3.10.45 \(\int e^{c+d x} \cosh (a+b x) \sinh ^3(a+b x) \, dx\) [945]

Optimal. Leaf size=137 \[ -\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}+\frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )} \]

[Out]

-1/2*b*exp(d*x+c)*cosh(2*b*x+2*a)/(4*b^2-d^2)+1/2*b*exp(d*x+c)*cosh(4*b*x+4*a)/(16*b^2-d^2)+1/4*d*exp(d*x+c)*s
inh(2*b*x+2*a)/(4*b^2-d^2)-1/8*d*exp(d*x+c)*sinh(4*b*x+4*a)/(16*b^2-d^2)

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Rubi [A]
time = 0.07, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5620, 5582} \begin {gather*} \frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}-\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Cosh[a + b*x]*Sinh[a + b*x]^3,x]

[Out]

-1/2*(b*E^(c + d*x)*Cosh[2*a + 2*b*x])/(4*b^2 - d^2) + (b*E^(c + d*x)*Cosh[4*a + 4*b*x])/(2*(16*b^2 - d^2)) +
(d*E^(c + d*x)*Sinh[2*a + 2*b*x])/(4*(4*b^2 - d^2)) - (d*E^(c + d*x)*Sinh[4*a + 4*b*x])/(8*(16*b^2 - d^2))

Rule 5582

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a + b*x)
)*(Sinh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2)), x] + Simp[e*F^(c*(a + b*x))*(Cosh[d + e*x]/(e^2 - b^2*c^2*Log[F]^2
)), x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 - b^2*c^2*Log[F]^2, 0]

Rule 5620

Int[Cosh[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sinh[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol]
:> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sinh[d + e*x]^m*Cosh[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e,
f, g}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int e^{c+d x} \cosh (a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {1}{4} e^{c+d x} \sinh (2 a+2 b x)+\frac {1}{8} e^{c+d x} \sinh (4 a+4 b x)\right ) \, dx\\ &=\frac {1}{8} \int e^{c+d x} \sinh (4 a+4 b x) \, dx-\frac {1}{4} \int e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=-\frac {b e^{c+d x} \cosh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}+\frac {b e^{c+d x} \cosh (4 a+4 b x)}{2 \left (16 b^2-d^2\right )}+\frac {d e^{c+d x} \sinh (2 a+2 b x)}{4 \left (4 b^2-d^2\right )}-\frac {d e^{c+d x} \sinh (4 a+4 b x)}{8 \left (16 b^2-d^2\right )}\\ \end {align*}

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Mathematica [A]
time = 0.80, size = 86, normalized size = 0.63 \begin {gather*} \frac {1}{8} e^{c+d x} \left (\frac {-4 b \cosh (2 (a+b x))+2 d \sinh (2 (a+b x))}{4 b^2-d^2}+\frac {4 b \cosh (4 (a+b x))-d \sinh (4 (a+b x))}{16 b^2-d^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(c + d*x)*((-4*b*Cosh[2*(a + b*x)] + 2*d*Sinh[2*(a + b*x)])/(4*b^2 - d^2) + (4*b*Cosh[4*(a + b*x)] - d*Sinh
[4*(a + b*x)])/(16*b^2 - d^2)))/8

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Maple [A]
time = 1.72, size = 202, normalized size = 1.47

