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

Optimal. Leaf size=34 \[ \frac{\sinh (c+d x)}{b d}-\frac{a \log (a+b \sinh (c+d x))}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0569536, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{\sinh (c+d x)}{b d}-\frac{a \log (a+b \sinh (c+d x))}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Log[a + b*Sinh[c + d*x]])/(b^2*d)) + Sinh[c + d*x]/(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 (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{b (a+x)} \, dx,x,b \sinh (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a+x} \, dx,x,b \sinh (c+d x)\right )}{b^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{a+x}\right ) \, dx,x,b \sinh (c+d x)\right )}{b^2 d}\\ &=-\frac{a \log (a+b \sinh (c+d x))}{b^2 d}+\frac{\sinh (c+d x)}{b d}\\ \end{align*}

Mathematica [A]  time = 0.0386967, size = 33, normalized size = 0.97 \[ -\frac{\frac{a \log (a+b \sinh (c+d x))}{b^2}-\frac{\sinh (c+d x)}{b}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 35, normalized size = 1. \begin{align*} -{\frac{a\ln \left ( a+b\sinh \left ( dx+c \right ) \right ) }{{b}^{2}d}}+{\frac{\sinh \left ( dx+c \right ) }{bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.07533, size = 112, normalized size = 3.29 \begin{align*} -\frac{{\left (d x + c\right )} a}{b^{2} d} + \frac{e^{\left (d x + c\right )}}{2 \, b d} - \frac{e^{\left (-d x - c\right )}}{2 \, b d} - \frac{a \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.12315, size = 354, normalized size = 10.41 \begin{align*} \frac{2 \, a d x \cosh \left (d x + c\right ) + b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} - 2 \,{\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\frac{2 \,{\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \,{\left (a d x + b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - b}{2 \,{\left (b^{2} d \cosh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*d*x*cosh(d*x + c) + b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 - 2*(a*cosh(d*x + c) + a*sinh(d*x + c))*log
(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + 2*(a*d*x + b*cosh(d*x + c))*sinh(d*x + c) - b)/(b^
2*d*cosh(d*x + c) + b^2*d*sinh(d*x + c))

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Sympy [A]  time = 1.47607, size = 65, normalized size = 1.91 \begin{align*} \begin{cases} \frac{x \sinh{\left (c \right )} \cosh{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \sinh{\left (c \right )} \cosh{\left (c \right )}}{a + b \sinh{\left (c \right )}} & \text{for}\: d = 0 \\\frac{\sinh ^{2}{\left (c + d x \right )}}{2 a d} & \text{for}\: b = 0 \\- \frac{a \log{\left (\frac{a}{b} + \sinh{\left (c + d x \right )} \right )}}{b^{2} d} + \frac{\sinh{\left (c + d x \right )}}{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)/(a+b*sinh(d*x+c)),x)

[Out]

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

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Giac [A]  time = 1.14879, size = 84, normalized size = 2.47 \begin{align*} \frac{e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}}{2 \, b d} - \frac{a \log \left ({\left | b{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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