\(\int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [302]

Optimal result

Integrand size = 21, antiderivative size = 59 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^3 d}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{2 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2747, 711} \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^3 d}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{2 b d} \]

[In]

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

[Out]

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

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {-\left (\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))\right )+a b \sinh (c+d x)-\frac {1}{2} b^2 \sinh ^2(c+d x)}{b^3 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.86 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {-\frac {-\frac {\sinh \left (d x +c \right )^{2} b}{2}+a \sinh \left (d x +c \right )}{b^{2}}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{3}}}{d}\)	\(53\)
default	\(\frac {-\frac {-\frac {\sinh \left (d x +c \right )^{2} b}{2}+a \sinh \left (d x +c \right )}{b^{2}}+\frac {\left (a^{2}+b^{2}\right ) \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{3}}}{d}\)	\(53\)
risch	\(-\frac {x \,a^{2}}{b^{3}}-\frac {x}{b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {a \,{\mathrm e}^{d x +c}}{2 b^{2} d}+\frac {a \,{\mathrm e}^{-d x -c}}{2 b^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}-\frac {2 a^{2} c}{b^{3} d}-\frac {2 c}{b d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right ) a^{2}}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b d}\)	\(170\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (57) = 114\).

Time = 0.24 (sec) , antiderivative size = 327, normalized size of antiderivative = 5.54 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{2} \cosh \left (d x + c\right )^{4} + b^{2} \sinh \left (d x + c\right )^{4} - 8 \, {\left (a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (b^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} + 4 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - 4 \, {\left (a^{2} + b^{2}\right )} d x - 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 8 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - 4 \, {\left (a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right ) - 3 \, a b \cosh \left (d x + c\right )^{2} + a b\right )} \sinh \left (d x + c\right )}{8 \, {\left (b^{3} d \cosh \left (d x + c\right )^{2} + 2 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} d \sinh \left (d x + c\right )^{2}\right )}} \]

[In]

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

[Out]

1/8*(b^2*cosh(d*x + c)^4 + b^2*sinh(d*x + c)^4 - 8*(a^2 + b^2)*d*x*cosh(d*x + c)^2 - 4*a*b*cosh(d*x + c)^3 + 4
*(b^2*cosh(d*x + c) - a*b)*sinh(d*x + c)^3 + 4*a*b*cosh(d*x + c) + 2*(3*b^2*cosh(d*x + c)^2 - 4*(a^2 + b^2)*d*
x - 6*a*b*cosh(d*x + c))*sinh(d*x + c)^2 + b^2 + 8*((a^2 + b^2)*cosh(d*x + c)^2 + 2*(a^2 + b^2)*cosh(d*x + c)*
sinh(d*x + c) + (a^2 + b^2)*sinh(d*x + c)^2)*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + 4*
(b^2*cosh(d*x + c)^3 - 4*(a^2 + b^2)*d*x*cosh(d*x + c) - 3*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c))/(b^3*d*co
sh(d*x + c)^2 + 2*b^3*d*cosh(d*x + c)*sinh(d*x + c) + b^3*d*sinh(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (57) = 114\).

Time = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.15 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, b^{2} d} + \frac {{\left (a^{2} + b^{2}\right )} {\left (d x + c\right )}}{b^{3} d} + \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, b^{2} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{3} d} \]

[In]

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

[Out]

-1/8*(4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + (a^2 + b^2)*(d*x + c)/(b^3*d) + 1/8*(4*a*e^(-d*x - c) +
b*e^(-2*d*x - 2*c))/(b^2*d) + (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^3*d)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.56 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{2}} + \frac {8 \, {\left (a^{2} + b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{3}}}{8 \, d} \]

[In]

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

[Out]

1/8*((b*(e^(d*x + c) - e^(-d*x - c))^2 - 4*a*(e^(d*x + c) - e^(-d*x - c)))/b^2 + 8*(a^2 + b^2)*log(abs(b*(e^(d
*x + c) - e^(-d*x - c)) + 2*a))/b^3)/d

Mupad [B] (verification not implemented)

Time = 1.05 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.03 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}-\frac {x\,\left (a^2+b^2\right )}{b^3}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}+\frac {\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^2+b^2\right )}{b^3\,d}+\frac {a\,{\mathrm {e}}^{-c-d\,x}}{2\,b^2\,d}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{2\,b^2\,d} \]

[In]

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

[Out]

exp(- 2*c - 2*d*x)/(8*b*d) - (x*(a^2 + b^2))/b^3 + exp(2*c + 2*d*x)/(8*b*d) + (log(2*a*exp(d*x)*exp(c) - b + b
*exp(2*c)*exp(2*d*x))*(a^2 + b^2))/(b^3*d) + (a*exp(- c - d*x))/(2*b^2*d) - (a*exp(c + d*x))/(2*b^2*d)