method	result	size
default	\(\frac {\sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 b -8 d}-\frac {\sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 \left (2 b +d \right )}-\frac {\sinh \left (\left (4 b -d \right ) x +4 a -c \right )}{16 \left (4 b -d \right )}+\frac {\sinh \left (\left (4 b +d \right ) x +4 a +c \right )}{64 b +16 d}-\frac {\cosh \left (2 a -c +\left (2 b -d \right ) x \right )}{8 \left (2 b -d \right )}-\frac {\cosh \left (2 a +c +\left (2 b +d \right ) x \right )}{8 \left (2 b +d \right )}+\frac {\cosh \left (\left (4 b -d \right ) x +4 a -c \right )}{64 b -16 d}+\frac {\cosh \left (\left (4 b +d \right ) x +4 a +c \right )}{64 b +16 d}\)	\(202\)
risch	\(\frac {\left (16 b^{3} {\mathrm e}^{8 b x +8 a}-4 b^{2} d \,{\mathrm e}^{8 b x +8 a}-4 b \,d^{2} {\mathrm e}^{8 b x +8 a}+d^{3} {\mathrm e}^{8 b x +8 a}-64 b^{3} {\mathrm e}^{6 b x +6 a}+32 b^{2} d \,{\mathrm e}^{6 b x +6 a}+4 b \,d^{2} {\mathrm e}^{6 b x +6 a}-2 d^{3} {\mathrm e}^{6 b x +6 a}-64 b^{3} {\mathrm e}^{2 b x +2 a}-32 b^{2} d \,{\mathrm e}^{2 b x +2 a}+4 b \,d^{2} {\mathrm e}^{2 b x +2 a}+2 d^{3} {\mathrm e}^{2 b x +2 a}+16 b^{3}+4 b^{2} d -4 d^{2} b -d^{3}\right ) {\mathrm e}^{-4 b x +d x -4 a +c}}{16 \left (4 b +d \right ) \left (2 b +d \right ) \left (4 b -d \right ) \left (2 b -d \right )}\)	\(244\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sinh(2*a-c+(2*b-d)*x)/(2*b-d)-1/8*sinh(2*a+c+(2*b+d)*x)/(2*b+d)-1/16/(4*b-d)*sinh((4*b-d)*x+4*a-c)+1/16/(4
*b+d)*sinh((4*b+d)*x+4*a+c)-1/8*cosh(2*a-c+(2*b-d)*x)/(2*b-d)-1/8*cosh(2*a+c+(2*b+d)*x)/(2*b+d)+1/16*cosh((4*b
-d)*x+4*a-c)/(4*b-d)+1/16*cosh((4*b+d)*x+4*a+c)/(4*b+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(exp(d*x+c)*cosh(b*x+a)*sinh(b*x+a)^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(1-d/b>0)', see `assume?` for m
ore details)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (125) = 250\).
time = 0.36, size = 505, normalized size = 3.69 \begin {gather*} -\frac {{\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{4} + {\left (16 \, b^{3} - b d^{2} - 6 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} - {\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - {\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} - {\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{4} - {\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (4 \, b^{3} - b d^{2}\right )} \sinh \left (b x + a\right )^{4} - {\left (16 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (16 \, b^{3} - b d^{2} - 6 \, {\left (4 \, b^{3} - b d^{2}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} - {\left ({\left (4 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )^{3} - {\left (16 \, b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (64 \, b^{4} - 20 \, b^{2} d^{2} + d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((4*b^2*d - d^3)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)^3 - (4*b^3 - b*d^2)*cosh(d*x + c)*sinh(b*x + a
)^4 + (16*b^3 - b*d^2 - 6*(4*b^3 - b*d^2)*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a)^2 + ((4*b^2*d - d^3)*co
sh(b*x + a)^3 - (16*b^2*d - d^3)*cosh(b*x + a))*cosh(d*x + c)*sinh(b*x + a) - ((4*b^3 - b*d^2)*cosh(b*x + a)^4
- (16*b^3 - b*d^2)*cosh(b*x + a)^2)*cosh(d*x + c) - ((4*b^3 - b*d^2)*cosh(b*x + a)^4 - (4*b^2*d - d^3)*cosh(b
*x + a)*sinh(b*x + a)^3 + (4*b^3 - b*d^2)*sinh(b*x + a)^4 - (16*b^3 - b*d^2)*cosh(b*x + a)^2 - (16*b^3 - b*d^2
- 6*(4*b^3 - b*d^2)*cosh(b*x + a)^2)*sinh(b*x + a)^2 - ((4*b^2*d - d^3)*cosh(b*x + a)^3 - (16*b^2*d - d^3)*co
sh(b*x + a))*sinh(b*x + a))*sinh(d*x + c))/((64*b^4 - 20*b^2*d^2 + d^4)*cosh(b*x + a)^4 - 2*(64*b^4 - 20*b^2*d
