### 3.169 $$\int \sinh (a+b x) \sinh ^3(c+d x) \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0746128, 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 = {5613, 2637} $-\frac{\sinh (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sinh (a+x (b-d)-c)}{8 (b-d)}-\frac{3 \sinh (a+x (b+d)+c)}{8 (b+d)}+\frac{\sinh (a+x (b+3 d)+3 c)}{8 (b+3 d)}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sinh[a + b*x]*Sinh[c + d*x]^3,x]

[Out]

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

Rule 5613

Int[Sinh[v_]^(p_.)*Sinh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Sinh[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 2637

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

Rubi steps

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

Mathematica [A]  time = 0.465432, size = 86, normalized size = 0.95 $\frac{1}{8} \left (-\frac{\sinh (a+b x-3 c-3 d x)}{b-3 d}+\frac{3 \sinh (a+b x-c-d x)}{b-d}+\frac{\sinh (a+b x+3 c+3 d x)}{b+3 d}-\frac{3 \sinh (a+x (b+d)+c)}{b+d}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.015, size = 84, normalized size = 0.9 \begin{align*} -{\frac{\sinh \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{8\,b-24\,d}}+{\frac{3\,\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}-{\frac{3\,\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\sinh \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(sinh(b*x+a)*sinh(d*x+c)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90875, size = 516, normalized size = 5.67 \begin{align*} -\frac{9 \,{\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} -{\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (d x + c\right )^{3} + 3 \,{\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right )^{3} -{\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \,{\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} -{\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (b x + a\right )\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(sinh(b*x+a)*sinh(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/4*(9*(b^2*d - d^3)*cosh(d*x + c)*sinh(b*x + a)*sinh(d*x + c)^2 - (b^3 - b*d^2)*cosh(b*x + a)*sinh(d*x + c)^
3 + 3*((b^2*d - d^3)*cosh(d*x + c)^3 - (b^2*d - 9*d^3)*cosh(d*x + c))*sinh(b*x + a) - 3*((b^3 - b*d^2)*cosh(b*
x + a)*cosh(d*x + c)^2 - (b^3 - 9*b*d^2)*cosh(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 = 50.452, size = 932, normalized size = 10.24 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(a)*sinh(c)**3, Eq(b, 0) & Eq(d, 0)), (x*sinh(a - 3*d*x)*sinh(c + d*x)**3/8 + 3*x*sinh(a - 3*
d*x)*sinh(c + d*x)*cosh(c + d*x)**2/8 + 3*x*sinh(c + d*x)**2*cosh(a - 3*d*x)*cosh(c + d*x)/8 + x*cosh(a - 3*d*
x)*cosh(c + d*x)**3/8 + sinh(a - 3*d*x)*cosh(c + d*x)**3/(8*d) - 7*sinh(c + d*x)**3*cosh(a - 3*d*x)/(24*d) + s
inh(c + d*x)*cosh(a - 3*d*x)*cosh(c + d*x)**2/(4*d), Eq(b, -3*d)), (3*x*sinh(a - d*x)*sinh(c + d*x)**3/8 - 3*x
*sinh(a - d*x)*sinh(c + d*x)*cosh(c + d*x)**2/8 + 3*x*sinh(c + d*x)**2*cosh(a - d*x)*cosh(c + d*x)/8 - 3*x*cos
h(a - d*x)*cosh(c + d*x)**3/8 + 3*sinh(a - d*x)*cosh(c + d*x)**3/(8*d) - 5*sinh(c + d*x)**3*cosh(a - d*x)/(8*d
) + 3*sinh(c + d*x)*cosh(a - d*x)*cosh(c + d*x)**2/(4*d), Eq(b, -d)), (3*x*sinh(a + d*x)*sinh(c + d*x)**3/8 -
3*x*sinh(a + d*x)*sinh(c + d*x)*cosh(c + d*x)**2/8 - 3*x*sinh(c + d*x)**2*cosh(a + d*x)*cosh(c + d*x)/8 + 3*x*
cosh(a + d*x)*cosh(c + d*x)**3/8 + 3*sinh(a + d*x)*cosh(c + d*x)**3/(8*d) + 5*sinh(c + d*x)**3*cosh(a + d*x)/(
8*d) - 3*sinh(c + d*x)*cosh(a + d*x)*cosh(c + d*x)**2/(4*d), Eq(b, d)), (x*sinh(a + 3*d*x)*sinh(c + d*x)**3/8
+ 3*x*sinh(a + 3*d*x)*sinh(c + d*x)*cosh(c + d*x)**2/8 - 3*x*sinh(c + d*x)**2*cosh(a + 3*d*x)*cosh(c + d*x)/8
- x*cosh(a + 3*d*x)*cosh(c + d*x)**3/8 + 7*sinh(a + 3*d*x)*sinh(c + d*x)**2*cosh(c + d*x)/(8*d) + 5*sinh(a + 3
*d*x)*cosh(c + d*x)**3/(12*d) - 9*sinh(c + d*x)*cosh(a + 3*d*x)*cosh(c + d*x)**2/(8*d), Eq(b, 3*d)), (b**3*sin
h(c + d*x)**3*cosh(a + b*x)/(b**4 - 10*b**2*d**2 + 9*d**4) - 3*b**2*d*sinh(a + b*x)*sinh(c + d*x)**2*cosh(c +
d*x)/(b**4 - 10*b**2*d**2 + 9*d**4) - 7*b*d**2*sinh(c + d*x)**3*cosh(a + b*x)/(b**4 - 10*b**2*d**2 + 9*d**4) +
6*b*d**2*sinh(c + d*x)*cosh(a + b*x)*cosh(c + d*x)**2/(b**4 - 10*b**2*d**2 + 9*d**4) + 9*d**3*sinh(a + b*x)*s
inh(c + d*x)**2*cosh(c + d*x)/(b**4 - 10*b**2*d**2 + 9*d**4) - 6*d**3*sinh(a + b*x)*cosh(c + d*x)**3/(b**4 - 1
0*b**2*d**2 + 9*d**4), True))

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Giac [B]  time = 1.27518, 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(sinh(b*x+a)*sinh(d*x+c)^3,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)