### 3.186 $$\int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx$$

Optimal. Leaf size=138 $-\frac{3 \cosh (a+x (b-2 d)-2 c)}{16 (b-2 d)}+\frac{\cosh (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}-\frac{3 \cosh (a+x (b+2 d)+2 c)}{16 (b+2 d)}+\frac{\cosh (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac{3 \cosh (a+b x)}{8 b}+\frac{\cosh (3 a+3 b x)}{24 b}$

[Out]

(-3*Cosh[a + b*x])/(8*b) + Cosh[3*a + 3*b*x]/(24*b) - (3*Cosh[a - 2*c + (b - 2*d)*x])/(16*(b - 2*d)) + Cosh[3*
a - 2*c + (3*b - 2*d)*x]/(16*(3*b - 2*d)) - (3*Cosh[a + 2*c + (b + 2*d)*x])/(16*(b + 2*d)) + Cosh[3*a + 2*c +
(3*b + 2*d)*x]/(16*(3*b + 2*d))

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Rubi [A]  time = 0.113771, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {5618, 2638} $-\frac{3 \cosh (a+x (b-2 d)-2 c)}{16 (b-2 d)}+\frac{\cosh (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}-\frac{3 \cosh (a+x (b+2 d)+2 c)}{16 (b+2 d)}+\frac{\cosh (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac{3 \cosh (a+b x)}{8 b}+\frac{\cosh (3 a+3 b x)}{24 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[c + d*x]^2*Sinh[a + b*x]^3,x]

[Out]

(-3*Cosh[a + b*x])/(8*b) + Cosh[3*a + 3*b*x]/(24*b) - (3*Cosh[a - 2*c + (b - 2*d)*x])/(16*(b - 2*d)) + Cosh[3*
a - 2*c + (3*b - 2*d)*x]/(16*(3*b - 2*d)) - (3*Cosh[a + 2*c + (b + 2*d)*x])/(16*(b + 2*d)) + Cosh[3*a + 2*c +
(3*b + 2*d)*x]/(16*(3*b + 2*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 ^2(c+d x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac{3}{8} \sinh (a+b x)+\frac{1}{8} \sinh (3 a+3 b x)-\frac{3}{16} \sinh (a-2 c+(b-2 d) x)+\frac{1}{16} \sinh (3 a-2 c+(3 b-2 d) x)-\frac{3}{16} \sinh (a+2 c+(b+2 d) x)+\frac{1}{16} \sinh (3 a+2 c+(3 b+2 d) x)\right ) \, dx\\ &=\frac{1}{16} \int \sinh (3 a-2 c+(3 b-2 d) x) \, dx+\frac{1}{16} \int \sinh (3 a+2 c+(3 b+2 d) x) \, dx+\frac{1}{8} \int \sinh (3 a+3 b x) \, dx-\frac{3}{16} \int \sinh (a-2 c+(b-2 d) x) \, dx-\frac{3}{16} \int \sinh (a+2 c+(b+2 d) x) \, dx-\frac{3}{8} \int \sinh (a+b x) \, dx\\ &=-\frac{3 \cosh (a+b x)}{8 b}+\frac{\cosh (3 a+3 b x)}{24 b}-\frac{3 \cosh (a-2 c+(b-2 d) x)}{16 (b-2 d)}+\frac{\cosh (3 a-2 c+(3 b-2 d) x)}{16 (3 b-2 d)}-\frac{3 \cosh (a+2 c+(b+2 d) x)}{16 (b+2 d)}+\frac{\cosh (3 a+2 c+(3 b+2 d) x)}{16 (3 b+2 d)}\\ \end{align*}

Mathematica [A]  time = 1.64109, size = 153, normalized size = 1.11 $\frac{1}{48} \left (-\frac{9 \cosh (a+b x-2 c-2 d x)}{b-2 d}+\frac{3 \cosh (3 a+3 b x-2 c-2 d x)}{3 b-2 d}-\frac{9 \cosh (a+b x+2 c+2 d x)}{b+2 d}+\frac{3 \cosh (3 a+3 b x+2 c+2 d x)}{3 b+2 d}-\frac{18 \sinh (a) \sinh (b x)}{b}+\frac{2 \sinh (3 a) \sinh (3 b x)}{b}-\frac{18 \cosh (a) \cosh (b x)}{b}+\frac{2 \cosh (3 a) \cosh (3 b x)}{b}\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[c + d*x]^2*Sinh[a + b*x]^3,x]

[Out]

((-18*Cosh[a]*Cosh[b*x])/b + (2*Cosh[3*a]*Cosh[3*b*x])/b - (9*Cosh[a - 2*c + b*x - 2*d*x])/(b - 2*d) + (3*Cosh
[3*a - 2*c + 3*b*x - 2*d*x])/(3*b - 2*d) - (9*Cosh[a + 2*c + b*x + 2*d*x])/(b + 2*d) + (3*Cosh[3*a + 2*c + 3*b
*x + 2*d*x])/(3*b + 2*d) - (18*Sinh[a]*Sinh[b*x])/b + (2*Sinh[3*a]*Sinh[3*b*x])/b)/48

