Integrand size = 19, antiderivative size = 58 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=-\frac {3 b \arctan (\sinh (c+d x))}{2 d}+\frac {a \cosh (c+d x)}{d}+\frac {b \sinh (c+d x)}{d}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \] Output:
-3/2*b*arctan(sinh(d*x+c))/d+a*cosh(d*x+c)/d+b*sinh(d*x+c)/d+1/2*b*sech(d* x+c)*tanh(d*x+c)/d
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.33 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=-\frac {3 b \arctan (\sinh (c+d x))}{2 d}+\frac {a \cosh (c) \cosh (d x)}{d}+\frac {a \sinh (c) \sinh (d x)}{d}+\frac {3 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {b \sinh (c+d x) \tanh ^2(c+d x)}{d} \] Input:
Integrate[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^3),x]
Output:
(-3*b*ArcTan[Sinh[c + d*x]])/(2*d) + (a*Cosh[c]*Cosh[d*x])/d + (a*Sinh[c]* Sinh[d*x])/d + (3*b*Sech[c + d*x]*Tanh[c + d*x])/(2*d) + (b*Sinh[c + d*x]* Tanh[c + d*x]^2)/d
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.31, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, 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.
\(\displaystyle \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \sin (i c+i d x) \left (a+i b \tan (i c+i d x)^3\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \sin (i c+i d x) \left (i b \tan (i c+i d x)^3+a\right )dx\) |
\(\Big \downarrow \) 4149 |
\(\displaystyle -i \int \left (i b \sinh (c+d x) \tanh ^3(c+d x)+i a \sinh (c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -i \left (\frac {i a \cosh (c+d x)}{d}-\frac {3 i b \arctan (\sinh (c+d x))}{2 d}+\frac {3 i b \sinh (c+d x)}{2 d}-\frac {i b \sinh (c+d x) \tanh ^2(c+d x)}{2 d}\right )\) |
Input:
Int[Sinh[c + d*x]*(a + b*Tanh[c + d*x]^3),x]
Output:
(-I)*((((-3*I)/2)*b*ArcTan[Sinh[c + d*x]])/d + (I*a*Cosh[c + d*x])/d + ((( 3*I)/2)*b*Sinh[c + d*x])/d - ((I/2)*b*Sinh[c + d*x]*Tanh[c + d*x]^2)/d)
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 = 1.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {a \cosh \left (d x +c \right )+b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(73\) |
default | \(\frac {a \cosh \left (d x +c \right )+b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(73\) |
risch | \(\frac {{\mathrm e}^{d x +c} a}{2 d}+\frac {{\mathrm e}^{d x +c} b}{2 d}+\frac {{\mathrm e}^{-d x -c} a}{2 d}-\frac {{\mathrm e}^{-d x -c} b}{2 d}+\frac {b \,{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}\) | \(125\) |
Input:
int(sinh(d*x+c)*(a+b*tanh(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/d*(a*cosh(d*x+c)+b*(sinh(d*x+c)^3/cosh(d*x+c)^2+3*sinh(d*x+c)/cosh(d*x+c )^2-3/2*sech(d*x+c)*tanh(d*x+c)-3*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (54) = 108\).
