Integrand size = 21, antiderivative size = 55 \[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {(a-b)^2 \arctan (\sinh (c+d x))}{d}+\frac {(2 a-b) b \sinh (c+d x)}{d}+\frac {b^2 \sinh ^3(c+d x)}{3 d} \] Output:
(a-b)^2*arctan(sinh(d*x+c))/d+(2*a-b)*b*sinh(d*x+c)/d+1/3*b^2*sinh(d*x+c)^ 3/d
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27 \[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\sinh (c+d x) \left (\frac {3 (a-b)^2 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right )}{\sqrt {-\sinh ^2(c+d x)}}+b \left (6 a+b \left (-3+\sinh ^2(c+d x)\right )\right )\right )}{3 d} \] Input:
Integrate[Sech[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(Sinh[c + d*x]*((3*(a - b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]])/Sqrt[-Sinh[c + d*x]^2] + b*(6*a + b*(-3 + Sinh[c + d*x]^2))))/(3*d)
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3669, 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.
\(\displaystyle \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a-b \sin (i c+i d x)^2\right )^2}{\cos (i c+i d x)}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {\left (b \sinh ^2(c+d x)+a\right )^2}{\sinh ^2(c+d x)+1}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \frac {\int \left (\frac {(a-b)^2}{\sinh ^2(c+d x)+1}+b^2 \sinh ^2(c+d x)+(2 a-b) b\right )d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a-b)^2 \arctan (\sinh (c+d x))+b (2 a-b) \sinh (c+d x)+\frac {1}{3} b^2 \sinh ^3(c+d x)}{d}\) |
Input:
Int[Sech[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]
Output:
((a - b)^2*ArcTan[Sinh[c + d*x]] + (2*a - b)*b*Sinh[c + d*x] + (b^2*Sinh[c + d*x]^3)/3)/d
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[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 77.58 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {2 a^{2} \arctan \left ({\mathrm e}^{d x +c}\right )+2 a b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{3}-\sinh \left (d x +c \right )+2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(70\) |
default | \(\frac {2 a^{2} \arctan \left ({\mathrm e}^{d x +c}\right )+2 a b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{3}-\sinh \left (d x +c \right )+2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(70\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}+\frac {{\mathrm e}^{d x +c} a b}{d}-\frac {5 \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {{\mathrm e}^{-d x -c} a b}{d}+\frac {5 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}-\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{d}-\frac {2 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a b}{d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{2}}{d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{d}+\frac {2 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a b}{d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{2}}{d}\) | \(207\) |
Input:
int(sech(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(2*a^2*arctan(exp(d*x+c))+2*a*b*(sinh(d*x+c)-2*arctan(exp(d*x+c)))+b^2 *(1/3*sinh(d*x+c)^3-sinh(d*x+c)+2*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (53) = 106\).
Time = 0.10 (sec) , antiderivative size = 446, normalized size of antiderivative = 8.11 \[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \, {\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 8 \, a b + 5 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - b^{2} + 48 \, {\left ({\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a^{2} - 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3}\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 6 \, {\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (8 \, a b - 5 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \] Input:
integrate(sech(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/24*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh (d*x + c)^6 + 3*(8*a*b - 5*b^2)*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + 8*a*b - 5*b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*(8*a*b - 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(8*a*b - 5*b^2)*cosh(d*x + c)^2 + 3*(5*b^2*cosh(d*x + c)^4 + 6*(8*a*b - 5*b^2)*cosh(d*x + c)^2 - 8*a*b + 5 *b^2)*sinh(d*x + c)^2 - b^2 + 48*((a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + 3* (a^2 - 2*a*b + b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2 - 2*a*b + b^2)* cosh(d*x + c)*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*sinh(d*x + c)^3)*arcta n(cosh(d*x + c) + sinh(d*x + c)) + 6*(b^2*cosh(d*x + c)^5 + 2*(8*a*b - 5*b ^2)*cosh(d*x + c)^3 - (8*a*b - 5*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cos h(d*x + c)^3 + 3*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*d*cosh(d*x + c)*sinh( d*x + c)^2 + d*sinh(d*x + c)^3)
\[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {sech}{\left (c + d x \right )}\, dx \] Input:
integrate(sech(d*x+c)*(a+b*sinh(d*x+c)**2)**2,x)
Output:
Integral((a + b*sinh(c + d*x)**2)**2*sech(c + d*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (53) = 106\).
Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.42 \[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {1}{24} \, b^{2} {\left (\frac {{\left (15 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )} e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {15 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {48 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} + a b {\left (\frac {4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{2} \arctan \left (\sinh \left (d x + c\right )\right )}{d} \] Input:
integrate(sech(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
-1/24*b^2*((15*e^(-2*d*x - 2*c) - 1)*e^(3*d*x + 3*c)/d - (15*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + 48*arctan(e^(-d*x - c))/d) + a*b*(4*arctan(e^(-d*x - c))/d + e^(d*x + c)/d - e^(-d*x - c)/d) + a^2*arctan(sinh(d*x + c))/d
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.85 \[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b e^{\left (d x + c\right )} - 15 \, b^{2} e^{\left (d x + c\right )} + 48 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) - {\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \] Input:
integrate(sech(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/24*(b^2*e^(3*d*x + 3*c) + 24*a*b*e^(d*x + c) - 15*b^2*e^(d*x + c) + 48*( a^2 - 2*a*b + b^2)*arctan(e^(d*x + c)) - (24*a*b*e^(2*d*x + 2*c) - 15*b^2* e^(2*d*x + 2*c) + b^2)*e^(-3*d*x - 3*c))/d
Time = 0.18 (sec) , antiderivative size = 182, normalized size of antiderivative = 3.31 \[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (8\,a\,b-5\,b^2\right )}{8\,d}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^2\,\sqrt {d^2}+b^2\,\sqrt {d^2}-2\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}}\right )\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}}{\sqrt {d^2}}+\frac {b\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a-5\,b\right )}{8\,d} \] Input:
int((a + b*sinh(c + d*x)^2)^2/cosh(c + d*x),x)
Output:
(b^2*exp(3*c + 3*d*x))/(24*d) - (b^2*exp(- 3*c - 3*d*x))/(24*d) - (exp(- c - d*x)*(8*a*b - 5*b^2))/(8*d) + (2*atan((exp(d*x)*exp(c)*(a^2*(d^2)^(1/2) + b^2*(d^2)^(1/2) - 2*a*b*(d^2)^(1/2)))/(d*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)^(1/2)))*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)^(1/2))/( d^2)^(1/2) + (b*exp(c + d*x)*(8*a - 5*b))/(8*d)
Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.96 \[ \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {48 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2}-96 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a b +48 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}+e^{6 d x +6 c} b^{2}+24 e^{4 d x +4 c} a b -15 e^{4 d x +4 c} b^{2}-24 e^{2 d x +2 c} a b +15 e^{2 d x +2 c} b^{2}-b^{2}}{24 e^{3 d x +3 c} d} \] Input:
int(sech(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x)
Output:
(48*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a**2 - 96*e**(3*c + 3*d*x)*atan(e* *(c + d*x))*a*b + 48*e**(3*c + 3*d*x)*atan(e**(c + d*x))*b**2 + e**(6*c + 6*d*x)*b**2 + 24*e**(4*c + 4*d*x)*a*b - 15*e**(4*c + 4*d*x)*b**2 - 24*e**( 2*c + 2*d*x)*a*b + 15*e**(2*c + 2*d*x)*b**2 - b**2)/(24*e**(3*c + 3*d*x)*d )