Integrand size = 21, antiderivative size = 41 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-\frac {b \text {arctanh}(\cosh (c+d x))}{d}+\frac {a \coth (c+d x)}{d}-\frac {a \coth ^3(c+d x)}{3 d} \]
Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.85 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\frac {2 a \coth (c+d x)}{3 d}-\frac {a \coth (c+d x) \text {csch}^2(c+d x)}{3 d}-\frac {b \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d} \]
(2*a*Coth[c + d*x])/(3*d) - (a*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (b*L og[Cosh[c/2 + (d*x)/2]])/d + (b*Log[Sinh[c/2 + (d*x)/2]])/d
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3699, 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.
\(\displaystyle \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+i b \sin (i c+i d x)^3}{\sin (i c+i d x)^4}dx\) |
\(\Big \downarrow \) 3699 |
\(\displaystyle \int \left (a \text {csch}^4(c+d x)+b \text {csch}(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \coth ^3(c+d x)}{3 d}+\frac {a \coth (c+d x)}{d}-\frac {b \text {arctanh}(\cosh (c+d x))}{d}\) |
3.2.49.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_ ))^(p_.), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n) ^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4] || Gt Q[p, 0] || (EqQ[p, -1] && IntegerQ[n]))
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {a \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-2 b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{d}\) | \(36\) |
default | \(\frac {a \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )-2 b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{d}\) | \(36\) |
risch | \(-\frac {4 a \left (3 \,{\mathrm e}^{2 d x +2 c}-1\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}\) | \(63\) |
parallelrisch | \(\frac {24 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\coth \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\coth \left (\frac {d x}{2}+\frac {c}{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-9\right )}{24 d}\) | \(86\) |
Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (39) = 78\).
Time = 0.26 (sec) , antiderivative size = 652, normalized size of antiderivative = 15.90 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-\frac {12 \, a \cosh \left (d x + c\right )^{2} + 24 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 12 \, a \sinh \left (d x + c\right )^{2} + 3 \, {\left (b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - 3 \, b \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{4} - 6 \, b \cosh \left (d x + c\right )^{2} + b\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (b \cosh \left (d x + c\right )^{5} - 2 \, b \cosh \left (d x + c\right )^{3} + b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - 3 \, {\left (b \cosh \left (d x + c\right )^{6} + 6 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b \sinh \left (d x + c\right )^{6} - 3 \, b \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{2} - b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 3 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b \cosh \left (d x + c\right )^{4} - 6 \, b \cosh \left (d x + c\right )^{2} + b\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (b \cosh \left (d x + c\right )^{5} - 2 \, b \cosh \left (d x + c\right )^{3} + b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) - 4 \, a}{3 \, {\left (d \cosh \left (d x + c\right )^{6} + 6 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + d \sinh \left (d x + c\right )^{6} - 3 \, d \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} - 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} - 6 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 6 \, {\left (d \cosh \left (d x + c\right )^{5} - 2 \, d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - d\right )}} \]
-1/3*(12*a*cosh(d*x + c)^2 + 24*a*cosh(d*x + c)*sinh(d*x + c) + 12*a*sinh( d*x + c)^2 + 3*(b*cosh(d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b* sinh(d*x + c)^6 - 3*b*cosh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d *x + c)^4 + 4*(5*b*cosh(d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 3*(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*si nh(d*x + c)^2 + 6*(b*cosh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 3*(b*cosh( d*x + c)^6 + 6*b*cosh(d*x + c)*sinh(d*x + c)^5 + b*sinh(d*x + c)^6 - 3*b*c osh(d*x + c)^4 + 3*(5*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^4 + 4*(5*b*cosh (d*x + c)^3 - 3*b*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 3 *(5*b*cosh(d*x + c)^4 - 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 6*(b*co sh(d*x + c)^5 - 2*b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) - b)* log(cosh(d*x + c) + sinh(d*x + c) - 1) - 4*a)/(d*cosh(d*x + c)^6 + 6*d*cos h(d*x + c)*sinh(d*x + c)^5 + d*sinh(d*x + c)^6 - 3*d*cosh(d*x + c)^4 + 3*( 5*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^4 + 4*(5*d*cosh(d*x + c)^3 - 3*d*co sh(d*x + c))*sinh(d*x + c)^3 + 3*d*cosh(d*x + c)^2 + 3*(5*d*cosh(d*x + c)^ 4 - 6*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 6*(d*cosh(d*x + c)^5 - 2*d* cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) - d)
Timed out. \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (39) = 78\).
Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 3.20 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} + \frac {4}{3} \, a {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} \]
-b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d) + 4/3*a*(3*e^(-2*d* x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1 )))
Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.51 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-\frac {3 \, b \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {4 \, {\left (3 \, a e^{\left (2 \, d x + 2 \, c\right )} - a\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
-1/3*(3*b*log(e^(d*x + c) + 1) - 3*b*log(abs(e^(d*x + c) - 1)) + 4*(3*a*e^ (2*d*x + 2*c) - a)/(e^(2*d*x + 2*c) - 1)^3)/d
Time = 1.46 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.68 \[ \int \text {csch}^4(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx=-\frac {4\,a}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {2\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {-d^2}} \]