### 3.174 $$\int \cosh (a+b x) \cosh ^2(c+d x) \, dx$$

Optimal. Leaf size=62 $\frac{\sinh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac{\sinh (a+x (b+2 d)+2 c)}{4 (b+2 d)}+\frac{\sinh (a+b x)}{2 b}$

[Out]

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

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Rubi [A]  time = 0.0550744, antiderivative size = 62, 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 = {5614, 2637} $\frac{\sinh (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac{\sinh (a+x (b+2 d)+2 c)}{4 (b+2 d)}+\frac{\sinh (a+b x)}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[a + b*x]*Cosh[c + d*x]^2,x]

[Out]

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

Rule 5614

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

Mathematica [A]  time = 0.688158, size = 69, normalized size = 1.11 $\frac{1}{4} \left (\frac{\sinh (a+b x-2 c-2 d x)}{b-2 d}+\frac{\sinh (a+b x+2 c+2 d x)}{b+2 d}+\frac{2 \sinh (a) \cosh (b x)}{b}+\frac{2 \cosh (a) \sinh (b x)}{b}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.019, size = 57, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( bx+a \right ) }{2\,b}}+{\frac{\sinh \left ( a-2\,c+ \left ( b-2\,d \right ) x \right ) }{4\,b-8\,d}}+{\frac{\sinh \left ( a+2\,c+ \left ( b+2\,d \right ) x \right ) }{4\,b+8\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90188, size = 286, normalized size = 4.61 \begin{align*} -\frac{4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) - b^{2} \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} -{\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} - 4 \, d^{2}\right )} \sinh \left (b x + a\right )}{2 \,{\left ({\left (b^{3} - 4 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2} -{\left (b^{3} - 4 \, b d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 8.7833, size = 398, normalized size = 6.42 \begin{align*} \begin{cases} x \cosh{\left (a \right )} \cosh ^{2}{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\left (- \frac{x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac{x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{\sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d}\right ) \cosh{\left (a \right )} & \text{for}\: b = 0 \\\frac{x \sinh{\left (a - 2 d x \right )} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2} + \frac{x \sinh ^{2}{\left (c + d x \right )} \cosh{\left (a - 2 d x \right )}}{4} + \frac{x \cosh{\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{\sinh{\left (a - 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{8 d} - \frac{3 \sinh{\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8 d} & \text{for}\: b = - 2 d \\- \frac{x \sinh{\left (a + 2 d x \right )} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2} + \frac{x \sinh ^{2}{\left (c + d x \right )} \cosh{\left (a + 2 d x \right )}}{4} + \frac{x \cosh{\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac{\sinh{\left (a + 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{8 d} + \frac{3 \sinh{\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8 d} & \text{for}\: b = 2 d \\\frac{b^{2} \sinh{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac{2 b d \sinh{\left (c + d x \right )} \cosh{\left (a + b x \right )} \cosh{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} + \frac{2 d^{2} \sinh{\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac{2 d^{2} \sinh{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*cosh(a)*cosh(c)**2, Eq(b, 0) & Eq(d, 0)), ((-x*sinh(c + d*x)**2/2 + x*cosh(c + d*x)**2/2 + sinh(c
+ d*x)*cosh(c + d*x)/(2*d))*cosh(a), Eq(b, 0)), (x*sinh(a - 2*d*x)*sinh(c + d*x)*cosh(c + d*x)/2 + x*sinh(c +
d*x)**2*cosh(a - 2*d*x)/4 + x*cosh(a - 2*d*x)*cosh(c + d*x)**2/4 + sinh(a - 2*d*x)*sinh(c + d*x)**2/(8*d) - 3
*sinh(a - 2*d*x)*cosh(c + d*x)**2/(8*d), Eq(b, -2*d)), (-x*sinh(a + 2*d*x)*sinh(c + d*x)*cosh(c + d*x)/2 + x*s
inh(c + d*x)**2*cosh(a + 2*d*x)/4 + x*cosh(a + 2*d*x)*cosh(c + d*x)**2/4 - sinh(a + 2*d*x)*sinh(c + d*x)**2/(8
*d) + 3*sinh(a + 2*d*x)*cosh(c + d*x)**2/(8*d), Eq(b, 2*d)), (b**2*sinh(a + b*x)*cosh(c + d*x)**2/(b**3 - 4*b*
d**2) - 2*b*d*sinh(c + d*x)*cosh(a + b*x)*cosh(c + d*x)/(b**3 - 4*b*d**2) + 2*d**2*sinh(a + b*x)*sinh(c + d*x)
**2/(b**3 - 4*b*d**2) - 2*d**2*sinh(a + b*x)*cosh(c + d*x)**2/(b**3 - 4*b*d**2), True))

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Giac [B]  time = 1.21939, size = 162, normalized size = 2.61 \begin{align*} \frac{e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{8 \,{\left (b + 2 \, d\right )}} + \frac{e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{8 \,{\left (b - 2 \, d\right )}} + \frac{e^{\left (b x + a\right )}}{4 \, b} - \frac{e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{8 \,{\left (b - 2 \, d\right )}} - \frac{e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{8 \,{\left (b + 2 \, d\right )}} - \frac{e^{\left (-b x - a\right )}}{4 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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