### 3.185 $$\int \cosh (c+d x) \sinh ^3(a+b x) \, dx$$

Optimal. Leaf size=97 $-\frac{3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac{\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac{\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)}$

[Out]

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

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Rubi [A]  time = 0.0805228, antiderivative size = 97, 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} $-\frac{3 \cosh (a+x (b-d)-c)}{8 (b-d)}+\frac{\cosh (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \cosh (a+x (b+d)+c)}{8 (b+d)}+\frac{\cosh (3 a+x (3 b+d)+c)}{8 (3 b+d)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[c + d*x]*Sinh[a + b*x]^3,x]

[Out]

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

Mathematica [A]  time = 0.495546, size = 90, normalized size = 0.93 $\frac{1}{8} \left (-\frac{3 \cosh (a+b x-c-d x)}{b-d}+\frac{\cosh (3 a+3 b x-c-d x)}{3 b-d}+\frac{\cosh (3 a+3 b x+c+d x)}{3 b+d}-\frac{3 \cosh (a+x (b+d)+c)}{b+d}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 90, normalized size = 0.9 \begin{align*} -{\frac{3\,\cosh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{\cosh \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}-{\frac{3\,\cosh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\cosh \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*cosh(a-c+(b-d)*x)/(b-d)+1/8*cosh(3*a-c+(3*b-d)*x)/(3*b-d)-3/8*cosh(a+c+(b+d)*x)/(b+d)+1/8*cosh(3*a+c+(3*b
+d)*x)/(3*b+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)*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.14674, size = 562, normalized size = 5.79 \begin{align*} \frac{9 \,{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + 3 \,{\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{3} -{\left (9 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) -{\left ({\left (b^{2} d - d^{3}\right )} \sinh \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{2} d - d^{3} -{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 46.8247, size = 942, normalized size = 9.71 \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)*sinh(b*x+a)**3,x)

[Out]

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

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Giac [B]  time = 1.17978, size = 247, normalized size = 2.55 \begin{align*} \frac{e^{\left (3 \, b x + d x + 3 \, a + c\right )}}{16 \,{\left (3 \, b + d\right )}} + \frac{e^{\left (3 \, b x - d x + 3 \, a - c\right )}}{16 \,{\left (3 \, b - 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{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 (-3 \, b x + d x - 3 \, a + c\right )}}{16 \,{\left (3 \, b - d\right )}} + \frac{e^{\left (-3 \, b x - d x - 3 \, a - c\right )}}{16 \,{\left (3 \, b + d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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