Integrand size = 15, antiderivative size = 67 \[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=-\frac {\cosh (a-c) \text {sech}^3(c+b x)}{3 b}+\frac {\arctan (\sinh (c+b x)) \sinh (a-c)}{2 b}+\frac {\text {sech}(c+b x) \sinh (a-c) \tanh (c+b x)}{2 b} \] Output:
-1/3*cosh(a-c)*sech(b*x+c)^3/b+1/2*arctan(sinh(b*x+c))*sinh(a-c)/b+1/2*sec h(b*x+c)*sinh(a-c)*tanh(b*x+c)/b
Time = 0.23 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.57 \[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=\frac {-3 \cosh (a-c-b x) \text {sech}(c) \text {sech}^2(c+b x)+3 \cosh (a-c+b x) \text {sech}(c) \text {sech}^2(c+b x)-4 \cosh (a-c) \text {sech}^3(c+b x)+12 \arctan \left (\sinh (c)+\cosh (c) \tanh \left (\frac {b x}{2}\right )\right ) \sinh (a-c)+6 \text {sech}(c+b x) \sinh (a-c) \tanh (c)}{12 b} \] Input:
Integrate[Sech[c + b*x]^4*Sinh[a + b*x],x]
Output:
(-3*Cosh[a - c - b*x]*Sech[c]*Sech[c + b*x]^2 + 3*Cosh[a - c + b*x]*Sech[c ]*Sech[c + b*x]^2 - 4*Cosh[a - c]*Sech[c + b*x]^3 + 12*ArcTan[Sinh[c] + Co sh[c]*Tanh[(b*x)/2]]*Sinh[a - c] + 6*Sech[c + b*x]*Sinh[a - c]*Tanh[c])/(1 2*b)
Time = 0.41 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6158, 3042, 26, 3086, 15, 4255, 3042, 4257}
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.
\(\displaystyle \int \sinh (a+b x) \text {sech}^4(b x+c) \, dx\) |
\(\Big \downarrow \) 6158 |
\(\displaystyle \sinh (a-c) \int \text {sech}^3(c+b x)dx+\cosh (a-c) \int \text {sech}^3(c+b x) \tanh (c+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )^3dx+\cosh (a-c) \int -i \sec (i c+i b x)^3 \tan (i c+i b x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )^3dx-i \cosh (a-c) \int \sec (i c+i b x)^3 \tan (i c+i b x)dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {\cosh (a-c) \int \text {sech}^2(c+b x)d\text {sech}(c+b x)}{b}+\sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\cosh (a-c) \text {sech}^3(b x+c)}{3 b}+\sinh (a-c) \int \csc \left (i c+i b x+\frac {\pi }{2}\right )^3dx\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \sinh (a-c) \left (\frac {1}{2} \int \text {sech}(c+b x)dx+\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {sech}^3(b x+c)}{3 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cosh (a-c) \text {sech}^3(b x+c)}{3 b}+\sinh (a-c) \left (\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}+\frac {1}{2} \int \csc \left (i c+i b x+\frac {\pi }{2}\right )dx\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \sinh (a-c) \left (\frac {\arctan (\sinh (b x+c))}{2 b}+\frac {\tanh (b x+c) \text {sech}(b x+c)}{2 b}\right )-\frac {\cosh (a-c) \text {sech}^3(b x+c)}{3 b}\) |
Input:
Int[Sech[c + b*x]^4*Sinh[a + b*x],x]
Output:
-1/3*(Cosh[a - c]*Sech[c + b*x]^3)/b + Sinh[a - c]*(ArcTan[Sinh[c + b*x]]/ (2*b) + (Sech[c + b*x]*Tanh[c + b*x])/(2*b))
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Sech[w_]^(n_.)*Sinh[v_], x_Symbol] :> Simp[Cosh[v - w] Int[Tanh[w]*Se ch[w]^(n - 1), x], x] + Simp[Sinh[v - w] Int[Sech[w]^(n - 1), x], x] /; G tQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]
Result contains complex when optimal does not.
