3.181 \(\int \cosh ^3(c+d x) \sinh (a+b x) \, dx\)

Optimal. Leaf size=91 \[ \frac {\cosh (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac {3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac {3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac {\cosh (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]

[Out]

1/8*cosh(a-3*c+(b-3*d)*x)/(b-3*d)+3/8*cosh(a-c+(b-d)*x)/(b-d)+3/8*cosh(a+c+(b+d)*x)/(b+d)+1/8*cosh(a+3*c+(b+3*
d)*x)/(b+3*d)

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Rubi [A]  time = 0.08, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5618, 2638} \[ \frac {\cosh (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac {3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac {3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac {\cosh (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^3*Sinh[a + b*x],x]

[Out]

Cosh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Cosh[a - c + (b - d)*x])/(8*(b - d)) + (3*Cosh[a + c + (b + d)*
x])/(8*(b + d)) + Cosh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))

Rule 2638

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

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]))

Rubi steps

\begin {align*} \int \cosh ^3(c+d x) \sinh (a+b x) \, dx &=\int \left (\frac {1}{8} \sinh (a-3 c+(b-3 d) x)+\frac {3}{8} \sinh (a-c+(b-d) x)+\frac {3}{8} \sinh (a+c+(b+d) x)+\frac {1}{8} \sinh (a+3 c+(b+3 d) x)\right ) \, dx\\ &=\frac {1}{8} \int \sinh (a-3 c+(b-3 d) x) \, dx+\frac {1}{8} \int \sinh (a+3 c+(b+3 d) x) \, dx+\frac {3}{8} \int \sinh (a-c+(b-d) x) \, dx+\frac {3}{8} \int \sinh (a+c+(b+d) x) \, dx\\ &=\frac {\cosh (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac {3 \cosh (a-c+(b-d) x)}{8 (b-d)}+\frac {3 \cosh (a+c+(b+d) x)}{8 (b+d)}+\frac {\cosh (a+3 c+(b+3 d) x)}{8 (b+3 d)}\\ \end {align*}

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Mathematica [A]  time = 0.46, size = 85, normalized size = 0.93 \[ \frac {1}{8} \left (\frac {\cosh (a+b x-3 c-3 d x)}{b-3 d}+\frac {3 \cosh (a+b x-c-d x)}{b-d}+\frac {\cosh (a+b x+3 c+3 d x)}{b+3 d}+\frac {3 \cosh (a+x (b+d)+c)}{b+d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^3*Sinh[a + b*x],x]

[Out]

(Cosh[a - 3*c + b*x - 3*d*x]/(b - 3*d) + (3*Cosh[a - c + b*x - d*x])/(b - d) + Cosh[a + 3*c + b*x + 3*d*x]/(b
+ 3*d) + (3*Cosh[a + c + (b + d)*x])/(b + d))/8

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fricas [B]  time = 0.48, size = 213, normalized size = 2.34 \[ \frac {{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} + 3 \, {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - 3 \, {\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{3} + 3 \, {\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) - 3 \, {\left (b^{2} d - 9 \, d^{3} + 3 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right ) \sinh \left (d x + c\right )}{4 \, {\left ({\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*sinh(b*x+a),x, algorithm="fricas")

[Out]

1/4*((b^3 - b*d^2)*cosh(b*x + a)*cosh(d*x + c)^3 + 3*(b^3 - b*d^2)*cosh(b*x + a)*cosh(d*x + c)*sinh(d*x + c)^2
- 3*(b^2*d - d^3)*sinh(b*x + a)*sinh(d*x + c)^3 + 3*(b^3 - 9*b*d^2)*cosh(b*x + a)*cosh(d*x + c) - 3*(b^2*d -
9*d^3 + 3*(b^2*d - d^3)*cosh(d*x + c)^2)*sinh(b*x + a)*sinh(d*x + c))/((b^4 - 10*b^2*d^2 + 9*d^4)*cosh(b*x + a
)^2 - (b^4 - 10*b^2*d^2 + 9*d^4)*sinh(b*x + a)^2)

