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]

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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Rubi [A]  time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 2637

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

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]))

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*}

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Mathematica [A]  time = 0.72, 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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fricas [B]  time = 0.44, size = 115, normalized size = 1.85 \[ -\frac {4 \, b d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) - b^{2} \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - {\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} - 4 \, d^{2}\right )} \sinh \left (b x + a\right )}{2 \, {\left ({\left (b^{3} - 4 \, b d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{3} - 4 \, b d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]

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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giac [B]  time = 0.13, size = 120, normalized size = 1.94 \[ \frac {e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} + \frac {e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} + \frac {e^{\left (b x + a\right )}}{4 \, b} - \frac {e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{8 \, {\left (b - 2 \, d\right )}} - \frac {e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{8 \, {\left (b + 2 \, d\right )}} - \frac {e^{\left (-b x - a\right )}}{4 \, b} \]

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

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maple [A]  time = 0.21, size = 57, normalized size = 0.92 \[ \frac {\sinh \left (b x +a \right )}{2 b}+\frac {\sinh \left (a -2 c +\left (b -2 d \right ) x \right )}{4 b -8 d}+\frac {\sinh \left (a +2 c +\left (b +2 d \right ) x \right )}{4 b +8 d} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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

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mupad [B]  time = 0.23, size = 68, normalized size = 1.10 \[ \frac {2\,d^2\,\mathrm {sinh}\left (a+b\,x\right )-b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )+2\,b\,d\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{4\,b\,d^2-b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)*cosh(c + d*x)^2,x)

[Out]

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

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sympy [A]  time = 6.29, size = 408, normalized size = 6.58 \[ \begin {cases} x \cosh {\relax (a )} \cosh ^{2}{\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\left (- \frac {x \sinh ^{2}{\left (c + d x \right )}}{2} + \frac {x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d}\right ) \cosh {\relax (a )} & \text {for}\: b = 0 \\\frac {x \sinh {\left (a - 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a - 2 d x \right )}}{4} + \frac {x \cosh {\left (a - 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac {\sinh {\left (a - 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 \sinh {\left (c + d x \right )} \cosh {\left (a - 2 d x \right )} \cosh {\left (c + d x \right )}}{4 d} & \text {for}\: b = - 2 d \\- \frac {x \sinh {\left (a + 2 d x \right )} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + 2 d x \right )}}{4} + \frac {x \cosh {\left (a + 2 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} - \frac {\sinh {\left (a + 2 d x \right )} \sinh ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 \sinh {\left (c + d x \right )} \cosh {\left (a + 2 d x \right )} \cosh {\left (c + d x \right )}}{4 d} & \text {for}\: b = 2 d \\\frac {b^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 b d \sinh {\left (c + d x \right )} \cosh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{b^{3} - 4 b d^{2}} + \frac {2 d^{2} \sinh {\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac {2 d^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \text {otherwise} \end {cases} \]

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