Integrand size = 21, antiderivative size = 98 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {5 b \arctan (\sinh (c+d x))}{2 d}-\frac {a \cosh (c+d x)}{d}+\frac {a \cosh ^3(c+d x)}{3 d}-\frac {5 b \sinh (c+d x)}{2 d}+\frac {5 b \sinh ^3(c+d x)}{6 d}-\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d} \]
5/2*b*arctan(sinh(d*x+c))/d-a*cosh(d*x+c)/d+1/3*a*cosh(d*x+c)^3/d-5/2*b*si nh(d*x+c)/d+5/6*b*sinh(d*x+c)^3/d-1/2*b*sinh(d*x+c)^3*tanh(d*x+c)^2/d
Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.14 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {5 b \arctan (\sinh (c+d x))}{2 d}-\frac {3 a \cosh (c+d x)}{4 d}+\frac {a \cosh (3 (c+d x))}{12 d}-\frac {5 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {5 b \sinh (c+d x) \tanh ^2(c+d x)}{3 d}+\frac {b \sinh ^3(c+d x) \tanh ^2(c+d x)}{3 d} \]
(5*b*ArcTan[Sinh[c + d*x]])/(2*d) - (3*a*Cosh[c + d*x])/(4*d) + (a*Cosh[3* (c + d*x)])/(12*d) - (5*b*Sech[c + d*x]*Tanh[c + d*x])/(2*d) - (5*b*Sinh[c + d*x]*Tanh[c + d*x]^2)/(3*d) + (b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/(3*d)
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 26, 4149, 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.
\(\displaystyle \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin (i c+i d x)^3 \left (a+i b \tan (i c+i d x)^3\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \sin (i c+i d x)^3 \left (i b \tan (i c+i d x)^3+a\right )dx\) |
\(\Big \downarrow \) 4149 |
\(\displaystyle i \int \left (-i b \tanh ^3(c+d x) \sinh ^3(c+d x)-i a \sinh ^3(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (-\frac {i a \cosh ^3(c+d x)}{3 d}+\frac {i a \cosh (c+d x)}{d}-\frac {5 i b \arctan (\sinh (c+d x))}{2 d}-\frac {5 i b \sinh ^3(c+d x)}{6 d}+\frac {5 i b \sinh (c+d x)}{2 d}+\frac {i b \sinh ^3(c+d x) \tanh ^2(c+d x)}{2 d}\right )\) |
I*((((-5*I)/2)*b*ArcTan[Sinh[c + d*x]])/d + (I*a*Cosh[c + d*x])/d - ((I/3) *a*Cosh[c + d*x]^3)/d + (((5*I)/2)*b*Sinh[c + d*x])/d - (((5*I)/6)*b*Sinh[ c + d*x]^3)/d + ((I/2)*b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/d)
3.1.50.3.1 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[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
Time = 0.98 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(104\) |
default | \(\frac {a \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(104\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c} a}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} b}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a}{8 d}-\frac {9 \,{\mathrm e}^{d x +c} b}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 d}+\frac {9 \,{\mathrm e}^{-d x -c} b}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} b}{24 d}-\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {5 i b \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {5 i b \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\) | \(186\) |
1/d*(a*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+b*(1/3*sinh(d*x+c)^5/cosh(d*x+ c)^2-5/3*sinh(d*x+c)^3/cosh(d*x+c)^2-5*sinh(d*x+c)/cosh(d*x+c)^2+5/2*sech( d*x+c)*tanh(d*x+c)+5*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 1070 vs. \(2 (88) = 176\).
