3.394 \(\int \frac{\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=76 \[ \frac{a^2 \sinh (c+d x)}{b^3 d}-\frac{a^3 \log (a+b \sinh (c+d x))}{b^4 d}-\frac{a \sinh ^2(c+d x)}{2 b^2 d}+\frac{\sinh ^3(c+d x)}{3 b d} \]

[Out]

-((a^3*Log[a + b*Sinh[c + d*x]])/(b^4*d)) + (a^2*Sinh[c + d*x])/(b^3*d) - (a*Sinh[c + d*x]^2)/(2*b^2*d) + Sinh
[c + d*x]^3/(3*b*d)

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Rubi [A]  time = 0.0966879, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac{a^2 \sinh (c+d x)}{b^3 d}-\frac{a^3 \log (a+b \sinh (c+d x))}{b^4 d}-\frac{a \sinh ^2(c+d x)}{2 b^2 d}+\frac{\sinh ^3(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*Log[a + b*Sinh[c + d*x]])/(b^4*d)) + (a^2*Sinh[c + d*x])/(b^3*d) - (a*Sinh[c + d*x]^2)/(2*b^2*d) + Sinh
[c + d*x]^3/(3*b*d)

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.15399, size = 66, normalized size = 0.87 \[ \frac{6 a^2 b \sinh (c+d x)-6 a^3 \log (a+b \sinh (c+d x))-3 a b^2 \sinh ^2(c+d x)+2 b^3 \sinh ^3(c+d x)}{6 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^3*Log[a + b*Sinh[c + d*x]] + 6*a^2*b*Sinh[c + d*x] - 3*a*b^2*Sinh[c + d*x]^2 + 2*b^3*Sinh[c + d*x]^3)/(6
*b^4*d)

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Maple [A]  time = 0.016, size = 73, normalized size = 1. \begin{align*} -{\frac{{a}^{3}\ln \left ( a+b\sinh \left ( dx+c \right ) \right ) }{{b}^{4}d}}+{\frac{{a}^{2}\sinh \left ( dx+c \right ) }{{b}^{3}d}}-{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}a}{2\,{b}^{2}d}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{3\,bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*ln(a+b*sinh(d*x+c))/b^4/d+a^2*sinh(d*x+c)/b^3/d-1/2*a*sinh(d*x+c)^2/b^2/d+1/3*sinh(d*x+c)^3/b/d

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Maxima [B]  time = 1.12151, size = 231, normalized size = 3.04 \begin{align*} -\frac{{\left (d x + c\right )} a^{3}}{b^{4} d} - \frac{a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{4} d} - \frac{{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \,{\left (4 \, a^{2} - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, b^{3} d} - \frac{3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \,{\left (4 \, a^{2} - b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, b^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*x + c)*a^3/(b^4*d) - a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^4*d) - 1/24*(3*a*b*e^(-d*x - c
) - b^2 - 3*(4*a^2 - b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - 1/24*(3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3
*d*x - 3*c) + 3*(4*a^2 - b^2)*e^(-d*x - c))/(b^3*d)

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Fricas [B]  time = 2.49711, size = 1484, normalized size = 19.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b^3*cosh(d*x + c)^6 + b^3*sinh(d*x + c)^6 + 24*a^3*d*x*cosh(d*x + c)^3 - 3*a*b^2*cosh(d*x + c)^5 + 3*(2*
b^3*cosh(d*x + c) - a*b^2)*sinh(d*x + c)^5 + 3*(4*a^2*b - b^3)*cosh(d*x + c)^4 + 3*(5*b^3*cosh(d*x + c)^2 - 5*
a*b^2*cosh(d*x + c) + 4*a^2*b - b^3)*sinh(d*x + c)^4 - 3*a*b^2*cosh(d*x + c) + 2*(10*b^3*cosh(d*x + c)^3 + 12*
a^3*d*x - 15*a*b^2*cosh(d*x + c)^2 + 6*(4*a^2*b - b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - b^3 - 3*(4*a^2*b - b^3
)*cosh(d*x + c)^2 + 3*(5*b^3*cosh(d*x + c)^4 + 24*a^3*d*x*cosh(d*x + c) - 10*a*b^2*cosh(d*x + c)^3 - 4*a^2*b +
b^3 + 6*(4*a^2*b - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 24*(a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c)^2*si
nh(d*x + c) + 3*a^3*cosh(d*x + c)*sinh(d*x + c)^2 + a^3*sinh(d*x + c)^3)*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x
+ c) - sinh(d*x + c))) + 3*(2*b^3*cosh(d*x + c)^5 + 24*a^3*d*x*cosh(d*x + c)^2 - 5*a*b^2*cosh(d*x + c)^4 + 4*
(4*a^2*b - b^3)*cosh(d*x + c)^3 - a*b^2 - 2*(4*a^2*b - b^3)*cosh(d*x + c))*sinh(d*x + c))/(b^4*d*cosh(d*x + c)
^3 + 3*b^4*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^4*d*cosh(d*x + c)*sinh(d*x + c)^2 + b^4*d*sinh(d*x + c)^3)

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Sympy [A]  time = 2.83729, size = 105, normalized size = 1.38 \begin{align*} \begin{cases} \frac{x \sinh ^{3}{\left (c \right )} \cosh{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sinh ^{4}{\left (c + d x \right )}}{4 a d} & \text{for}\: b = 0 \\\frac{x \sinh ^{3}{\left (c \right )} \cosh{\left (c \right )}}{a + b \sinh{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{a^{3} \log{\left (\frac{a}{b} + \sinh{\left (c + d x \right )} \right )}}{b^{4} d} + \frac{a^{2} \sinh{\left (c + d x \right )}}{b^{3} d} - \frac{a \sinh ^{2}{\left (c + d x \right )}}{2 b^{2} d} + \frac{\sinh ^{3}{\left (c + d x \right )}}{3 b d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(c)**3*cosh(c)/a, Eq(b, 0) & Eq(d, 0)), (sinh(c + d*x)**4/(4*a*d), Eq(b, 0)), (x*sinh(c)**3*c
osh(c)/(a + b*sinh(c)), Eq(d, 0)), (-a**3*log(a/b + sinh(c + d*x))/(b**4*d) + a**2*sinh(c + d*x)/(b**3*d) - a*
sinh(c + d*x)**2/(2*b**2*d) + sinh(c + d*x)**3/(3*b*d), True))

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Giac [A]  time = 1.25732, size = 171, normalized size = 2.25 \begin{align*} -\frac{a^{3} \log \left ({\left | b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{4} d} + \frac{b^{2} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 3 \, a b d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 12 \, a^{2} d^{2}{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{24 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(b^4*d) + 1/24*(b^2*d^2*(e^(d*x + c) - e^(-d*x - c))^3 - 3
*a*b*d^2*(e^(d*x + c) - e^(-d*x - c))^2 + 12*a^2*d^2*(e^(d*x + c) - e^(-d*x - c)))/(b^3*d^3)