### 3.176 $$\int \cosh ^2(a+b x) \cosh ^2(c+d x) \, dx$$

Optimal. Leaf size=88 $\frac{\sinh (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac{\sinh (2 (a+c)+2 x (b+d))}{16 (b+d)}+\frac{\sinh (2 a+2 b x)}{8 b}+\frac{\sinh (2 c+2 d x)}{8 d}+\frac{x}{4}$

[Out]

x/4 + Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) + Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*
(a + c) + 2*(b + d)*x]/(16*(b + d))

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Rubi [A]  time = 0.0668682, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {5614, 2637} $\frac{\sinh (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac{\sinh (2 (a+c)+2 x (b+d))}{16 (b+d)}+\frac{\sinh (2 a+2 b x)}{8 b}+\frac{\sinh (2 c+2 d x)}{8 d}+\frac{x}{4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x/4 + Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*(a - c) + 2*(b - d)*x]/(16*(b - d)) + Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*
(a + c) + 2*(b + d)*x]/(16*(b + 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 ^2(a+b x) \cosh ^2(c+d x) \, dx &=\int \left (\frac{1}{4}+\frac{1}{4} \cosh (2 a+2 b x)+\frac{1}{8} \cosh (2 (a-c)+2 (b-d) x)+\frac{1}{4} \cosh (2 c+2 d x)+\frac{1}{8} \cosh (2 (a+c)+2 (b+d) x)\right ) \, dx\\ &=\frac{x}{4}+\frac{1}{8} \int \cosh (2 (a-c)+2 (b-d) x) \, dx+\frac{1}{8} \int \cosh (2 (a+c)+2 (b+d) x) \, dx+\frac{1}{4} \int \cosh (2 a+2 b x) \, dx+\frac{1}{4} \int \cosh (2 c+2 d x) \, dx\\ &=\frac{x}{4}+\frac{\sinh (2 a+2 b x)}{8 b}+\frac{\sinh (2 (a-c)+2 (b-d) x)}{16 (b-d)}+\frac{\sinh (2 c+2 d x)}{8 d}+\frac{\sinh (2 (a+c)+2 (b+d) x)}{16 (b+d)}\\ \end{align*}

Mathematica [A]  time = 0.68262, size = 105, normalized size = 1.19 $\frac{2 d \left (b^2-d^2\right ) \sinh (2 (a+b x))+b d (b+d) \sinh (2 (a+x (b-d)-c))+b (b-d) (d (\sinh (2 (a+x (b+d)+c))+4 x (b+d))+2 (b+d) \sinh (2 (c+d x)))}{16 b d (b-d) (b+d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*d*(b^2 - d^2)*Sinh[2*(a + b*x)] + b*d*(b + d)*Sinh[2*(a - c + (b - d)*x)] + b*(b - d)*(2*(b + d)*Sinh[2*(c
+ d*x)] + d*(4*(b + d)*x + Sinh[2*(a + c + (b + d)*x)])))/(16*b*(b - d)*d*(b + d))

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Maple [A]  time = 0.012, size = 83, normalized size = 0.9 \begin{align*}{\frac{x}{4}}+{\frac{\sinh \left ( 2\,bx+2\,a \right ) }{8\,b}}+{\frac{\sinh \left ( 2\,dx+2\,c \right ) }{8\,d}}+{\frac{\sinh \left ( \left ( 2\,b-2\,d \right ) x+2\,a-2\,c \right ) }{16\,b-16\,d}}+{\frac{\sinh \left ( \left ( 2\,b+2\,d \right ) x+2\,a+2\,c \right ) }{16\,b+16\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x+1/8*sinh(2*b*x+2*a)/b+1/8*sinh(2*d*x+2*c)/d+1/16/(b-d)*sinh((2*b-2*d)*x+2*a-2*c)+1/16/(b+d)*sinh((2*b+2*
d)*x+2*a+2*c)

