ &=\frac{\sinh (2 a-c+(2 b-d) x)}{4 (2 b-d)}-\frac{\sinh (c+d x)}{2 d}+\frac{\sinh (2 a+c+(2 b+d) x)}{4 (2 b+d)}\\ \end{align*}

Mathematica [A]  time = 0.720775, size = 74, normalized size = 1.09 $\frac{1}{4} \left (\frac{\sinh (2 a+2 b x-c-d x)}{2 b-d}+\frac{\sinh (2 a+2 b x+c+d x)}{2 b+d}-\frac{2 \sinh (c) \cosh (d x)}{d}-\frac{2 \cosh (c) \sinh (d x)}{d}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[c + d*x]*Sinh[a + b*x]^2,x]

[Out]

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

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Maple [A]  time = 0.01, size = 63, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{8\,b-4\,d}}-{\frac{\sinh \left ( dx+c \right ) }{2\,d}}+{\frac{\sinh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{8\,b+4\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)*sinh(b*x+a)^2,x)

[Out]

1/4*sinh(2*a-c+(2*b-d)*x)/(2*b-d)-1/2*sinh(d*x+c)/d+1/4*sinh(2*a+c+(2*b+d)*x)/(2*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(cosh(d*x+c)*sinh(b*x+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.83368, size = 266, normalized size = 3.91 \begin{align*} \frac{4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) -{\left (d^{2} \cosh \left (b x + a\right )^{2} + d^{2} \sinh \left (b x + a\right )^{2} + 4 \, b^{2} - d^{2}\right )} \sinh \left (d x + c\right )}{2 \,{\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2} -{\left (4 \, b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)*sinh(b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(4*b*d*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a) - (d^2*cosh(b*x + a)^2 + d^2*sinh(b*x + a)^2 + 4*b^2 - d^
2)*sinh(d*x + c))/((4*b^2*d - d^3)*cosh(b*x + a)^2 - (4*b^2*d - d^3)*sinh(b*x + a)^2)

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Sympy [A]  time = 9.19713, size = 405, normalized size = 5.96 \begin{align*} \begin{cases} x \sinh ^{2}{\left (a \right )} \cosh{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \sinh ^{2}{\left (a - \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} + \frac{x \sinh{\left (a - \frac{d x}{2} \right )} \sinh{\left (c + d x \right )} \cosh{\left (a - \frac{d x}{2} \right )}}{2} + \frac{x \cosh ^{2}{\left (a - \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} + \frac{\sinh ^{2}{\left (a - \frac{d x}{2} \right )} \sinh{\left (c + d x \right )}}{d} + \frac{\sinh{\left (a - \frac{d x}{2} \right )} \cosh{\left (a - \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = - \frac{d}{2} \\\frac{x \sinh ^{2}{\left (a + \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} - \frac{x \sinh{\left (a + \frac{d x}{2} \right )} \sinh{\left (c + d x \right )} \cosh{\left (a + \frac{d x}{2} \right )}}{2} + \frac{x \cosh ^{2}{\left (a + \frac{d x}{2} \right )} \cosh{\left (c + d x \right )}}{4} + \frac{3 \sinh ^{2}{\left (a + \frac{d x}{2} \right )} \sinh{\left (c + d x \right )}}{4 d} - \frac{\sinh{\left (c + d x \right )} \cosh ^{2}{\left (a + \frac{d x}{2} \right )}}{4 d} & \text{for}\: b = \frac{d}{2} \\\left (\frac{x \sinh ^{2}{\left (a + b x \right )}}{2} - \frac{x \cosh ^{2}{\left (a + b x \right )}}{2} + \frac{\sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{2 b}\right ) \cosh{\left (c \right )} & \text{for}\: d = 0 \\\frac{2 b^{2} \sinh ^{2}{\left (a + b x \right )} \sinh{\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac{2 b^{2} \sinh{\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b^{2} d - d^{3}} + \frac{2 b d \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )} \cosh{\left (c + d x \right )}}{4 b^{2} d - d^{3}} - \frac{d^{2} \sinh ^{2}{\left (a + b x \right )} \sinh{\left (c + d x \right )}}{4 b^{2} d - d^{3}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)*sinh(b*x+a)**2,x)

[Out]

Piecewise((x*sinh(a)**2*cosh(c), Eq(b, 0) & Eq(d, 0)), (x*sinh(a - d*x/2)**2*cosh(c + d*x)/4 + x*sinh(a - d*x/
2)*sinh(c + d*x)*cosh(a - d*x/2)/2 + x*cosh(a - d*x/2)**2*cosh(c + d*x)/4 + sinh(a - d*x/2)**2*sinh(c + d*x)/d
+ sinh(a - d*x/2)*cosh(a - d*x/2)*cosh(c + d*x)/(2*d), Eq(b, -d/2)), (x*sinh(a + d*x/2)**2*cosh(c + d*x)/4 -
x*sinh(a + d*x/2)*sinh(c + d*x)*cosh(a + d*x/2)/2 + x*cosh(a + d*x/2)**2*cosh(c + d*x)/4 + 3*sinh(a + d*x/2)**
2*sinh(c + d*x)/(4*d) - sinh(c + d*x)*cosh(a + d*x/2)**2/(4*d), Eq(b, d/2)), ((x*sinh(a + b*x)**2/2 - x*cosh(a
+ b*x)**2/2 + sinh(a + b*x)*cosh(a + b*x)/(2*b))*cosh(c), Eq(d, 0)), (2*b**2*sinh(a + b*x)**2*sinh(c + d*x)/(
4*b**2*d - d**3) - 2*b**2*sinh(c + d*x)*cosh(a + b*x)**2/(4*b**2*d - d**3) + 2*b*d*sinh(a + b*x)*cosh(a + b*x)
*cosh(c + d*x)/(4*b**2*d - d**3) - d**2*sinh(a + b*x)**2*sinh(c + d*x)/(4*b**2*d - d**3), True))

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Giac [A]  time = 1.16759, size = 167, normalized size = 2.46 \begin{align*} \frac{e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \,{\left (2 \, b + d\right )}} + \frac{e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{8 \,{\left (2 \, b - d\right )}} - \frac{e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \,{\left (2 \, b - d\right )}} - \frac{e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{8 \,{\left (2 \, b + d\right )}} - \frac{e^{\left (d x + c\right )}}{4 \, d} + \frac{e^{\left (-d x - c\right )}}{4 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)*sinh(b*x+a)^2,x, algorithm="giac")

[Out]

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