Time = 7.20 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.66
method | result | size |
risch | \(-\frac {{\mathrm e}^{b x +a} \left (-3 \,{\mathrm e}^{4 b x +6 a +4 c}+3 \,{\mathrm e}^{4 b x +4 a +6 c}+8 \,{\mathrm e}^{2 b x +6 a +2 c}+8 \,{\mathrm e}^{2 b x +4 a +4 c}+3 \,{\mathrm e}^{6 a}-3 \,{\mathrm e}^{4 a +2 c}\right )}{6 b \left ({\mathrm e}^{2 b x +2 a +2 c}+{\mathrm e}^{2 a}\right )^{3}}+\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}+i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}-\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{4 b}+\frac {i \ln \left ({\mathrm e}^{b x +a}-i {\mathrm e}^{a -c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{4 b}\) | \(245\) |
Input:
int(sech(b*x+c)^4*sinh(b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/6*exp(b*x+a)*(-3*exp(4*b*x+6*a+4*c)+3*exp(4*b*x+4*a+6*c)+8*exp(2*b*x+6* a+2*c)+8*exp(2*b*x+4*a+4*c)+3*exp(6*a)-3*exp(4*a+2*c))/b/(exp(2*b*x+2*a+2* c)+exp(2*a))^3+1/4*I*ln(exp(b*x+a)+I*exp(a-c))/b*exp(-a-c)*exp(a)^2-1/4*I* ln(exp(b*x+a)+I*exp(a-c))/b*exp(-a-c)*exp(c)^2-1/4*I*ln(exp(b*x+a)-I*exp(a -c))/b*exp(-a-c)*exp(a)^2+1/4*I*ln(exp(b*x+a)-I*exp(a-c))/b*exp(-a-c)*exp( c)^2
Leaf count of result is larger than twice the leaf count of optimal. 1881 vs. \(2 (61) = 122\).
Time = 0.10 (sec) , antiderivative size = 1881, normalized size of antiderivative = 28.07 \[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=\text {Too large to display} \] Input:
integrate(sech(b*x+c)^4*sinh(b*x+a),x, algorithm="fricas")
Output:
1/6*(3*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^5 + 3*(cosh(-a + c)^2 - 2*cosh(- a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^5 - 15*(2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^4 - 8*(cosh(-a + c)^2 + 1)*cosh(b *x + c)^3 + 2*(15*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (15*cosh(b*x + c) ^2 - 4)*sinh(-a + c)^2 - 4*cosh(-a + c)^2 - 2*(15*cosh(b*x + c)^2*cosh(-a + c) - 4*cosh(-a + c))*sinh(-a + c) - 4)*sinh(b*x + c)^3 + 6*(5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^3 + (5*cosh(b*x + c)^3 - 4*cosh(b*x + c))*sinh(-a + c)^2 - 4*(cosh(-a + c)^2 + 1)*cosh(b*x + c) - 2*(5*cosh(b*x + c)^3*cosh (-a + c) - 4*cosh(b*x + c)*cosh(-a + c))*sinh(-a + c))*sinh(b*x + c)^2 + ( 3*cosh(b*x + c)^5 - 8*cosh(b*x + c)^3 - 3*cosh(b*x + c))*sinh(-a + c)^2 + 3*((cosh(-a + c)^2 - 1)*cosh(b*x + c)^6 + (cosh(-a + c)^2 - 2*cosh(-a + c) *sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c)^6 - 6*(2*cosh(b*x + c)*c osh(-a + c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 - 1)*cosh(b*x + c))*sinh(b*x + c)^5 + 3*(cosh(-a + c)^2 - 1)*cosh(b*x + c) ^4 + 3*(5*(cosh(-a + c)^2 - 1)*cosh(b*x + c)^2 + (5*cosh(b*x + c)^2 + 1)*s inh(-a + c)^2 + cosh(-a + c)^2 - 2*(5*cosh(b*x + c)^2*cosh(-a + c) + cosh( -a + c))*sinh(-a + c) - 1)*sinh(b*x + c)^4 + 4*(5*(cosh(-a + c)^2 - 1)*cos h(b*x + c)^3 + (5*cosh(b*x + c)^3 + 3*cosh(b*x + c))*sinh(-a + c)^2 + 3*(c osh(-a + c)^2 - 1)*cosh(b*x + c) - 2*(5*cosh(b*x + c)^3*cosh(-a + c) + ...
\[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=\int \sinh {\left (a + b x \right )} \operatorname {sech}^{4}{\left (b x + c \right )}\, dx \] Input:
integrate(sech(b*x+c)**4*sinh(b*x+a),x)
Output:
Integral(sinh(a + b*x)*sech(b*x + c)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (61) = 122\).
Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.31 \[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=-\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (-b x - c\right )}\right ) e^{\left (-a - c\right )}}{2 \, b} + \frac {3 \, {\left (e^{\left (2 \, a + 4 \, c\right )} - e^{\left (6 \, c\right )}\right )} e^{\left (-b x - a\right )} - 8 \, {\left (e^{\left (4 \, a + 2 \, c\right )} + e^{\left (2 \, a + 4 \, c\right )}\right )} e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, {\left (e^{\left (6 \, a\right )} - e^{\left (4 \, a + 2 \, c\right )}\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{6 \, b {\left (3 \, e^{\left (-2 \, b x + 4 \, c\right )} + 3 \, e^{\left (-4 \, b x + 2 \, c\right )} + e^{\left (-6 \, b x\right )} + e^{\left (6 \, c\right )}\right )}} \] Input:
integrate(sech(b*x+c)^4*sinh(b*x+a),x, algorithm="maxima")
Output:
-1/2*(e^(2*a) - e^(2*c))*arctan(e^(-b*x - c))*e^(-a - c)/b + 1/6*(3*(e^(2* a + 4*c) - e^(6*c))*e^(-b*x - a) - 8*(e^(4*a + 2*c) + e^(2*a + 4*c))*e^(-3 *b*x - 3*a) - 3*(e^(6*a) - e^(4*a + 2*c))*e^(-5*b*x - 5*a))/(b*(3*e^(-2*b* x + 4*c) + 3*e^(-4*b*x + 2*c) + e^(-6*b*x) + e^(6*c)))
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (61) = 122\).
Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.87 \[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=\frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} \arctan \left (e^{\left (b x + c\right )}\right ) e^{\left (-a - c\right )}}{2 \, b} + \frac {{\left (3 \, e^{\left (5 \, b x + 2 \, a + 4 \, c\right )} - 3 \, e^{\left (5 \, b x + 6 \, c\right )} - 8 \, e^{\left (3 \, b x + 2 \, a + 2 \, c\right )} - 8 \, e^{\left (3 \, b x + 4 \, c\right )} - 3 \, e^{\left (b x + 2 \, a\right )} + 3 \, e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{6 \, b {\left (e^{\left (2 \, b x + 2 \, c\right )} + 1\right )}^{3}} \] Input:
integrate(sech(b*x+c)^4*sinh(b*x+a),x, algorithm="giac")
Output:
1/2*(e^(2*a) - e^(2*c))*arctan(e^(b*x + c))*e^(-a - c)/b + 1/6*(3*e^(5*b*x + 2*a + 4*c) - 3*e^(5*b*x + 6*c) - 8*e^(3*b*x + 2*a + 2*c) - 8*e^(3*b*x + 4*c) - 3*e^(b*x + 2*a) + 3*e^(b*x + 2*c))*e^(-a)/(b*(e^(2*b*x + 2*c) + 1) ^3)
Timed out. \[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=\int \frac {\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (c+b\,x\right )}^4} \,d x \] Input:
int(sinh(a + b*x)/cosh(c + b*x)^4,x)
Output:
int(sinh(a + b*x)/cosh(c + b*x)^4, x)
Time = 0.27 (sec) , antiderivative size = 287, normalized size of antiderivative = 4.28 \[ \int \text {sech}^4(c+b x) \sinh (a+b x) \, dx=\frac {3 e^{6 b x +2 a +6 c} \mathit {atan} \left (e^{b x +c}\right )-3 e^{6 b x +8 c} \mathit {atan} \left (e^{b x +c}\right )+9 e^{4 b x +2 a +4 c} \mathit {atan} \left (e^{b x +c}\right )-9 e^{4 b x +6 c} \mathit {atan} \left (e^{b x +c}\right )+9 e^{2 b x +2 a +2 c} \mathit {atan} \left (e^{b x +c}\right )-9 e^{2 b x +4 c} \mathit {atan} \left (e^{b x +c}\right )+3 e^{2 a} \mathit {atan} \left (e^{b x +c}\right )-3 e^{2 c} \mathit {atan} \left (e^{b x +c}\right )+3 e^{5 b x +2 a +5 c}-3 e^{5 b x +7 c}-8 e^{3 b x +2 a +3 c}-8 e^{3 b x +5 c}-3 e^{b x +2 a +c}+3 e^{b x +3 c}}{6 e^{a +c} b \left (e^{6 b x +6 c}+3 e^{4 b x +4 c}+3 e^{2 b x +2 c}+1\right )} \] Input:
int(sech(b*x+c)^4*sinh(b*x+a),x)
Output:
(3*e**(2*a + 6*b*x + 6*c)*atan(e**(b*x + c)) - 3*e**(6*b*x + 8*c)*atan(e** (b*x + c)) + 9*e**(2*a + 4*b*x + 4*c)*atan(e**(b*x + c)) - 9*e**(4*b*x + 6 *c)*atan(e**(b*x + c)) + 9*e**(2*a + 2*b*x + 2*c)*atan(e**(b*x + c)) - 9*e **(2*b*x + 4*c)*atan(e**(b*x + c)) + 3*e**(2*a)*atan(e**(b*x + c)) - 3*e** (2*c)*atan(e**(b*x + c)) + 3*e**(2*a + 5*b*x + 5*c) - 3*e**(5*b*x + 7*c) - 8*e**(2*a + 3*b*x + 3*c) - 8*e**(3*b*x + 5*c) - 3*e**(2*a + b*x + c) + 3* e**(b*x + 3*c))/(6*e**(a + c)*b*(e**(6*b*x + 6*c) + 3*e**(4*b*x + 4*c) + 3 *e**(2*b*x + 2*c) + 1))