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giac [B]  time = 0.13, size = 179, normalized size = 1.97 \[ \frac {e^{\left (b x + 3 \, d x + a + 3 \, c\right )}}{16 \, {\left (b + 3 \, d\right )}} + \frac {3 \, e^{\left (b x + d x + a + c\right )}}{16 \, {\left (b + d\right )}} + \frac {3 \, e^{\left (b x - d x + a - c\right )}}{16 \, {\left (b - d\right )}} + \frac {e^{\left (b x - 3 \, d x + a - 3 \, c\right )}}{16 \, {\left (b - 3 \, d\right )}} + \frac {e^{\left (-b x + 3 \, d x - a + 3 \, c\right )}}{16 \, {\left (b - 3 \, d\right )}} + \frac {3 \, e^{\left (-b x + d x - a + c\right )}}{16 \, {\left (b - d\right )}} + \frac {3 \, e^{\left (-b x - d x - a - c\right )}}{16 \, {\left (b + d\right )}} + \frac {e^{\left (-b x - 3 \, d x - a - 3 \, c\right )}}{16 \, {\left (b + 3 \, d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3*sinh(b*x+a),x, algorithm="giac")

[Out]

1/16*e^(b*x + 3*d*x + a + 3*c)/(b + 3*d) + 3/16*e^(b*x + d*x + a + c)/(b + d) + 3/16*e^(b*x - d*x + a - c)/(b
- d) + 1/16*e^(b*x - 3*d*x + a - 3*c)/(b - 3*d) + 1/16*e^(-b*x + 3*d*x - a + 3*c)/(b - 3*d) + 3/16*e^(-b*x + d
*x - a + c)/(b - d) + 3/16*e^(-b*x - d*x - a - c)/(b + d) + 1/16*e^(-b*x - 3*d*x - a - 3*c)/(b + 3*d)

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maple [A]  time = 0.07, size = 84, normalized size = 0.92 \[ \frac {\cosh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 b -24 d}+\frac {3 \cosh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {3 \cosh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\cosh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^3*sinh(b*x+a),x)

[Out]

1/8*cosh(a-3*c+(b-3*d)*x)/(b-3*d)+3/8*cosh(a-c+(b-d)*x)/(b-d)+3/8*cosh(a+c+(b+d)*x)/(b+d)+1/8*cosh(a+3*c+(b+3*
d)*x)/(b+3*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(d*x+c)^3*sinh(b*x+a),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(-(3*d)/b>0)', see `assume?` fo
r more details)Is -(3*d)/b equal to -1?

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mupad [B]  time = 1.87, size = 182, normalized size = 2.00 \[ \frac {6\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {6\,d^3\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {3\,d\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,\left (b^2-3\,d^2\right )}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (7\,b\,d^2-b^3\right )}{b^4-10\,b^2\,d^2+9\,d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^3*sinh(a + b*x),x)

[Out]

(6*b*d^2*cosh(a + b*x)*cosh(c + d*x)*sinh(c + d*x)^2)/(b^4 + 9*d^4 - 10*b^2*d^2) - (6*d^3*sinh(a + b*x)*sinh(c
+ d*x)^3)/(b^4 + 9*d^4 - 10*b^2*d^2) - (3*d*cosh(c + d*x)^2*sinh(a + b*x)*sinh(c + d*x)*(b^2 - 3*d^2))/(b^4 +
9*d^4 - 10*b^2*d^2) - (cosh(a + b*x)*cosh(c + d*x)^3*(7*b*d^2 - b^3))/(b^4 + 9*d^4 - 10*b^2*d^2)

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sympy [A]  time = 31.09, size = 921, normalized size = 10.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**3*sinh(b*x+a),x)

[Out]

