Integrand size = 21, antiderivative size = 47 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {(a+b) \coth (c+d x)}{d}+\frac {2 a \coth ^3(c+d x)}{3 d}-\frac {a \coth ^5(c+d x)}{5 d} \]
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.51 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {8 a \coth (c+d x)}{15 d}-\frac {b \coth (c+d x)}{d}+\frac {4 a \coth (c+d x) \text {csch}^2(c+d x)}{15 d}-\frac {a \coth (c+d x) \text {csch}^4(c+d x)}{5 d} \]
(-8*a*Coth[c + d*x])/(15*d) - (b*Coth[c + d*x])/d + (4*a*Coth[c + d*x]*Csc h[c + d*x]^2)/(15*d) - (a*Coth[c + d*x]*Csch[c + d*x]^4)/(5*d)
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 25, 3696, 1433, 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.
\(\displaystyle \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {a+b \sin (i c+i d x)^4}{\sin (i c+i d x)^6}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {b \sin (i c+i d x)^4+a}{\sin (i c+i d x)^6}dx\) |
\(\Big \downarrow \) 3696 |
\(\displaystyle \frac {\int \coth ^6(c+d x) \left ((a+b) \tanh ^4(c+d x)-2 a \tanh ^2(c+d x)+a\right )d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 1433 |
\(\displaystyle \frac {\int \left (a \coth ^6(c+d x)-2 a \coth ^4(c+d x)+(a+b) \coth ^2(c+d x)\right )d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-(a+b) \coth (c+d x)-\frac {1}{5} a \coth ^5(c+d x)+\frac {2}{3} a \coth ^3(c+d x)}{d}\) |
3.2.94.3.1 Defintions of rubi rules used
Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || !IntegerQ[(m + 1)/2])
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1 )/f Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2) ^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] & & IntegerQ[m/2] && IntegerQ[p]
Time = 0.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )-b \coth \left (d x +c \right )}{d}\) | \(45\) |
default | \(\frac {a \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )-b \coth \left (d x +c \right )}{d}\) | \(45\) |
parallelrisch | \(\frac {-3 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a +25 a \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-150 a -240 b \right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a -\frac {25 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{3}+50 a +80 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{480 d}\) | \(97\) |
risch | \(-\frac {2 \left (15 b \,{\mathrm e}^{8 d x +8 c}-60 b \,{\mathrm e}^{6 d x +6 c}+80 \,{\mathrm e}^{4 d x +4 c} a +90 b \,{\mathrm e}^{4 d x +4 c}-40 a \,{\mathrm e}^{2 d x +2 c}-60 b \,{\mathrm e}^{2 d x +2 c}+8 a +15 b \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}\) | \(98\) |
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (43) = 86\).
Time = 0.26 (sec) , antiderivative size = 333, normalized size of antiderivative = 7.09 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {4 \, {\left ({\left (4 \, a + 15 \, b\right )} \cosh \left (d x + c\right )^{4} - 16 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (4 \, a + 15 \, b\right )} \sinh \left (d x + c\right )^{4} - 20 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (4 \, a + 15 \, b\right )} \cosh \left (d x + c\right )^{2} - 10 \, a - 30 \, b\right )} \sinh \left (d x + c\right )^{2} - 8 \, {\left (2 \, a \cosh \left (d x + c\right )^{3} - 5 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 40 \, a + 45 \, b\right )}}{15 \, {\left (d \cosh \left (d x + c\right )^{6} + 6 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + d \sinh \left (d x + c\right )^{6} - 6 \, d \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} - 2 \, d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} - 4 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 15 \, d \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, d \cosh \left (d x + c\right )^{4} - 12 \, d \cosh \left (d x + c\right )^{2} + 5 \, d\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} - 8 \, d \cosh \left (d x + c\right )^{3} + 5 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 10 \, d\right )}} \]
-4/15*((4*a + 15*b)*cosh(d*x + c)^4 - 16*a*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a + 15*b)*sinh(d*x + c)^4 - 20*(a + 3*b)*cosh(d*x + c)^2 + 2*(3*(4*a + 15*b)*cosh(d*x + c)^2 - 10*a - 30*b)*sinh(d*x + c)^2 - 8*(2*a*cosh(d*x + c)^3 - 5*a*cosh(d*x + c))*sinh(d*x + c) + 40*a + 45*b)/(d*cosh(d*x + c)^6 + 6*d*cosh(d*x + c)*sinh(d*x + c)^5 + d*sinh(d*x + c)^6 - 6*d*cosh(d*x + c )^4 + 3*(5*d*cosh(d*x + c)^2 - 2*d)*sinh(d*x + c)^4 + 4*(5*d*cosh(d*x + c) ^3 - 4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 15*d*cosh(d*x + c)^2 + 3*(5*d*co sh(d*x + c)^4 - 12*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^2 + 2*(3*d*cosh( d*x + c)^5 - 8*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c) - 10*d )
Timed out. \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (43) = 86\).
Time = 0.19 (sec) , antiderivative size = 228, normalized size of antiderivative = 4.85 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {16}{15} \, a {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + \frac {2 \, b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
-16/15*a*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10 *e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6* d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 1/(d*(5*e^(-2 *d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8* c) + e^(-10*d*x - 10*c) - 1))) + 2*b/(d*(e^(-2*d*x - 2*c) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (43) = 86\).
Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.06 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 \, b e^{\left (8 \, d x + 8 \, c\right )} - 60 \, b e^{\left (6 \, d x + 6 \, c\right )} + 80 \, a e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b e^{\left (4 \, d x + 4 \, c\right )} - 40 \, a e^{\left (2 \, d x + 2 \, c\right )} - 60 \, b e^{\left (2 \, d x + 2 \, c\right )} + 8 \, a + 15 \, b\right )}}{15 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}} \]
-2/15*(15*b*e^(8*d*x + 8*c) - 60*b*e^(6*d*x + 6*c) + 80*a*e^(4*d*x + 4*c) + 90*b*e^(4*d*x + 4*c) - 40*a*e^(2*d*x + 2*c) - 60*b*e^(2*d*x + 2*c) + 8*a + 15*b)/(d*(e^(2*d*x + 2*c) - 1)^5)
Time = 1.48 (sec) , antiderivative size = 337, normalized size of antiderivative = 7.17 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx=\frac {\frac {2\,b}{5\,d}+\frac {6\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}-\frac {2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a+3\,b\right )}{5\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {2\,\left (8\,a+3\,b\right )}{15\,d}-\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}+\frac {2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {\frac {2\,b}{5\,d}-\frac {8\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}-\frac {8\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}+\frac {2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a+3\,b\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {4\,b}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
((2*b)/(5*d) + (6*b*exp(4*c + 4*d*x))/(5*d) - (2*b*exp(6*c + 6*d*x))/(5*d) - (2*exp(2*c + 2*d*x)*(8*a + 3*b))/(5*d))/(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((2*(8*a + 3*b))/ (15*d) - (4*b*exp(2*c + 2*d*x))/(5*d) + (2*b*exp(4*c + 4*d*x))/(5*d))/(3*e xp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((2*b)/(5*d ) - (8*b*exp(2*c + 2*d*x))/(5*d) - (8*b*exp(6*c + 6*d*x))/(5*d) + (2*b*exp (8*c + 8*d*x))/(5*d) + (4*exp(4*c + 4*d*x)*(8*a + 3*b))/(5*d))/(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1) - (4*b)/(5*d*(exp(2*c + 2*d*x) - 1))