Integrand size = 21, antiderivative size = 59 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^3 d}-\frac {a \sinh (c+d x)}{b^2 d}+\frac {\sinh ^2(c+d x)}{2 b d} \] Output:
(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
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {-\left (\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))\right )+a b \sinh (c+d x)-\frac {1}{2} b^2 \sinh ^2(c+d x)}{b^3 d} \] Input:
Integrate[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]),x]
Output:
-((-((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))
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3147, 25, 476, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i c+i d x)^3}{a-i b \sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 3147 |
\(\displaystyle -\frac {\int -\frac {\sinh ^2(c+d x) b^2+b^2}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b^3 d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\sinh ^2(c+d x) b^2+b^2}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b^3 d}\) |
\(\Big \downarrow \) 476 |
\(\displaystyle \frac {\int \left (-a+b \sinh (c+d x)+\frac {a^2+b^2}{a+b \sinh (c+d x)}\right )d(b \sinh (c+d x))}{b^3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))+a b \sinh (c+d x)-\frac {1}{2} b^2 \sinh ^2(c+d x)}{b^3 d}\) |
Input:
Int[Cosh[c + d*x]^3/(a + b*Sinh[c + d*x]),x]
Output:
-((-((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))
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[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] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 5.30 (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 | \(-\frac {x \,a^{2}}{b^{3}}-\frac {x}{b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {a \,{\mathrm e}^{d x +c}}{2 b^{2} d}+\frac {a \,{\mathrm e}^{-d x -c}}{2 b^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}-\frac {2 a^{2} c}{b^{3} d}-\frac {2 c}{b d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right ) a^{2}}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b d}\) | \(170\) |
Input:
int(cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
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)))
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (57) = 114\).
Time = 0.11 (sec) , antiderivative size = 327, normalized size of antiderivative = 5.54 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{2} \cosh \left (d x + c\right )^{4} + b^{2} \sinh \left (d x + c\right )^{4} - 8 \, {\left (a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (b^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} + 4 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - 4 \, {\left (a^{2} + b^{2}\right )} d x - 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 8 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - 4 \, {\left (a^{2} + b^{2}\right )} d x \cosh \left (d x + c\right ) - 3 \, a b \cosh \left (d x + c\right )^{2} + a b\right )} \sinh \left (d x + c\right )}{8 \, {\left (b^{3} d \cosh \left (d x + c\right )^{2} + 2 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} d \sinh \left (d x + c\right )^{2}\right )}} \] Input:
integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
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*cosh(d*x + c)^2 + 2*b^3*d*cosh(d*x + c)*sinh (d*x + c) + b^3*d*sinh(d*x + c)^2)
Timed out. \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (57) = 114\).
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.15 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, b^{2} d} + \frac {{\left (a^{2} + b^{2}\right )} {\left (d x + c\right )}}{b^{3} d} + \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, b^{2} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{3} d} \] Input:
integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-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)
Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.56 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{2}} + \frac {8 \, {\left (a^{2} + b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{3}}}{8 \, d} \] Input:
integrate(cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
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
Time = 1.24 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.03 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}-\frac {x\,\left (a^2+b^2\right )}{b^3}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}+\frac {\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^2+b^2\right )}{b^3\,d}+\frac {a\,{\mathrm {e}}^{-c-d\,x}}{2\,b^2\,d}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{2\,b^2\,d} \] Input:
int(cosh(c + d*x)^3/(a + b*sinh(c + d*x)),x)
Output:
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)
Time = 0.22 (sec) , antiderivative size = 181, normalized size of antiderivative = 3.07 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e^{4 d x +4 c} b^{2}-4 e^{3 d x +3 c} a b +8 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) a^{2}+8 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2}-8 e^{2 d x +2 c} a^{2} d x -8 e^{2 d x +2 c} b^{2} d x +4 e^{d x +c} a b +b^{2}}{8 e^{2 d x +2 c} b^{3} d} \] Input:
int(cosh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(e**(4*c + 4*d*x)*b**2 - 4*e**(3*c + 3*d*x)*a*b + 8*e**(2*c + 2*d*x)*log(e **(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**2 + 8*e**(2*c + 2*d*x)*log(e* *(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*b**2 - 8*e**(2*c + 2*d*x)*a**2*d* x - 8*e**(2*c + 2*d*x)*b**2*d*x + 4*e**(c + d*x)*a*b + b**2)/(8*e**(2*c + 2*d*x)*b**3*d)