### 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]

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))

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Rubi [A]  time = 0.0817464, antiderivative size = 91, normalized size of antiderivative = 1., 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 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]))

Rule 2638

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

Mathematica [A]  time = 0.44746, 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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Maple [A]  time = 0.012, size = 84, normalized size = 0.9 \begin{align*}{\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\,b-8\,d}}+{\frac{3\,\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\cosh \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{8\,b+24\,d}} \end{align*}

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., 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(d*x+c)^3*sinh(b*x+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.85347, size = 514, normalized size = 5.65 \begin{align*} \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 )}} \end{align*}

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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Sympy [A]  time = 46.3861, size = 940, normalized size = 10.33 \begin{align*} \text{result too large to display} \end{align*}

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 - 3*sinh(a - 3*d*x)*sinh(c + d*x)*cosh(c + d*x)**2/(8*d) - sinh(c + d*x)**2*cosh(a - 3*d
*x)*cosh(c + d*x)/(8*d) - 5*cosh(a - 3*d*x)*cosh(c + d*x)**3/(12*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)*cosh(a - d*x)*cosh(c + d*x)**2/8 + 3*sinh(a - d*x)*sinh(c + d*x)*cosh(c + d*x)**2/(8*d) + 3*sinh
(c + d*x)**2*cosh(a - d*x)*cosh(c + d*x)/(8*d) - cosh(a - d*x)*cosh(c + d*x)**3/(4*d), Eq(b, -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)*cosh(a + d*x)*cosh(c + d*x)**2/8 + 3*sinh(a + d*x)*sinh(c + d*x)*cosh(c + d*x)**
2/(8*d) - 3*sinh(c + d*x)**2*cosh(a + d*x)*cosh(c + d*x)/(8*d) + cosh(a + d*x)*cosh(c + d*x)**3/(4*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)*si
nh(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)*co
sh(c + d*x)**2/(b**4 - 10*b**2*d**2 + 9*d**4), True))

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Giac [B]  time = 1.19169, size = 242, normalized size = 2.66 \begin{align*} \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 )}} \end{align*}

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)