Integrand size = 21, antiderivative size = 83 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=-\frac {a^3 \text {arctanh}(\cosh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)}{d}+\frac {(3 a-2 b) b^2 \cosh ^3(c+d x)}{3 d}+\frac {b^3 \cosh ^5(c+d x)}{5 d} \]
-a^3*arctanh(cosh(d*x+c))/d+b*(3*a^2-3*a*b+b^2)*cosh(d*x+c)/d+1/3*(3*a-2*b )*b^2*cosh(d*x+c)^3/d+1/5*b^3*cosh(d*x+c)^5/d
Time = 2.88 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.14 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {30 b \left (24 a^2-18 a b+5 b^2\right ) \cosh (c+d x)+5 (12 a-5 b) b^2 \cosh (3 (c+d x))+3 b^3 \cosh (5 (c+d x))+240 a^3 \left (-\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{240 d} \]
(30*b*(24*a^2 - 18*a*b + 5*b^2)*Cosh[c + d*x] + 5*(12*a - 5*b)*b^2*Cosh[3* (c + d*x)] + 3*b^3*Cosh[5*(c + d*x)] + 240*a^3*(-Log[Cosh[(c + d*x)/2]] + Log[Sinh[(c + d*x)/2]]))/(240*d)
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 3665, 300, 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.
\(\displaystyle \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \left (a-b \sin (i c+i d x)^2\right )^3}{\sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\left (a-b \sin (i c+i d x)^2\right )^3}{\sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \frac {\left (b \cosh ^2(c+d x)+a-b\right )^3}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle -\frac {\int \left (-b^3 \cosh ^4(c+d x)-(3 a-2 b) b^2 \cosh ^2(c+d x)-b \left (3 a^2-3 b a+b^2\right )+\frac {a^3}{1-\cosh ^2(c+d x)}\right )d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^3 \text {arctanh}(\cosh (c+d x))-b \left (3 a^2-3 a b+b^2\right ) \cosh (c+d x)-\frac {1}{3} b^2 (3 a-2 b) \cosh ^3(c+d x)-\frac {1}{5} b^3 \cosh ^5(c+d x)}{d}\) |
-((a^3*ArcTanh[Cosh[c + d*x]] - b*(3*a^2 - 3*a*b + b^2)*Cosh[c + d*x] - (( 3*a - 2*b)*b^2*Cosh[c + d*x]^3)/3 - (b^3*Cosh[c + d*x]^5)/5)/d)
3.1.24.3.1 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_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.45 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \cosh \left (d x +c \right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(86\) |
default | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \cosh \left (d x +c \right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (\frac {8}{15}+\frac {\sinh \left (d x +c \right )^{4}}{5}-\frac {4 \sinh \left (d x +c \right )^{2}}{15}\right ) \cosh \left (d x +c \right )}{d}\) | \(86\) |
parallelrisch | \(\frac {a^{3} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\frac {\left (a -\frac {5 b}{12}\right ) b \cosh \left (3 d x +3 c \right )}{12}+\frac {b^{2} \cosh \left (5 d x +5 c \right )}{240}+\left (a^{2}-\frac {3}{4} a b +\frac {5}{24} b^{2}\right ) \cosh \left (d x +c \right )+a^{2}-\frac {2 a b}{3}+\frac {8 b^{2}}{45}\right ) b}{d}\) | \(87\) |
risch | \(\frac {b^{3} {\mathrm e}^{5 d x +5 c}}{160 d}+\frac {{\mathrm e}^{3 d x +3 c} a \,b^{2}}{8 d}-\frac {5 \,{\mathrm e}^{3 d x +3 c} b^{3}}{96 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}-\frac {9 \,{\mathrm e}^{d x +c} a \,b^{2}}{8 d}+\frac {5 \,{\mathrm e}^{d x +c} b^{3}}{16 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}-\frac {9 \,{\mathrm e}^{-d x -c} a \,b^{2}}{8 d}+\frac {5 \,{\mathrm e}^{-d x -c} b^{3}}{16 d}+\frac {{\mathrm e}^{-3 d x -3 c} a \,b^{2}}{8 d}-\frac {5 \,{\mathrm e}^{-3 d x -3 c} b^{3}}{96 d}+\frac {b^{3} {\mathrm e}^{-5 d x -5 c}}{160 d}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\) | \(236\) |
1/d*(-2*a^3*arctanh(exp(d*x+c))+3*a^2*b*cosh(d*x+c)+3*a*b^2*(-2/3+1/3*sinh (d*x+c)^2)*cosh(d*x+c)+b^3*(8/15+1/5*sinh(d*x+c)^4-4/15*sinh(d*x+c)^2)*cos h(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 1128 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 1128, normalized size of antiderivative = 13.59 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/480*(3*b^3*cosh(d*x + c)^10 + 30*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 3*b ^3*sinh(d*x + c)^10 + 5*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^8 + 5*(27*b^3*cos h(d*x + c)^2 + 12*a*b^2 - 5*b^3)*sinh(d*x + c)^8 + 40*(9*b^3*cosh(d*x + c) ^3 + (12*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 30*(24*a^2*b - 18 *a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 10*(63*b^3*cosh(d*x + c)^4 + 72*a^2*b - 54*a*b^2 + 15*b^3 + 14*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(189*b^3*cosh(d*x + c)^5 + 70*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^3 + 45 *(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 30*(24*a^2 *b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 10*(63*b^3*cosh(d*x + c)^6 + 35*( 12*a*b^2 - 5*b^3)*cosh(d*x + c)^4 + 72*a^2*b - 54*a*b^2 + 15*b^3 + 45*(24* a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 40*(9*b^3*cos h(d*x + c)^7 + 7*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^5 + 15*(24*a^2*b - 18*a* b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*b^3 + 5*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^2 + 5*(27 *b^3*cosh(d*x + c)^8 + 28*(12*a*b^2 - 5*b^3)*cosh(d*x + c)^6 + 90*(24*a^2* b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 12*a*b^2 - 5*b^3 + 36*(24*a^2*b - 18*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 480*(a^3*cosh(d*x + c )^5 + 5*a^3*cosh(d*x + c)^4*sinh(d*x + c) + 10*a^3*cosh(d*x + c)^3*sinh(d* x + c)^2 + 10*a^3*cosh(d*x + c)^2*sinh(d*x + c)^3 + 5*a^3*cosh(d*x + c)*si nh(d*x + c)^4 + a^3*sinh(d*x + c)^5)*log(cosh(d*x + c) + sinh(d*x + c) ...
Timed out. \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (79) = 158\).
Time = 0.19 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.33 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {1}{480} \, b^{3} {\left (\frac {3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac {25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {150 \, e^{\left (d x + c\right )}}{d} + \frac {150 \, e^{\left (-d x - c\right )}}{d} - \frac {25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{8} \, a b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{3} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
1/480*b^3*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d) + 1/8 *a*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + 3/2*a^2*b*(e^(d*x + c)/d + e^(-d*x - c)/d) + a^3*log(tanh(1/2 *d*x + 1/2*c))/d
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (79) = 158\).
Time = 0.32 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.43 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 60 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 25 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 720 \, a^{2} b e^{\left (d x + c\right )} - 540 \, a b^{2} e^{\left (d x + c\right )} + 150 \, b^{3} e^{\left (d x + c\right )} - 480 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 480 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + {\left (720 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 540 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 25 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \]
1/480*(3*b^3*e^(5*d*x + 5*c) + 60*a*b^2*e^(3*d*x + 3*c) - 25*b^3*e^(3*d*x + 3*c) + 720*a^2*b*e^(d*x + c) - 540*a*b^2*e^(d*x + c) + 150*b^3*e^(d*x + c) - 480*a^3*log(e^(d*x + c) + 1) + 480*a^3*log(abs(e^(d*x + c) - 1)) + (7 20*a^2*b*e^(4*d*x + 4*c) - 540*a*b^2*e^(4*d*x + 4*c) + 150*b^3*e^(4*d*x + 4*c) + 60*a*b^2*e^(2*d*x + 2*c) - 25*b^3*e^(2*d*x + 2*c) + 3*b^3)*e^(-5*d* x - 5*c))/d
Time = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.22 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b-18\,a\,b^2+5\,b^3\right )}{16\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (24\,a^2\,b-18\,a\,b^2+5\,b^3\right )}{16\,d}+\frac {b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}+\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (12\,a-5\,b\right )}{96\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (12\,a-5\,b\right )}{96\,d} \]
(exp(c + d*x)*(24*a^2*b - 18*a*b^2 + 5*b^3))/(16*d) - (2*atan((a^3*exp(d*x )*exp(c)*(-d^2)^(1/2))/(d*(a^6)^(1/2)))*(a^6)^(1/2))/(-d^2)^(1/2) + (exp(- c - d*x)*(24*a^2*b - 18*a*b^2 + 5*b^3))/(16*d) + (b^3*exp(- 5*c - 5*d*x)) /(160*d) + (b^3*exp(5*c + 5*d*x))/(160*d) + (b^2*exp(- 3*c - 3*d*x)*(12*a - 5*b))/(96*d) + (b^2*exp(3*c + 3*d*x)*(12*a - 5*b))/(96*d)