Integrand size = 12, antiderivative size = 73 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=a^3 x+\frac {b \left (6 a^2+b^2\right ) \arctan (\sinh (c+d x))}{2 d}+\frac {5 a b^2 \tanh (c+d x)}{2 d}+\frac {b^2 (a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d} \] Output:
a^3*x+1/2*b*(6*a^2+b^2)*arctan(sinh(d*x+c))/d+5/2*a*b^2*tanh(d*x+c)/d+1/2* b^2*(a+b*sech(d*x+c))*tanh(d*x+c)/d
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=\frac {2 a^3 d x-6 a^2 b \cot ^{-1}(\sinh (c+d x))+b^3 \arctan (\sinh (c+d x))+b^2 (6 a+b \text {sech}(c+d x)) \tanh (c+d x)}{2 d} \] Input:
Integrate[(a + b*Sech[c + d*x])^3,x]
Output:
(2*a^3*d*x - 6*a^2*b*ArcCot[Sinh[c + d*x]] + b^3*ArcTan[Sinh[c + d*x]] + b ^2*(6*a + b*Sech[c + d*x])*Tanh[c + d*x])/(2*d)
Time = 0.45 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4269, 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.
| \(\displaystyle \int (a+b \text {sech}(c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4269 |
| \(\displaystyle \frac {1}{2} \int \left (2 a^3+5 b^2 \text {sech}^2(c+d x) a+b \left (6 a^2+b^2\right ) \text {sech}(c+d x)\right )dx+\frac {b^2 \tanh (c+d x) (a+b \text {sech}(c+d x))}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2 a^3 x+\frac {b \left (6 a^2+b^2\right ) \arctan (\sinh (c+d x))}{d}+\frac {5 a b^2 \tanh (c+d x)}{d}\right )+\frac {b^2 \tanh (c+d x) (a+b \text {sech}(c+d x))}{2 d}\) |
Input:
Int[(a + b*Sech[c + d*x])^3,x]
Output:
(b^2*(a + b*Sech[c + d*x])*Tanh[c + d*x])/(2*d) + (2*a^3*x + (b*(6*a^2 + b ^2)*ArcTan[Sinh[c + d*x]])/d + (5*a*b^2*Tanh[c + d*x])/d)/2
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[1/(n - 1) Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3* a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] / ; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
Time = 0.66 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (d x +c \right )+6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )+3 \tanh \left (d x +c \right ) a \,b^{2}+b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(66\) |
| default | \(\frac {a^{3} \left (d x +c \right )+6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )+3 \tanh \left (d x +c \right ) a \,b^{2}+b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(66\) |
| parts | \(a^{3} x +\frac {b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}+\frac {3 a \,b^{2} \tanh \left (d x +c \right )}{d}+\frac {3 a^{2} b \arctan \left (\sinh \left (d x +c \right )\right )}{d}\) | \(67\) |
| parallelrisch | \(\frac {-3 i \left (a^{2}+\frac {b^{2}}{6}\right ) b \left (\cosh \left (2 d x +2 c \right )+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+3 i \left (a^{2}+\frac {b^{2}}{6}\right ) b \left (\cosh \left (2 d x +2 c \right )+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+a^{3} d x \cosh \left (2 d x +2 c \right )+a^{3} d x +3 a \,b^{2} \sinh \left (2 d x +2 c \right )+b^{3} \sinh \left (d x +c \right )}{d \left (\cosh \left (2 d x +2 c \right )+1\right )}\) | \(139\) |
| risch | \(a^{3} x -\frac {b^{2} \left (-b \,{\mathrm e}^{3 d x +3 c}+6 a \,{\mathrm e}^{2 d x +2 c}+b \,{\mathrm e}^{d x +c}+6 a \right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{d}+\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{d}-\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\) | \(142\) |
Input:
int((a+b*sech(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(d*x+c)+6*a^2*b*arctan(exp(d*x+c))+3*tanh(d*x+c)*a*b^2+b^3*(1/2*s ech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (67) = 134\).
