3.2.69 \(\int \sinh (a+b x) \sinh ^3(c+d x) \, dx\) [169]

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

[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 = 0.06, 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 = {5732, 2717} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]*Sinh[c + d*x]^3,x]

[Out]

-1/8*Sinh[a - 3*c + (b - 3*d)*x]/(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 2717

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

Rule 5732

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 = 0.33, size = 86, normalized size = 0.95 \begin {gather*} \frac {1}{8} \left (-\frac {\sinh (a-3 c+b x-3 d x)}{b-3 d}+\frac {3 \sinh (a-c+b x-d x)}{b-d}+\frac {\sinh (a+3 c+b x+3 d x)}{b+3 d}-\frac {3 \sinh (a+c+(b+d) x)}{b+d}\right ) \end {gather*}

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 = 2.07, size = 84, normalized size = 0.92

method	result	size
default	\(-\frac {\sinh \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}+\frac {3 \sinh \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \sinh \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\sinh \left (a +3 c +\left (b +3 d \right ) x \right )}{8 b +24 d}\)	\(84\)
risch	\(\frac {\left (b \,{\mathrm e}^{2 b x +2 a}-3 \,{\mathrm e}^{2 b x +2 a} d +b +3 d \right ) {\mathrm e}^{-b x +3 d x -a +3 c}}{16 \left (b +3 d \right ) \left (b -3 d \right )}-\frac {3 \left (b \,{\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 b x +2 a} d +b +d \right ) {\mathrm e}^{-b x +d x -a +c}}{16 \left (b +d \right ) \left (b -d \right )}+\frac {3 \left (b \,{\mathrm e}^{2 b x +2 a}+{\mathrm e}^{2 b x +2 a} d +b -d \right ) {\mathrm e}^{-b x -d x -a -c}}{16 \left (b +d \right ) \left (b -d \right )}-\frac {\left (b \,{\mathrm e}^{2 b x +2 a}+3 \,{\mathrm e}^{2 b x +2 a} d +b -3 d \right ) {\mathrm e}^{-b x -3 d x -a -3 c}}{16 \left (b +3 d \right ) \left (b -3 d \right )}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(b*x+a)*sinh(d*x+c)^3,x,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (83) = 166\).
time = 0.43, size = 218, normalized size = 2.40 \begin {gather*} -\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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 921 vs. \(2 (76) = 152\).
time = 2.42, size = 921, normalized size = 10.12 \begin {gather*} \begin {cases} x \sinh {\left (a \right )} \sinh ^{3}{\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh {\left (a - 3 d x \right )} \sinh ^{3}{\left (c + d x \right )}}{8} + \frac {3 x \sinh {\left (a - 3 d x \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {3 x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a - 3 d x \right )} \cosh {\left (c + d x \right )}}{8} + \frac {x \cosh {\left (a - 3 d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8} + \frac {\sinh {\left (a - 3 d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} - \frac {7 \sinh ^{3}{\left (c + d x \right )} \cosh {\left (a - 3 d x \right )}}{24 d} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (a - 3 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: b = - 3 d \\\frac {3 x \sinh {\left (a - d x \right )} \sinh ^{3}{\left (c + d x \right )}}{8} - \frac {3 x \sinh {\left (a - d x \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} + \frac {3 x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a - d x \right )} \cosh {\left (c + d x \right )}}{8} - \frac {3 x \cosh {\left (a - d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8} + \frac {3 \sinh {\left (a - d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} - \frac {5 \sinh ^{3}{\left (c + d x \right )} \cosh {\left (a - d x \right )}}{8 d} + \frac {3 \sinh {\left (c + d x \right )} \cosh {\left (a - d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: b = - d \\\frac {3 x \sinh {\left (a + d x \right )} \sinh ^{3}{\left (c + d x \right )}}{8} - \frac {3 x \sinh {\left (a + d x \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} - \frac {3 x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + d x \right )} \cosh {\left (c + d x \right )}}{8} + \frac {3 x \cosh {\left (a + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8} + \frac {3 \sinh {\left (a + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {5 \sinh ^{3}{\left (c + d x \right )} \cosh {\left (a + d x \right )}}{8 d} - \frac {3 \sinh {\left (c + d x \right )} \cosh {\left (a + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: b = d \\\frac {x \sinh {\left (a + 3 d x \right )} \sinh ^{3}{\left (c + d x \right )}}{8} + \frac {3 x \sinh {\left (a + 3 d x \right )} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{8} - \frac {3 x \sinh ^{2}{\left (c + d x \right )} \cosh {\left (a + 3 d x \right )} \cosh {\left (c + d x \right )}}{8} - \frac {x \cosh {\left (a + 3 d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8} + \frac {\sinh {\left (a + 3 d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac {7 \sinh ^{3}{\left (c + d x \right )} \cosh {\left (a + 3 d x \right )}}{24 d} - \frac {\sinh {\left (c + d x \right )} \cosh {\left (a + 3 d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: b = 3 d \\\frac {b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} - \frac {3 b^{2} d \sinh {\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} - \frac {7 b d^{2} \sinh ^{3}{\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} + \frac {6 b d^{2} \sinh {\left (c + d x \right )} \cosh {\left (a + b x \right )} \cosh ^{2}{\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} + \frac {9 d^{3} \sinh {\left (a + b x \right )} \sinh ^{2}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} - \frac {6 d^{3} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (c + d x \right )}}{b^{4} - 10 b^{2} d^{2} + 9 d^{4}} & \text {otherwise} \end {cases} \end {gather*}

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 + 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) - sinh(c + d*x)*cosh(a + 3*d*x)*cosh(c + d*x)**2/(4*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*co
sh(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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (83) = 166\).
time = 0.39, size = 179, normalized size = 1.97 \begin {gather*} \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 {gather*}

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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Mupad [B]
time = 0.51, size = 182, normalized size = 2.00 \begin {gather*} \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} \end {gather*}

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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