Integrand size = 23, antiderivative size = 88 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=b^2 x+\frac {a b \text {arctanh}(\cosh (c+d x))}{d}-\frac {a^2 \coth (c+d x)}{d}+\frac {2 a^2 \coth ^3(c+d x)}{3 d}-\frac {a^2 \coth ^5(c+d x)}{5 d}-\frac {a b \coth (c+d x) \text {csch}(c+d x)}{d} \]
b^2*x+a*b*arctanh(cosh(d*x+c))/d-a^2*coth(d*x+c)/d+2/3*a^2*coth(d*x+c)^3/d -1/5*a^2*coth(d*x+c)^5/d-a*b*coth(d*x+c)*csch(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(216\) vs. \(2(88)=176\).
Time = 0.70 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.45 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {-128 a^2 \coth \left (\frac {1}{2} (c+d x)\right )-120 a b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+\frac {19}{2} a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)-\frac {3}{2} a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)+8 \left (60 b^2 c+60 b^2 d x+60 a b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-60 a b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-15 a b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-19 a^2 \text {csch}^3(c+d x) \sinh ^4\left (\frac {1}{2} (c+d x)\right )-12 a^2 \text {csch}^5(c+d x) \sinh ^6\left (\frac {1}{2} (c+d x)\right )-16 a^2 \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{480 d} \]
(-128*a^2*Coth[(c + d*x)/2] - 120*a*b*Csch[(c + d*x)/2]^2 + (19*a^2*Csch[( c + d*x)/2]^4*Sinh[c + d*x])/2 - (3*a^2*Csch[(c + d*x)/2]^6*Sinh[c + d*x]) /2 + 8*(60*b^2*c + 60*b^2*d*x + 60*a*b*Log[Cosh[(c + d*x)/2]] - 60*a*b*Log [Sinh[(c + d*x)/2]] - 15*a*b*Sech[(c + d*x)/2]^2 - 19*a^2*Csch[c + d*x]^3* Sinh[(c + d*x)/2]^4 - 12*a^2*Csch[c + d*x]^5*Sinh[(c + d*x)/2]^6 - 16*a^2* Tanh[(c + d*x)/2]))/(480*d)
Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 25, 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.
\(\displaystyle \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\left (a+i b \sin (i c+i d x)^3\right )^2}{\sin (i c+i d x)^6}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\left (i b \sin (i c+i d x)^3+a\right )^2}{\sin (i c+i d x)^6}dx\) |
\(\Big \downarrow \) 3699 |
\(\displaystyle -\int \left (-a^2 \text {csch}^6(c+d x)-2 a b \text {csch}^3(c+d x)-b^2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 \coth ^5(c+d x)}{5 d}+\frac {2 a^2 \coth ^3(c+d x)}{3 d}-\frac {a^2 \coth (c+d x)}{d}+\frac {a b \text {arctanh}(\cosh (c+d x))}{d}-\frac {a b \coth (c+d x) \text {csch}(c+d x)}{d}+b^2 x\) |
b^2*x + (a*b*ArcTanh[Cosh[c + d*x]])/d - (a^2*Coth[c + d*x])/d + (2*a^2*Co th[c + d*x]^3)/(3*d) - (a^2*Coth[c + d*x]^5)/(5*d) - (a*b*Coth[c + d*x]*Cs ch[c + d*x])/d
3.2.59.3.1 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 = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )+2 a b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (d x +c \right )}{d}\) | \(73\) |
default | \(\frac {a^{2} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )+2 a b \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (d x +c \right )}{d}\) | \(73\) |
risch | \(b^{2} x -\frac {2 a \left (15 b \,{\mathrm e}^{9 d x +9 c}-30 b \,{\mathrm e}^{7 d x +7 c}+80 \,{\mathrm e}^{4 d x +4 c} a +30 b \,{\mathrm e}^{3 d x +3 c}-40 a \,{\mathrm e}^{2 d x +2 c}-15 \,{\mathrm e}^{d x +c} b +8 a \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}+\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d}-\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d}\) | \(130\) |
parallelrisch | \(\frac {-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{2}-3 a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+25 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}+25 a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+120 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a b -120 a b \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+480 b^{2} d x -150 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}-150 a^{2} \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-480 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b}{480 d}\) | \(150\) |
1/d*(a^2*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c)+2*a*b*(- 1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+b^2*(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 2310 vs. \(2 (84) = 168\).