^2 + d^4)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (64*b^4 - 20*b^2*d^2 + d^4)*sinh(b*x + a)^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1292 vs. \(2 (114) = 228\).
time = 8.85, size = 1292, normalized size = 9.43 \begin {gather*} \begin {cases} x e^{c} \sinh ^{3}{\left (a \right )} \cosh {\left (a \right )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{2} \right )}}{8} + \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{2} \right )} \cosh {\left (a - \frac {d x}{2} \right )}}{4} - \frac {x e^{c} e^{d x} \sinh {\left (a - \frac {d x}{2} \right )} \cosh ^{3}{\left (a - \frac {d x}{2} \right )}}{4} - \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{2} \right )}}{8} - \frac {e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{2} \right )}}{8 d} - \frac {e^{c} e^{d x} \sinh ^{2}{\left (a - \frac {d x}{2} \right )} \cosh ^{2}{\left (a - \frac {d x}{2} \right )}}{2 d} - \frac {e^{c} e^{d x} \sinh {\left (a - \frac {d x}{2} \right )} \cosh ^{3}{\left (a - \frac {d x}{2} \right )}}{3 d} - \frac {e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{2} \right )}}{24 d} & \text {for}\: b = - \frac {d}{2} \\\frac {x e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{4} \right )}}{16} + \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{4} \right )} \cosh {\left (a - \frac {d x}{4} \right )}}{4} + \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a - \frac {d x}{4} \right )} \cosh ^{2}{\left (a - \frac {d x}{4} \right )}}{8} + \frac {x e^{c} e^{d x} \sinh {\left (a - \frac {d x}{4} \right )} \cosh ^{3}{\left (a - \frac {d x}{4} \right )}}{4} + \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{4} \right )}}{16} + \frac {e^{c} e^{d x} \sinh ^{4}{\left (a - \frac {d x}{4} \right )}}{6 d} + \frac {11 e^{c} e^{d x} \sinh ^{3}{\left (a - \frac {d x}{4} \right )} \cosh {\left (a - \frac {d x}{4} \right )}}{12 d} - \frac {5 e^{c} e^{d x} \sinh {\left (a - \frac {d x}{4} \right )} \cosh ^{3}{\left (a - \frac {d x}{4} \right )}}{12 d} - \frac {e^{c} e^{d x} \cosh ^{4}{\left (a - \frac {d x}{4} \right )}}{6 d} & \text {for}\: b = - \frac {d}{4} \\- \frac {x e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{4} \right )}}{16} + \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{4} \right )} \cosh {\left (a + \frac {d x}{4} \right )}}{4} - \frac {3 x e^{c} e^{d x} \sinh ^{2}{\left (a + \frac {d x}{4} \right )} \cosh ^{2}{\left (a + \frac {d x}{4} \right )}}{8} + \frac {x e^{c} e^{d x} \sinh {\left (a + \frac {d x}{4} \right )} \cosh ^{3}{\left (a + \frac {d x}{4} \right )}}{4} - \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{4} \right )}}{16} - \frac {e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{4} \right )}}{6 d} + \frac {11 e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{4} \right )} \cosh {\left (a + \frac {d x}{4} \right )}}{12 d} - \frac {5 e^{c} e^{d x} \sinh {\left (a + \frac {d x}{4} \right )} \cosh ^{3}{\left (a + \frac {d x}{4} \right )}}{12 d} + \frac {e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{4} \right )}}{6 d} & \text {for}\: b = \frac {d}{4} \\- \frac {x e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{2} \right )}}{8} + \frac {x e^{c} e^{d x} \sinh ^{3}{\left (a + \frac {d x}{2} \right )} \cosh {\left (a + \frac {d x}{2} \right )}}{4} - \frac {x e^{c} e^{d x} \sinh {\left (a + \frac {d x}{2} \right )} \cosh ^{3}{\left (a + \frac {d x}{2} \right )}}{4} + \frac {x e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{2} \right )}}{8} + \frac {e^{c} e^{d x} \sinh ^{4}{\left (a + \frac {d x}{2} \right )}}{8 d} + \frac {e^{c} e^{d x} \sinh ^{2}{\left (a + \frac {d x}{2} \right )} \cosh ^{2}{\left (a + \frac {d x}{2} \right )}}{2 d} - \frac {e^{c} e^{d x} \sinh {\left (a + \frac {d x}{2} \right )} \cosh ^{3}{\left (a + \frac {d x}{2} \right )}}{3 d} + \frac {e^{c} e^{d x} \cosh ^{4}{\left (a + \frac {d x}{2} \right )}}{24 d} & \text {for}\: b = \frac {d}{2} \\\frac {10 b^{3} e^{c} e^{d x} \sinh ^{4}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} + \frac {12 b^{3} e^{c} e^{d x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} - \frac {6 b^{3} e^{c} e^{d x} \cosh ^{4}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} - \frac {10 b^{2} d e^{c} e^{d x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} + \frac {6 b^{2} d e^{c} e^{d x} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} - \frac {b d^{2} e^{c} e^{d x} \sinh ^{4}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} - \frac {3 b d^{2} e^{c} e^{d x} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} + \frac {d^{3} e^{c} e^{d x} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{4} - 20 b^{2} d^{2} + d^{4}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*exp(c)*sinh(a)**3*cosh(a), Eq(b, 0) & Eq(d, 0)), (x*exp(c)*exp(d*x)*sinh(a - d*x/2)**4/8 + x*exp(
c)*exp(d*x)*sinh(a - d*x/2)**3*cosh(a - d*x/2)/4 - x*exp(c)*exp(d*x)*sinh(a - d*x/2)*cosh(a - d*x/2)**3/4 - x*
exp(c)*exp(d*x)*cosh(a - d*x/2)**4/8 - exp(c)*exp(d*x)*sinh(a - d*x/2)**4/(8*d) - exp(c)*exp(d*x)*sinh(a - d*x
/2)**2*cosh(a - d*x/2)**2/(2*d) - exp(c)*exp(d*x)*sinh(a - d*x/2)*cosh(a - d*x/2)**3/(3*d) - exp(c)*exp(d*x)*c
osh(a - d*x/2)**4/(24*d), Eq(b, -d/2)), (x*exp(c)*exp(d*x)*sinh(a - d*x/4)**4/16 + x*exp(c)*exp(d*x)*sinh(a -
d*x/4)**3*cosh(a - d*x/4)/4 + 3*x*exp(c)*exp(d*x)*sinh(a - d*x/4)**2*cosh(a - d*x/4)**2/8 + x*exp(c)*exp(d*x)*
sinh(a - d*x/4)*cosh(a - d*x/4)**3/4 + x*exp(c)*exp(d*x)*cosh(a - d*x/4)**4/16 + exp(c)*exp(d*x)*sinh(a - d*x/
4)**4/(6*d) + 11*exp(c)*exp(d*x)*sinh(a - d*x/4)**3*cosh(a - d*x/4)/(12*d) - 5*exp(c)*exp(d*x)*sinh(a - d*x/4)
*cosh(a - d*x/4)**3/(12*d) - exp(c)*exp(d*x)*cosh(a - d*x/4)**4/(6*d), Eq(b, -d/4)), (-x*exp(c)*exp(d*x)*sinh(
a + d*x/4)**4/16 + x*exp(c)*exp(d*x)*sinh(a + d*x/4)**3*cosh(a + d*x/4)/4 - 3*x*exp(c)*exp(d*x)*sinh(a + d*x/4
)**2*cosh(a + d*x/4)**2/8 + x*exp(c)*exp(d*x)*sinh(a + d*x/4)*cosh(a + d*x/4)**3/4 - x*exp(c)*exp(d*x)*cosh(a
+ d*x/4)**4/16 - exp(c)*exp(d*x)*sinh(a + d*x/4)**4/(6*d) + 11*exp(c)*exp(d*x)*sinh(a + d*x/4)**3*cosh(a + d*x
/4)/(12*d) - 5*exp(c)*exp(d*x)*sinh(a + d*x/4)*cosh(a + d*x/4)**3/(12*d) + exp(c)*exp(d*x)*cosh(a + d*x/4)**4/
(6*d), Eq(b, d/4)), (-x*exp(c)*exp(d*x)*sinh(a + d*x/2)**4/8 + x*exp(c)*exp(d*x)*sinh(a + d*x/2)**3*cosh(a + d
*x/2)/4 - x*exp(c)*exp(d*x)*sinh(a + d*x/2)*cosh(a + d*x/2)**3/4 + x*exp(c)*exp(d*x)*cosh(a + d*x/2)**4/8 + ex
p(c)*exp(d*x)*sinh(a + d*x/2)**4/(8*d) + exp(c)*exp(d*x)*sinh(a + d*x/2)**2*cosh(a + d*x/2)**2/(2*d) - exp(c)*
exp(d*x)*sinh(a + d*x/2)*cosh(a + d*x/2)**3/(3*d) + exp(c)*exp(d*x)*cosh(a + d*x/2)**4/(24*d), Eq(b, d/2)), (1
0*b**3*exp(c)*exp(d*x)*sinh(a + b*x)**4/(64*b**4 - 20*b**2*d**2 + d**4) + 12*b**3*exp(c)*exp(d*x)*sinh(a + b*x
)**2*cosh(a + b*x)**2/(64*b**4 - 20*b**2*d**2 + d**4) - 6*b**3*exp(c)*exp(d*x)*cosh(a + b*x)**4/(64*b**4 - 20*
b**2*d**2 + d**4) - 10*b**2*d*exp(c)*exp(d*x)*sinh(a + b*x)**3*cosh(a + b*x)/(64*b**4 - 20*b**2*d**2 + d**4) +
6*b**2*d*exp(c)*exp(d*x)*sinh(a + b*x)*cosh(a + b*x)**3/(64*b**4 - 20*b**2*d**2 + d**4) - b*d**2*exp(c)*exp(d
*x)*sinh(a + b*x)**4/(64*b**4 - 20*b**2*d**2 + d**4) - 3*b*d**2*exp(c)*exp(d*x)*sinh(a + b*x)**2*cosh(a + b*x)
**2/(64*b**4 - 20*b**2*d**2 + d**4) + d**3*exp(c)*exp(d*x)*sinh(a + b*x)**3*cosh(a + b*x)/(64*b**4 - 20*b**2*d
**2 + d**4), True))

