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]

-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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Rubi [A]  time = 0.07, 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 = {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 2637

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

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]))

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*}

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Mathematica [A]  time = 0.48, 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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fricas [B]  time = 0.44, size = 218, normalized size = 2.40 \[ -\frac {9 \, {\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} - {\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (d x + c\right )^{3} + 3 \, {\left ({\left (b^{2} d - d^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (b^{2} d - 9 \, d^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) - 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - {\left (b^{3} - 9 \, b d^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]

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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giac [B]  time = 0.15, 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(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)

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maple [A]  time = 0.30, size = 84, normalized size = 0.92 \[ -\frac {\sinh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> 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 = 0.51, size = 182, normalized size = 2.00 \[ \frac {6\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {6\,d^3\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {3\,d\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (b^2-3\,d^2\right )}{b^4-10\,b^2\,d^2+9\,d^4}-\frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (7\,b\,d^2-b^3\right )}{b^4-10\,b^2\,d^2+9\,d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)*sinh(c + d*x)^3,x)

[Out]

(6*b*d^2*cosh(a + b*x)*cosh(c + d*x)^2*sinh(c + d*x))/(b^4 + 9*d^4 - 10*b^2*d^2) - (6*d^3*cosh(c + d*x)^3*sinh
(a + b*x))/(b^4 + 9*d^4 - 10*b^2*d^2) - (3*d*cosh(c + d*x)*sinh(a + b*x)*sinh(c + d*x)^2*(b^2 - 3*d^2))/(b^4 +
9*d^4 - 10*b^2*d^2) - (cosh(a + b*x)*sinh(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 = 30.31, size = 918, normalized size = 10.09 \[ \text {result too large to display} \]

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)*sinh(c + d*x)**2*cosh(c + d*x)/(4*d) + sinh(a - 3*d*x)*cosh(c + d*x)**
3/(24*d) - 3*sinh(c + d*x)**3*cosh(a - 3*d*x)/(8*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)*sinh(c + d*x)**2*cosh(c + d*x)/(4*d) - 3*sinh(a - d*x)*cosh(c
+ d*x)**3/(8*d) + sinh(c + d*x)**3*cosh(a - d*x)/(8*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*co
sh(a + d*x)*cosh(c + d*x)**3/8 + 3*sinh(a + d*x)*sinh(c + d*x)**2*cosh(c + d*x)/(4*d) - 3*sinh(a + d*x)*cosh(c
+ d*x)**3/(8*d) - sinh(c + d*x)**3*cosh(a + d*x)/(8*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 - sinh(a + 3*d*x)*sinh(c + d*x)**2*cosh(c + d*x)/(4*d) + sinh(a + 3*d*x)*co
sh(c + d*x)**3/(24*d) + 3*sinh(c + d*x)**3*cosh(a + 3*d*x)/(8*d), Eq(b, 3*d)), (b**3*sinh(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)*sinh(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 - 10*b**2*d**2 + 9*d**4),
True))

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