Time = 0.28 (sec) , antiderivative size = 1070, normalized size of antiderivative = 10.92 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\text {Too large to display} \]
1/24*((a + b)*cosh(d*x + c)^10 + 10*(a + b)*cosh(d*x + c)*sinh(d*x + c)^9 + (a + b)*sinh(d*x + c)^10 - (7*a + 25*b)*cosh(d*x + c)^8 + (45*(a + b)*co sh(d*x + c)^2 - 7*a - 25*b)*sinh(d*x + c)^8 + 8*(15*(a + b)*cosh(d*x + c)^ 3 - (7*a + 25*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(13*a + 25*b)*cosh(d*x + c)^6 + 2*(105*(a + b)*cosh(d*x + c)^4 - 14*(7*a + 25*b)*cosh(d*x + c)^2 - 13*a - 25*b)*sinh(d*x + c)^6 + 4*(63*(a + b)*cosh(d*x + c)^5 - 14*(7*a + 25*b)*cosh(d*x + c)^3 - 3*(13*a + 25*b)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(13*a - 25*b)*cosh(d*x + c)^4 + 2*(105*(a + b)*cosh(d*x + c)^6 - 35*(7* a + 25*b)*cosh(d*x + c)^4 - 15*(13*a + 25*b)*cosh(d*x + c)^2 - 13*a + 25*b )*sinh(d*x + c)^4 + 8*(15*(a + b)*cosh(d*x + c)^7 - 7*(7*a + 25*b)*cosh(d* x + c)^5 - 5*(13*a + 25*b)*cosh(d*x + c)^3 - (13*a - 25*b)*cosh(d*x + c))* sinh(d*x + c)^3 - (7*a - 25*b)*cosh(d*x + c)^2 + (45*(a + b)*cosh(d*x + c) ^8 - 28*(7*a + 25*b)*cosh(d*x + c)^6 - 30*(13*a + 25*b)*cosh(d*x + c)^4 - 12*(13*a - 25*b)*cosh(d*x + c)^2 - 7*a + 25*b)*sinh(d*x + c)^2 + 120*(b*co sh(d*x + c)^7 + 7*b*cosh(d*x + c)*sinh(d*x + c)^6 + b*sinh(d*x + c)^7 + 2* b*cosh(d*x + c)^5 + (21*b*cosh(d*x + c)^2 + 2*b)*sinh(d*x + c)^5 + 5*(7*b* cosh(d*x + c)^3 + 2*b*cosh(d*x + c))*sinh(d*x + c)^4 + b*cosh(d*x + c)^3 + (35*b*cosh(d*x + c)^4 + 20*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^3 + (21*b *cosh(d*x + c)^5 + 20*b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c))*sinh(d*x + c) ^2 + (7*b*cosh(d*x + c)^6 + 10*b*cosh(d*x + c)^4 + 3*b*cosh(d*x + c)^2)...
\[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \sinh ^{3}{\left (c + d x \right )}\, dx \]
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.78 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {1}{24} \, b {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
1/24*b*((27*e^(-d*x - c) - e^(-3*d*x - 3*c))/d - 120*arctan(e^(-d*x - c))/ d - (25*e^(-2*d*x - 2*c) + 77*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) - 1)/( d*(e^(-3*d*x - 3*c) + 2*e^(-5*d*x - 5*c) + e^(-7*d*x - 7*c)))) + 1/24*a*(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 )
Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.35 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {120 \, b \arctan \left (e^{\left (d x + c\right )}\right ) + a e^{\left (3 \, d x + 3 \, c\right )} + b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, a e^{\left (d x + c\right )} - 27 \, b e^{\left (d x + c\right )} - {\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} - 27 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )} e^{\left (-3 \, d x - 3 \, c\right )} - \frac {24 \, {\left (b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{24 \, d} \]
1/24*(120*b*arctan(e^(d*x + c)) + a*e^(3*d*x + 3*c) + b*e^(3*d*x + 3*c) - 9*a*e^(d*x + c) - 27*b*e^(d*x + c) - (9*a*e^(2*d*x + 2*c) - 27*b*e^(2*d*x + 2*c) - a + b)*e^(-3*d*x - 3*c) - 24*(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/ (e^(2*d*x + 2*c) + 1)^2)/d
Time = 0.25 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.74 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+b\right )}{24\,d}+\frac {5\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (a-b\right )}{24\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+9\,b\right )}{8\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a-9\,b\right )}{8\,d}-\frac {b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(exp(3*c + 3*d*x)*(a + b))/(24*d) + (5*atan((b*exp(d*x)*exp(c)*(d^2)^(1/2) )/(d*(b^2)^(1/2)))*(b^2)^(1/2))/(d^2)^(1/2) + (exp(- 3*c - 3*d*x)*(a - b)) /(24*d) - (exp(c + d*x)*(3*a + 9*b))/(8*d) - (exp(- c - d*x)*(3*a - 9*b))/ (8*d) - (b*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)) + (2*b*exp(c + d*x))/( d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))