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Giac [A]
time = 0.40, size = 93, normalized size = 0.68 \begin {gather*} \frac {e^{\left (4 \, b x + d x + 4 \, a + c\right )}}{16 \, {\left (4 \, b + d\right )}} - \frac {e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{8 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{8 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (-4 \, b x + d x - 4 \, a + c\right )}}{16 \, {\left (4 \, b - d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*e^(4*b*x + d*x + 4*a + c)/(4*b + d) - 1/8*e^(2*b*x + d*x + 2*a + c)/(2*b + d) - 1/8*e^(-2*b*x + d*x - 2*a
+ c)/(2*b - d) + 1/16*e^(-4*b*x + d*x - 4*a + c)/(4*b - d)

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Mupad [B]
time = 1.14, size = 228, normalized size = 1.66 \begin {gather*} \frac {b\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^4\,\left (10\,b^2-d^2\right )}{64\,b^4-20\,b^2\,d^2+d^4}-\frac {6\,b^3\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,{\mathrm {e}}^{c+d\,x}}{64\,b^4-20\,b^2\,d^2+d^4}+\frac {3\,b\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{16\,b^2-d^2}-\frac {d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (10\,b^2-d^2\right )}{64\,b^4-20\,b^2\,d^2+d^4}+\frac {6\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {e}}^{c+d\,x}\,\mathrm {sinh}\left (a+b\,x\right )}{\left (4\,b^2-d^2\right )\,\left (16\,b^2-d^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*exp(c + d*x)*sinh(a + b*x)^4*(10*b^2 - d^2))/(64*b^4 + d^4 - 20*b^2*d^2) - (6*b^3*cosh(a + b*x)^4*exp(c + d
*x))/(64*b^4 + d^4 - 20*b^2*d^2) + (3*b*cosh(a + b*x)^2*exp(c + d*x)*sinh(a + b*x)^2)/(16*b^2 - d^2) - (d*cosh
(a + b*x)*exp(c + d*x)*sinh(a + b*x)^3*(10*b^2 - d^2))/(64*b^4 + d^4 - 20*b^2*d^2) + (6*b^2*d*cosh(a + b*x)^3*
exp(c + d*x)*sinh(a + b*x))/((4*b^2 - d^2)*(16*b^2 - d^2))

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