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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)^2*cosh(d*x+c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.84235, size = 455, normalized size = 5.17 \begin{align*} \frac{b^{2} d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} +{\left (b^{3} d - b d^{3}\right )} x +{\left (b^{2} d \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} +{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) -{\left (b d^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} +{\left (b d^{2} \cosh \left (b x + a\right )^{2} - b^{3} + b d^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right )^{2} -{\left (b^{3} d - b 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(b*x+a)^2*cosh(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 34.1966, size = 1027, normalized size = 11.67 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*cosh(a)**2*cosh(c)**2, Eq(b, 0) & Eq(d, 0)), ((-x*sinh(c + d*x)**2/2 + x*cosh(c + d*x)**2/2 + sin
h(c + d*x)*cosh(c + d*x)/(2*d))*cosh(a)**2, Eq(b, 0)), (3*x*sinh(a - d*x)**2*sinh(c + d*x)**2/8 - x*sinh(a - d
*x)**2*cosh(c + d*x)**2/8 + x*sinh(a - d*x)*sinh(c + d*x)*cosh(a - d*x)*cosh(c + d*x)/2 - x*sinh(c + d*x)**2*c
osh(a - d*x)**2/8 + 3*x*cosh(a - d*x)**2*cosh(c + d*x)**2/8 - sinh(a - d*x)**2*sinh(c + d*x)*cosh(c + d*x)/(2*
d) - sinh(a - d*x)*sinh(c + d*x)**2*cosh(a - d*x)/(8*d) - 5*sinh(a - d*x)*cosh(a - d*x)*cosh(c + d*x)**2/(8*d)
, Eq(b, -d)), (3*x*sinh(a + d*x)**2*sinh(c + d*x)**2/8 - x*sinh(a + d*x)**2*cosh(c + d*x)**2/8 - x*sinh(a + d*
x)*sinh(c + d*x)*cosh(a + d*x)*cosh(c + d*x)/2 - x*sinh(c + d*x)**2*cosh(a + d*x)**2/8 + 3*x*cosh(a + d*x)**2*
cosh(c + d*x)**2/8 - sinh(a + d*x)**2*sinh(c + d*x)*cosh(c + d*x)/(2*d) + sinh(a + d*x)*sinh(c + d*x)**2*cosh(
a + d*x)/(8*d) + 5*sinh(a + d*x)*cosh(a + d*x)*cosh(c + d*x)**2/(8*d), Eq(b, d)), ((-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)**2, Eq(d, 0)), (b**3*d*x*sinh(a + b*x)**2*sinh
(c + d*x)**2/(4*b**3*d - 4*b*d**3) - b**3*d*x*sinh(a + b*x)**2*cosh(c + d*x)**2/(4*b**3*d - 4*b*d**3) - b**3*d
*x*sinh(c + d*x)**2*cosh(a + b*x)**2/(4*b**3*d - 4*b*d**3) + b**3*d*x*cosh(a + b*x)**2*cosh(c + d*x)**2/(4*b**
3*d - 4*b*d**3) - b**3*sinh(a + b*x)**2*sinh(c + d*x)*cosh(c + d*x)/(4*b**3*d - 4*b*d**3) + b**3*sinh(c + d*x)
*cosh(a + b*x)**2*cosh(c + d*x)/(4*b**3*d - 4*b*d**3) + 2*b**2*d*sinh(a + b*x)*cosh(a + b*x)*cosh(c + d*x)**2/
(4*b**3*d - 4*b*d**3) - b*d**3*x*sinh(a + b*x)**2*sinh(c + d*x)**2/(4*b**3*d - 4*b*d**3) + b*d**3*x*sinh(a + b
*x)**2*cosh(c + d*x)**2/(4*b**3*d - 4*b*d**3) + b*d**3*x*sinh(c + d*x)**2*cosh(a + b*x)**2/(4*b**3*d - 4*b*d**
3) - b*d**3*x*cosh(a + b*x)**2*cosh(c + d*x)**2/(4*b**3*d - 4*b*d**3) - 2*b*d**2*sinh(c + d*x)*cosh(a + b*x)**
2*cosh(c + d*x)/(4*b**3*d - 4*b*d**3) + d**3*sinh(a + b*x)*sinh(c + d*x)**2*cosh(a + b*x)/(4*b**3*d - 4*b*d**3
) - d**3*sinh(a + b*x)*cosh(a + b*x)*cosh(c + d*x)**2/(4*b**3*d - 4*b*d**3), True))

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Giac [B]  time = 1.17973, size = 211, normalized size = 2.4 \begin{align*} \frac{1}{4} \, x + \frac{e^{\left (2 \, b x + 2 \, d x + 2 \, a + 2 \, c\right )}}{32 \,{\left (b + d\right )}} + \frac{e^{\left (2 \, b x - 2 \, d x + 2 \, a - 2 \, c\right )}}{32 \,{\left (b - d\right )}} + \frac{e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} - \frac{e^{\left (-2 \, b x + 2 \, d x - 2 \, a + 2 \, c\right )}}{32 \,{\left (b - d\right )}} - \frac{e^{\left (-2 \, b x - 2 \, d x - 2 \, a - 2 \, c\right )}}{32 \,{\left (b + d\right )}} - \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b} + \frac{e^{\left (2 \, d x + 2 \, c\right )}}{16 \, d} - \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x + 1/32*e^(2*b*x + 2*d*x + 2*a + 2*c)/(b + d) + 1/32*e^(2*b*x - 2*d*x + 2*a - 2*c)/(b - d) + 1/16*e^(2*b*
x + 2*a)/b - 1/32*e^(-2*b*x + 2*d*x - 2*a + 2*c)/(b - d) - 1/32*e^(-2*b*x - 2*d*x - 2*a - 2*c)/(b + d) - 1/16*
e^(-2*b*x - 2*a)/b + 1/16*e^(2*d*x + 2*c)/d - 1/16*e^(-2*d*x - 2*c)/d