Time = 0.10 (sec) , antiderivative size = 521, normalized size of antiderivative = 7.14 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=\frac {a^{3} d x \cosh \left (d x + c\right )^{4} + a^{3} d x \sinh \left (d x + c\right )^{4} + b^{3} \cosh \left (d x + c\right )^{3} + a^{3} d x - b^{3} \cosh \left (d x + c\right ) + {\left (4 \, a^{3} d x \cosh \left (d x + c\right ) + b^{3}\right )} \sinh \left (d x + c\right )^{3} - 6 \, a b^{2} + 2 \, {\left (a^{3} d x - 3 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (6 \, a^{3} d x \cosh \left (d x + c\right )^{2} + 2 \, a^{3} d x + 3 \, b^{3} \cosh \left (d x + c\right ) - 6 \, a b^{2}\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (6 \, a^{2} b + b^{3}\right )} \sinh \left (d x + c\right )^{4} + 6 \, a^{2} b + b^{3} + 2 \, {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{2} b + b^{3} + 3 \, {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (6 \, a^{2} b + b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + {\left (4 \, a^{3} d x \cosh \left (d x + c\right )^{3} + 3 \, b^{3} \cosh \left (d x + c\right )^{2} - b^{3} + 4 \, {\left (a^{3} d x - 3 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \] Input:
integrate((a+b*sech(d*x+c))^3,x, algorithm="fricas")
Output:
(a^3*d*x*cosh(d*x + c)^4 + a^3*d*x*sinh(d*x + c)^4 + b^3*cosh(d*x + c)^3 + a^3*d*x - b^3*cosh(d*x + c) + (4*a^3*d*x*cosh(d*x + c) + b^3)*sinh(d*x + c)^3 - 6*a*b^2 + 2*(a^3*d*x - 3*a*b^2)*cosh(d*x + c)^2 + (6*a^3*d*x*cosh(d *x + c)^2 + 2*a^3*d*x + 3*b^3*cosh(d*x + c) - 6*a*b^2)*sinh(d*x + c)^2 + ( (6*a^2*b + b^3)*cosh(d*x + c)^4 + 4*(6*a^2*b + b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (6*a^2*b + b^3)*sinh(d*x + c)^4 + 6*a^2*b + b^3 + 2*(6*a^2*b + b ^3)*cosh(d*x + c)^2 + 2*(6*a^2*b + b^3 + 3*(6*a^2*b + b^3)*cosh(d*x + c)^2 )*sinh(d*x + c)^2 + 4*((6*a^2*b + b^3)*cosh(d*x + c)^3 + (6*a^2*b + b^3)*c osh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) + (4*a^ 3*d*x*cosh(d*x + c)^3 + 3*b^3*cosh(d*x + c)^2 - b^3 + 4*(a^3*d*x - 3*a*b^2 )*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sin h(d*x + c)^3 + d*sinh(d*x + c)^4 + 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh( d*x + c) + d)
\[ \int (a+b \text {sech}(c+d x))^3 \, dx=\int \left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{3}\, dx \] Input:
integrate((a+b*sech(d*x+c))**3,x)
Output:
Integral((a + b*sech(c + d*x))**3, x)
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.56 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=a^{3} x - b^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {3 \, a^{2} b \arctan \left (\sinh \left (d x + c\right )\right )}{d} + \frac {6 \, a b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}} \] Input:
integrate((a+b*sech(d*x+c))^3,x, algorithm="maxima")
Output:
a^3*x - b^3*(arctan(e^(-d*x - c))/d - (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))) + 3*a^2*b*arctan(sinh(d*x + c))/d + 6*a*b^2/(d*(e^(-2*d*x - 2*c) + 1))
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.26 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=\frac {{\left (d x + c\right )} a^{3} + {\left (6 \, a^{2} b + b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + \frac {b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 6 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - b^{3} e^{\left (d x + c\right )} - 6 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{2}}}{d} \] Input:
integrate((a+b*sech(d*x+c))^3,x, algorithm="giac")
Output:
((d*x + c)*a^3 + (6*a^2*b + b^3)*arctan(e^(d*x + c)) + (b^3*e^(3*d*x + 3*c ) - 6*a*b^2*e^(2*d*x + 2*c) - b^3*e^(d*x + c) - 6*a*b^2)/(e^(2*d*x + 2*c) + 1)^2)/d
Time = 2.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.26 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=a^3\,x-\frac {\frac {6\,a\,b^2}{d}-\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+6\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {36\,a^4\,b^2+12\,a^2\,b^4+b^6}}\right )\,\sqrt {36\,a^4\,b^2+12\,a^2\,b^4+b^6}}{\sqrt {d^2}}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:
int((a + b/cosh(c + d*x))^3,x)
Output:
a^3*x - ((6*a*b^2)/d - (b^3*exp(c + d*x))/d)/(exp(2*c + 2*d*x) + 1) + (ata n((exp(d*x)*exp(c)*(b^3*(d^2)^(1/2) + 6*a^2*b*(d^2)^(1/2)))/(d*(b^6 + 12*a ^2*b^4 + 36*a^4*b^2)^(1/2)))*(b^6 + 12*a^2*b^4 + 36*a^4*b^2)^(1/2))/(d^2)^ (1/2) - (2*b^3*exp(c + d*x))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1 ))
Time = 0.21 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.25 \[ \int (a+b \text {sech}(c+d x))^3 \, dx=\frac {6 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+12 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +2 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+6 \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +\mathit {atan} \left (e^{d x +c}\right ) b^{3}+e^{4 d x +4 c} a^{3} d x +3 e^{4 d x +4 c} a \,b^{2}+e^{3 d x +3 c} b^{3}+2 e^{2 d x +2 c} a^{3} d x -e^{d x +c} b^{3}+a^{3} d x -3 a \,b^{2}}{d \left (e^{4 d x +4 c}+2 e^{2 d x +2 c}+1\right )} \] Input:
int((a+b*sech(d*x+c))^3,x)
Output:
(6*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b + e**(4*c + 4*d*x)*atan(e**( c + d*x))*b**3 + 12*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b + 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**3 + 6*atan(e**(c + d*x))*a**2*b + atan(e** (c + d*x))*b**3 + e**(4*c + 4*d*x)*a**3*d*x + 3*e**(4*c + 4*d*x)*a*b**2 + e**(3*c + 3*d*x)*b**3 + 2*e**(2*c + 2*d*x)*a**3*d*x - e**(c + d*x)*b**3 + a**3*d*x - 3*a*b**2)/(d*(e**(4*c + 4*d*x) + 2*e**(2*c + 2*d*x) + 1))