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Maple [A]  time = 0.017, size = 127, normalized size = 0.9 \begin{align*} -{\frac{3\,\cosh \left ( bx+a \right ) }{8\,b}}+{\frac{\cosh \left ( 3\,bx+3\,a \right ) }{24\,b}}-{\frac{3\,\cosh \left ( a-2\,c+ \left ( b-2\,d \right ) x \right ) }{16\,b-32\,d}}+{\frac{\cosh \left ( 3\,a-2\,c+ \left ( 3\,b-2\,d \right ) x \right ) }{48\,b-32\,d}}-{\frac{3\,\cosh \left ( a+2\,c+ \left ( b+2\,d \right ) x \right ) }{16\,b+32\,d}}+{\frac{\cosh \left ( 3\,a+2\,c+ \left ( 3\,b+2\,d \right ) x \right ) }{48\,b+32\,d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^2*sinh(b*x+a)^3,x)

[Out]

-3/8*cosh(b*x+a)/b+1/24*cosh(3*b*x+3*a)/b-3/16*cosh(a-2*c+(b-2*d)*x)/(b-2*d)+1/16*cosh(3*a-2*c+(3*b-2*d)*x)/(3
*b-2*d)-3/16*cosh(a+2*c+(b+2*d)*x)/(b+2*d)+1/16*cosh(3*a+2*c+(3*b+2*d)*x)/(3*b+2*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)^2*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.25953, size = 1054, normalized size = 7.64 \begin{align*} \frac{{\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right )^{3} + 9 \,{\left ({\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )^{3} -{\left (9 \, b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (9 \,{\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} +{\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 9 \,{\left ({\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \,{\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} -{\left (9 \, b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} - 9 \,{\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right ) - 12 \,{\left ({\left (b^{3} d - 4 \, b d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 3 \,{\left (9 \, b^{3} d - 4 \, b d^{3} -{\left (b^{3} d - 4 \, b d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{24 \,{\left ({\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \,{\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} +{\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b 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)^2*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/24*((9*b^4 - 40*b^2*d^2 + 16*d^4)*cosh(b*x + a)^3 + 9*((b^4 - 4*b^2*d^2)*cosh(b*x + a)^3 - (9*b^4 - 4*b^2*d^
2)*cosh(b*x + a))*cosh(d*x + c)^2 + 3*(9*(b^4 - 4*b^2*d^2)*cosh(b*x + a)*cosh(d*x + c)^2 + (9*b^4 - 40*b^2*d^2
+ 16*d^4)*cosh(b*x + a))*sinh(b*x + a)^2 + 9*((b^4 - 4*b^2*d^2)*cosh(b*x + a)^3 + 3*(b^4 - 4*b^2*d^2)*cosh(b*
x + a)*sinh(b*x + a)^2 - (9*b^4 - 4*b^2*d^2)*cosh(b*x + a))*sinh(d*x + c)^2 - 9*(9*b^4 - 40*b^2*d^2 + 16*d^4)*
cosh(b*x + a) - 12*((b^3*d - 4*b*d^3)*cosh(d*x + c)*sinh(b*x + a)^3 - 3*(9*b^3*d - 4*b*d^3 - (b^3*d - 4*b*d^3)
*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a))*sinh(d*x + c))/((9*b^5 - 40*b^3*d^2 + 16*b*d^4)*cosh(b*x + a)^4
- 2*(9*b^5 - 40*b^3*d^2 + 16*b*d^4)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (9*b^5 - 40*b^3*d^2 + 16*b*d^4)*sinh(b*
x + a)^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**2*sinh(b*x+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.18607, size = 346, normalized size = 2.51 \begin{align*} \frac{e^{\left (3 \, b x + 2 \, d x + 3 \, a + 2 \, c\right )}}{32 \,{\left (3 \, b + 2 \, d\right )}} + \frac{e^{\left (3 \, b x - 2 \, d x + 3 \, a - 2 \, c\right )}}{32 \,{\left (3 \, b - 2 \, d\right )}} + \frac{e^{\left (3 \, b x + 3 \, a\right )}}{48 \, b} - \frac{3 \, e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{32 \,{\left (b + 2 \, d\right )}} - \frac{3 \, e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{32 \,{\left (b - 2 \, d\right )}} - \frac{3 \, e^{\left (b x + a\right )}}{16 \, b} - \frac{3 \, e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{32 \,{\left (b - 2 \, d\right )}} - \frac{3 \, e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{32 \,{\left (b + 2 \, d\right )}} - \frac{3 \, e^{\left (-b x - a\right )}}{16 \, b} + \frac{e^{\left (-3 \, b x + 2 \, d x - 3 \, a + 2 \, c\right )}}{32 \,{\left (3 \, b - 2 \, d\right )}} + \frac{e^{\left (-3 \, b x - 2 \, d x - 3 \, a - 2 \, c\right )}}{32 \,{\left (3 \, b + 2 \, d\right )}} + \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

1/32*e^(3*b*x + 2*d*x + 3*a + 2*c)/(3*b + 2*d) + 1/32*e^(3*b*x - 2*d*x + 3*a - 2*c)/(3*b - 2*d) + 1/48*e^(3*b*
x + 3*a)/b - 3/32*e^(b*x + 2*d*x + a + 2*c)/(b + 2*d) - 3/32*e^(b*x - 2*d*x + a - 2*c)/(b - 2*d) - 3/16*e^(b*x
+ a)/b - 3/32*e^(-b*x + 2*d*x - a + 2*c)/(b - 2*d) - 3/32*e^(-b*x - 2*d*x - a - 2*c)/(b + 2*d) - 3/16*e^(-b*x
- a)/b + 1/32*e^(-3*b*x + 2*d*x - 3*a + 2*c)/(3*b - 2*d) + 1/32*e^(-3*b*x - 2*d*x - 3*a - 2*c)/(3*b + 2*d) +
1/48*e^(-3*b*x - 3*a)/b