Piecewise((x*sinh(a)*cosh(c)**3, Eq(b, 0) & Eq(d, 0)), (3*x*sinh(a - 3*d*x)*sinh(c + d*x)**2*cosh(c + d*x)/8 +
x*sinh(a - 3*d*x)*cosh(c + d*x)**3/8 + x*sinh(c + d*x)**3*cosh(a - 3*d*x)/8 + 3*x*sinh(c + d*x)*cosh(a - 3*d*
x)*cosh(c + d*x)**2/8 + sinh(a - 3*d*x)*sinh(c + d*x)**3/(8*d) + sinh(c + d*x)**2*cosh(a - 3*d*x)*cosh(c + d*x
)/(4*d) - 7*cosh(a - 3*d*x)*cosh(c + d*x)**3/(24*d), Eq(b, -3*d)), (-3*x*sinh(a - d*x)*sinh(c + d*x)**2*cosh(c
+ d*x)/8 + 3*x*sinh(a - d*x)*cosh(c + d*x)**3/8 - 3*x*sinh(c + d*x)**3*cosh(a - d*x)/8 + 3*x*sinh(c + d*x)*co
sh(a - d*x)*cosh(c + d*x)**2/8 + 3*sinh(a - d*x)*sinh(c + d*x)**3/(8*d) + 3*sinh(c + d*x)**2*cosh(a - d*x)*cos
h(c + d*x)/(4*d) - 5*cosh(a - d*x)*cosh(c + d*x)**3/(8*d), Eq(b, -d)), (-3*x*sinh(a + d*x)*sinh(c + d*x)**2*co
sh(c + d*x)/8 + 3*x*sinh(a + d*x)*cosh(c + d*x)**3/8 + 3*x*sinh(c + d*x)**3*cosh(a + d*x)/8 - 3*x*sinh(c + d*x
)*cosh(a + d*x)*cosh(c + d*x)**2/8 + 3*sinh(a + d*x)*sinh(c + d*x)**3/(8*d) - 3*sinh(c + d*x)**2*cosh(a + d*x)
*cosh(c + d*x)/(4*d) + 5*cosh(a + d*x)*cosh(c + d*x)**3/(8*d), Eq(b, d)), (3*x*sinh(a + 3*d*x)*sinh(c + d*x)**
2*cosh(c + d*x)/8 + x*sinh(a + 3*d*x)*cosh(c + d*x)**3/8 - x*sinh(c + d*x)**3*cosh(a + 3*d*x)/8 - 3*x*sinh(c +
d*x)*cosh(a + 3*d*x)*cosh(c + d*x)**2/8 + sinh(a + 3*d*x)*sinh(c + d*x)**3/(8*d) - sinh(c + d*x)**2*cosh(a +
3*d*x)*cosh(c + d*x)/(4*d) + 7*cosh(a + 3*d*x)*cosh(c + d*x)**3/(24*d), Eq(b, 3*d)), (b**3*cosh(a + b*x)*cosh(
c + d*x)**3/(b**4 - 10*b**2*d**2 + 9*d**4) - 3*b**2*d*sinh(a + b*x)*sinh(c + d*x)*cosh(c + d*x)**2/(b**4 - 10*
b**2*d**2 + 9*d**4) + 6*b*d**2*sinh(c + d*x)**2*cosh(a + b*x)*cosh(c + d*x)/(b**4 - 10*b**2*d**2 + 9*d**4) - 7
*b*d**2*cosh(a + b*x)*cosh(c + d*x)**3/(b**4 - 10*b**2*d**2 + 9*d**4) - 6*d**3*sinh(a + b*x)*sinh(c + d*x)**3/
(b**4 - 10*b**2*d**2 + 9*d**4) + 9*d**3*sinh(a + b*x)*sinh(c + d*x)*cosh(c + d*x)**2/(b**4 - 10*b**2*d**2 + 9*
d**4), True))

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