Integrand size = 27, antiderivative size = 76 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3 \log (a+b \sinh (c+d x))}{b^4 d}+\frac {a^2 \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \]
-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
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-6 a^3 \log (a+b \sinh (c+d x))+6 a^2 b \sinh (c+d x)-3 a b^2 \sinh ^2(c+d x)+2 b^3 \sinh ^3(c+d x)}{6 b^4 d} \]
(-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)
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3042, 26, 3312, 26, 27, 49, 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.
\(\displaystyle \int \frac {\sinh ^3(c+d x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sin (i c+i d x)^3 \cos (i c+i d x)}{a-i b \sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cos (i c+i d x) \sin (i c+i d x)^3}{a-i b \sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 3312 |
\(\displaystyle \frac {i \int -\frac {i \sinh ^3(c+d x)}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\int \frac {\sinh ^3(c+d x)}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {b^3 \sinh ^3(c+d x)}{a+b \sinh (c+d x)}d(b \sinh (c+d x))}{b^4 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (-\frac {a^3}{a+b \sinh (c+d x)}+a^2-b \sinh (c+d x) a+b^2 \sinh ^2(c+d x)\right )d(b \sinh (c+d x))}{b^4 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 (-\log (a+b \sinh (c+d x)))+a^2 b \sinh (c+d x)-\frac {1}{2} a b^2 \sinh ^2(c+d x)+\frac {1}{3} b^3 \sinh ^3(c+d x)}{b^4 d}\) |
(-(a^3*Log[a + b*Sinh[c + d*x]]) + a^2*b*Sinh[c + d*x] - (a*b^2*Sinh[c + d *x]^2)/2 + (b^3*Sinh[c + d*x]^3)/3)/(b^4*d)
3.4.94.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 13.60 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\sinh \left (d x +c \right )^{3} b^{2}}{3}-\frac {a \sinh \left (d x +c \right )^{2} b}{2}+a^{2} \sinh \left (d x +c \right )}{b^{3}}-\frac {a^{3} \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{4}}}{d}\) | \(65\) |
default | \(\frac {\frac {\frac {\sinh \left (d x +c \right )^{3} b^{2}}{3}-\frac {a \sinh \left (d x +c \right )^{2} b}{2}+a^{2} \sinh \left (d x +c \right )}{b^{3}}-\frac {a^{3} \ln \left (a +b \sinh \left (d x +c \right )\right )}{b^{4}}}{d}\) | \(65\) |
risch | \(\frac {a^{3} x}{b^{4}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}-\frac {{\mathrm e}^{d x +c}}{8 b d}-\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}+\frac {{\mathrm e}^{-d x -c}}{8 b d}-\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\frac {2 a^{3} c}{b^{4} d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{4} d}\) | \(195\) |
1/d*(1/b^3*(1/3*sinh(d*x+c)^3*b^2-1/2*a*sinh(d*x+c)^2*b+a^2*sinh(d*x+c))-a ^3/b^4*ln(a+b*sinh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (72) = 144\).
Time = 0.27 (sec) , antiderivative size = 602, normalized size of antiderivative = 7.92 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^{3} \cosh \left (d x + c\right )^{6} + b^{3} \sinh \left (d x + c\right )^{6} + 24 \, a^{3} d x \cosh \left (d x + c\right )^{3} - 3 \, a b^{2} \cosh \left (d x + c\right )^{5} + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right ) - a b^{2}\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{2} - 5 \, a b^{2} \cosh \left (d x + c\right ) + 4 \, a^{2} b - b^{3}\right )} \sinh \left (d x + c\right )^{4} - 3 \, a b^{2} \cosh \left (d x + c\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{3} + 12 \, a^{3} d x - 15 \, a b^{2} \cosh \left (d x + c\right )^{2} + 6 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - b^{3} - 3 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 24 \, a^{3} d x \cosh \left (d x + c\right ) - 10 \, a b^{2} \cosh \left (d x + c\right )^{3} - 4 \, a^{2} b + b^{3} + 6 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left (a^{3} \cosh \left (d x + c\right )^{3} + 3 \, a^{3} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{3} \sinh \left (d x + c\right )^{3}\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 ) + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right )^{5} + 24 \, a^{3} d x \cosh \left (d x + c\right )^{2} - 5 \, a b^{2} \cosh \left (d x + c\right )^{4} + 4 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{3} - a b^{2} - 2 \, {\left (4 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (b^{4} d \cosh \left (d x + c\right )^{3} + 3 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{3}\right )}} \]
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)*co sh(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*sinh(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)
Time = 0.92 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.38 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\begin {cases} \frac {x \sinh ^{3}{\left (c \right )} \cosh {\left (c \right )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sinh ^{4}{\left (c + d x \right )}}{4 a d} & \text {for}\: b = 0 \\\frac {x \sinh ^{3}{\left (c \right )} \cosh {\left (c \right )}}{a + b \sinh {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {a^{3} \log {\left (\frac {a}{b} + \sinh {\left (c + d x \right )} \right )}}{b^{4} d} + \frac {a^{2} \sinh {\left (c + d x \right )}}{b^{3} d} - \frac {a \cosh ^{2}{\left (c + d x \right )}}{2 b^{2} d} + \frac {\sinh ^{3}{\left (c + d x \right )}}{3 b d} & \text {otherwise} \end {cases} \]
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*cosh(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*co sh(c + d*x)**2/(2*b**2*d) + sinh(c + d*x)**3/(3*b*d), True))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (72) = 144\).
Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.25 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {{\left (d x + c\right )} a^{3}}{b^{4} d} - \frac {a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{4} d} - \frac {{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \, {\left (4 \, a^{2} - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, b^{3} d} - \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (4 \, a^{2} - b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, b^{3} d} \]
-(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)
Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.54 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {24 \, a^{3} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{4}} - \frac {b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 3 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 12 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{3}}}{24 \, d} \]
-1/24*(24*a^3*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/b^4 - (b^2*(e ^(d*x + c) - e^(-d*x - c))^3 - 3*a*b*(e^(d*x + c) - e^(-d*x - c))^2 + 12*a ^2*(e^(d*x + c) - e^(-d*x - c)))/b^3)/d
Time = 1.00 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \frac {\cosh (c+d x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^3\,\ln \left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )-\frac {b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{3}+\frac {a\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{2}-a^2\,b\,\mathrm {sinh}\left (c+d\,x\right )}{b^4\,d} \]