3.2.76 \(\int \cosh ^2(a+b x) \cosh ^2(c+d x) \, dx\) [176]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 = {5733, 2717} \begin {gather*} \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} \end {gather*}

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 2717

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

Rule 5733

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]))

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*}

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Mathematica [A]
time = 0.47, size = 105, normalized size = 1.19 \begin {gather*} \frac {2 d \left (b^2-d^2\right ) \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)} \end {gather*}

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 = 1.61, size = 83, normalized size = 0.94

method	result	size
default	\(\frac {x}{4}+\frac {\sinh \left (2 b x +2 a \right )}{8 b}+\frac {\sinh \left (2 d x +2 c \right )}{8 d}+\frac {\sinh \left (\left (2 b -2 d \right ) x +2 a -2 c \right )}{16 b -16 d}+\frac {\sinh \left (\left (2 b +2 d \right ) x +2 a +2 c \right )}{16 b +16 d}\)	\(83\)
risch	\(\frac {x}{4}+\frac {{\mathrm e}^{2 b x +2 a}}{16 b}-\frac {{\mathrm e}^{-2 b x -2 a}}{16 b}+\frac {\left (d \,{\mathrm e}^{4 b x +4 a} b -d^{2} {\mathrm e}^{4 b x +4 a}+2 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}-b d -d^{2}\right ) {\mathrm e}^{-2 b x +2 d x -2 a +2 c}}{32 \left (b +d \right ) \left (b -d \right ) d}-\frac {\left (-d \,{\mathrm e}^{4 b x +4 a} b -d^{2} {\mathrm e}^{4 b x +4 a}+2 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}+b d -d^{2}\right ) {\mathrm e}^{-2 b x -2 d x -2 a -2 c}}{32 \left (b +d \right ) \left (b -d \right ) d}\)	\(227\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(1-(2*d)/b>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (78) = 156\).
time = 0.34, size = 192, normalized size = 2.18 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (76) = 152\).
time = 1.96, size = 1027, normalized size = 11.67 \begin {gather*} \begin {cases} x \cosh ^{2}{\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 ^{2}{\left (a \right )} & \text {for}\: b = 0 \\\frac {3 x \sinh ^{2}{\left (a - d x \right )} \sinh ^{2}{\left (c + d x \right )}}{8} - \frac {x \sinh ^{2}{\left (a - d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {x \sinh {\left (a - d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (a - d x \right )} \cosh {\left (c + d x \right )}}{2} - \frac {x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (a - d x \right )}}{8} + \frac {3 x \cosh ^{2}{\left (a - d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} - \frac {\sinh ^{2}{\left (a - d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} - \frac {\sinh {\left (a - d x \right )} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a - d x \right )}}{8 d} - \frac {5 \sinh {\left (a - d x \right )} \cosh {\left (a - d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8 d} & \text {for}\: b = - d \\\frac {3 x \sinh ^{2}{\left (a + d x \right )} \sinh ^{2}{\left (c + d x \right )}}{8} - \frac {x \sinh ^{2}{\left (a + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} - \frac {x \sinh {\left (a + d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (a + d x \right )} \cosh {\left (c + d x \right )}}{2} - \frac {x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (a + d x \right )}}{8} + \frac {3 x \cosh ^{2}{\left (a + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} - \frac {3 \sinh {\left (a + d x \right )} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + d x \right )}}{8 d} + \frac {\sinh {\left (a + d x \right )} \cosh {\left (a + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8 d} + \frac {\sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + d x \right )} \cosh {\left (c + d x \right )}}{2 d} & \text {for}\: b = d \\\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 ^{2}{\left (c \right )} & \text {for}\: d = 0 \\\frac {b^{3} d x \sinh ^{2}{\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} - \frac {b^{3} d x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} - \frac {b^{3} d x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b^{3} d - 4 b d^{3}} + \frac {b^{3} d x \cosh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} - \frac {b^{3} \sinh ^{2}{\left (a + b x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} + \frac {b^{3} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )} \cosh {\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} + \frac {2 b^{2} d \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} - \frac {b d^{3} x \sinh ^{2}{\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} + \frac {b d^{3} x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} + \frac {b d^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )}}{4 b^{3} d - 4 b d^{3}} - \frac {b d^{3} x \cosh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} - \frac {2 b d^{2} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (a + b x \right )} \cosh {\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} + \frac {d^{3} \sinh {\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + b x \right )}}{4 b^{3} d - 4 b d^{3}} - \frac {d^{3} \sinh {\left (a + b x \right )} \cosh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 b^{3} d - 4 b d^{3}} & \text {otherwise} \end {cases} \end {gather*}

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 - 3*sinh(a + d*x)*sinh(c + d*x)**2*cosh(a + d*x)/(8*d) + sinh(a + d*x)*cosh(a + d*x)*cosh(c
+ d*x)**2/(8*d) + sinh(c + d*x)*cosh(a + d*x)**2*cosh(c + d*x)/(2*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 [A]
time = 0.40, size = 156, normalized size = 1.77 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 1.92, size = 115, normalized size = 1.31 \begin {gather*} \frac {d^3\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )-b^3\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )+b\,d^3\,x-b^3\,d\,x-2\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )+2\,b\,d^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{4\,b\,d^3-4\,b^3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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