Time = 0.12 (sec) , antiderivative size = 528, normalized size of antiderivative = 9.10 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(sinh(d*x+c)*(a+b*tanh(d*x+c)^3),x, algorithm="fricas")
Output:
1/2*((a + b)*cosh(d*x + c)^6 + 6*(a + b)*cosh(d*x + c)*sinh(d*x + c)^5 + ( a + b)*sinh(d*x + c)^6 + 3*(a + b)*cosh(d*x + c)^4 + 3*(5*(a + b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c)^4 + 4*(5*(a + b)*cosh(d*x + c)^3 + 3*(a + b )*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c)^2 + 3*(5*(a + b )*cosh(d*x + c)^4 + 6*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 - 6 *(b*cosh(d*x + c)^5 + 5*b*cosh(d*x + c)*sinh(d*x + c)^4 + b*sinh(d*x + c)^ 5 + 2*b*cosh(d*x + c)^3 + 2*(5*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^3 + 2* (5*b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c))*sinh(d*x + c)^2 + b*cosh(d*x + c ) + (5*b*cosh(d*x + c)^4 + 6*b*cosh(d*x + c)^2 + b)*sinh(d*x + c))*arctan( cosh(d*x + c) + sinh(d*x + c)) + 6*((a + b)*cosh(d*x + c)^5 + 2*(a + b)*co sh(d*x + c)^3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a - b)/(d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)*sinh(d*x + c)^4 + d*sinh(d*x + c)^5 + 2*d*cosh( d*x + c)^3 + 2*(5*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^3 + 2*(5*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^2 + d*cosh(d*x + c) + (5*d*cosh (d*x + c)^4 + 6*d*cosh(d*x + c)^2 + d)*sinh(d*x + c))
\[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right ) \sinh {\left (c + d x \right )}\, dx \] Input:
integrate(sinh(d*x+c)*(a+b*tanh(d*x+c)**3),x)
Output:
Integral((a + b*tanh(c + d*x)**3)*sinh(c + d*x), x)
Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.81 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {1}{2} \, b {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {a \cosh \left (d x + c\right )}{d} \] Input:
integrate(sinh(d*x+c)*(a+b*tanh(d*x+c)^3),x, algorithm="maxima")
Output:
1/2*b*(6*arctan(e^(-d*x - c))/d - e^(-d*x - c)/d + (4*e^(-2*d*x - 2*c) - e ^(-4*d*x - 4*c) + 1)/(d*(e^(-d*x - c) + 2*e^(-3*d*x - 3*c) + e^(-5*d*x - 5 *c)))) + a*cosh(d*x + c)/d
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.48 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=-\frac {6 \, b \arctan \left (e^{\left (d x + c\right )}\right ) - a e^{\left (d x + c\right )} - b e^{\left (d x + c\right )} - {\left (a - b\right )} e^{\left (-d x - c\right )} - \frac {2 \, {\left (b e^{\left (3 \, d x + 3 \, c\right )} - b e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{2 \, d} \] Input:
integrate(sinh(d*x+c)*(a+b*tanh(d*x+c)^3),x, algorithm="giac")
Output:
-1/2*(6*b*arctan(e^(d*x + c)) - a*e^(d*x + c) - b*e^(d*x + c) - (a - b)*e^ (-d*x - c) - 2*(b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^2 )/d
Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.21 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {{\mathrm {e}}^{-c-d\,x}\,\left (a-b\right )}{2\,d}-\frac {3\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{2\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:
int(sinh(c + d*x)*(a + b*tanh(c + d*x)^3),x)
Output:
(exp(- c - d*x)*(a - b))/(2*d) - (3*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(c + d*x)*(a + b))/(2*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))
Time = 0.24 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.19 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right ) \, dx=\frac {-12 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) b -2 e^{d x +c} \cosh \left (d x +c \right ) \tanh \left (d x +c \right ) b +4 e^{d x +c} \cosh \left (d x +c \right ) a +3 e^{2 d x +2 c} b -2 e^{d x +c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{2} b +2 e^{d x +c} \sinh \left (d x +c \right ) b -3 b}{4 e^{d x +c} d} \] Input:
int(sinh(d*x+c)*(a+b*tanh(d*x+c)^3),x)
Output:
( - 12*e**(c + d*x)*atan(e**(c + d*x))*b - 2*e**(c + d*x)*cosh(c + d*x)*ta nh(c + d*x)*b + 4*e**(c + d*x)*cosh(c + d*x)*a + 3*e**(2*c + 2*d*x)*b - 2* e**(c + d*x)*sinh(c + d*x)*tanh(c + d*x)**2*b + 2*e**(c + d*x)*sinh(c + d* x)*b - 3*b)/(4*e**(c + d*x)*d)