Time = 0.30 (sec) , antiderivative size = 2310, normalized size of antiderivative = 26.25 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\text {Too large to display} \]
1/15*(15*b^2*d*x*cosh(d*x + c)^10 + 15*b^2*d*x*sinh(d*x + c)^10 - 75*b^2*d *x*cosh(d*x + c)^8 - 30*a*b*cosh(d*x + c)^9 + 150*b^2*d*x*cosh(d*x + c)^6 + 30*(5*b^2*d*x*cosh(d*x + c) - a*b)*sinh(d*x + c)^9 + 60*a*b*cosh(d*x + c )^7 + 15*(45*b^2*d*x*cosh(d*x + c)^2 - 5*b^2*d*x - 18*a*b*cosh(d*x + c))*s inh(d*x + c)^8 + 60*(30*b^2*d*x*cosh(d*x + c)^3 - 10*b^2*d*x*cosh(d*x + c) - 18*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^7 + 30*(105*b^2*d*x*cosh(d* x + c)^4 - 70*b^2*d*x*cosh(d*x + c)^2 - 84*a*b*cosh(d*x + c)^3 + 5*b^2*d*x + 14*a*b*cosh(d*x + c))*sinh(d*x + c)^6 + 60*(63*b^2*d*x*cosh(d*x + c)^5 - 70*b^2*d*x*cosh(d*x + c)^3 - 63*a*b*cosh(d*x + c)^4 + 15*b^2*d*x*cosh(d* x + c) + 21*a*b*cosh(d*x + c)^2)*sinh(d*x + c)^5 - 60*a*b*cosh(d*x + c)^3 - 10*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^4 + 10*(315*b^2*d*x*cosh(d*x + c) ^6 - 525*b^2*d*x*cosh(d*x + c)^4 - 378*a*b*cosh(d*x + c)^5 + 225*b^2*d*x*c osh(d*x + c)^2 + 210*a*b*cosh(d*x + c)^3 - 15*b^2*d*x - 16*a^2)*sinh(d*x + c)^4 - 15*b^2*d*x + 20*(90*b^2*d*x*cosh(d*x + c)^7 - 210*b^2*d*x*cosh(d*x + c)^5 - 126*a*b*cosh(d*x + c)^6 + 150*b^2*d*x*cosh(d*x + c)^3 + 105*a*b* cosh(d*x + c)^4 - 3*a*b - 2*(15*b^2*d*x + 16*a^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 30*a*b*cosh(d*x + c) + 5*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^2 + 5*(135*b^2*d*x*cosh(d*x + c)^8 - 420*b^2*d*x*cosh(d*x + c)^6 - 216*a*b*cos h(d*x + c)^7 + 450*b^2*d*x*cosh(d*x + c)^4 + 252*a*b*cosh(d*x + c)^5 + 15* b^2*d*x - 36*a*b*cosh(d*x + c) - 12*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)...
Timed out. \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (84) = 168\).
Time = 0.20 (sec) , antiderivative size = 303, normalized size of antiderivative = 3.44 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=b^{2} x + a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {16}{15} \, a^{2} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} \]
b^2*x + a*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d* x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1)) ) - 16/15*a^2*(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)))
Time = 0.34 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.60 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=\frac {15 \, {\left (d x + c\right )} b^{2} + 15 \, a b \log \left (e^{\left (d x + c\right )} + 1\right ) - 15 \, a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} - 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, a b e^{\left (d x + c\right )} + 8 \, a^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \]
1/15*(15*(d*x + c)*b^2 + 15*a*b*log(e^(d*x + c) + 1) - 15*a*b*log(abs(e^(d *x + c) - 1)) - 2*(15*a*b*e^(9*d*x + 9*c) - 30*a*b*e^(7*d*x + 7*c) + 80*a^ 2*e^(4*d*x + 4*c) + 30*a*b*e^(3*d*x + 3*c) - 40*a^2*e^(2*d*x + 2*c) - 15*a *b*e^(d*x + c) + 8*a^2)/(e^(2*d*x + 2*c) - 1)^5)/d
Time = 1.42 (sec) , antiderivative size = 351, normalized size of antiderivative = 3.99 \[ \int \text {csch}^6(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx=b^2\,x-\frac {\frac {32\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}-\frac {8\,a\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d}+\frac {24\,a\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{5\,d}-\frac {24\,a\,b\,{\mathrm {e}}^{5\,c+5\,d\,x}}{5\,d}+\frac {8\,a\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}+\frac {2\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}}-\frac {64\,a^2}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {16\,a^2}{5\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {2\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {12\,a\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
b^2*x - ((32*a^2*exp(4*c + 4*d*x))/(5*d) - (8*a*b*exp(c + d*x))/(5*d) + (2 4*a*b*exp(3*c + 3*d*x))/(5*d) - (24*a*b*exp(5*c + 5*d*x))/(5*d) + (8*a*b*e xp(7*c + 7*d*x))/(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) + (2*atan((a* b*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(-d^ 2)^(1/2) - (64*a^2)/(15*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6 *c + 6*d*x) - 1)) - (16*a^2)/(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*a*b*exp(c + d*x))/(d*( exp(2*c + 2*d*x) - 1)) - (12*a*b*exp(c + d*x))/(5*d*(exp(4*c + 4*d*x) - 2* exp(2*c + 2*d*x) + 1))