Integrand size = 23, antiderivative size = 64 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {(a-b) (a+3 b) \arctan (\sinh (c+d x))}{2 d}+\frac {b^2 \sinh (c+d x)}{d}+\frac {(a-b)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
1/2*(a-b)*(a+3*b)*arctan(sinh(d*x+c))/d+b^2*sinh(d*x+c)/d+1/2*(a-b)^2*sech (d*x+c)*tanh(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.66 (sec) , antiderivative size = 233, normalized size of antiderivative = 3.64 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\text {csch}^3(c+d x) \left (-64 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^6(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2-35 \left (375 a^2+a (37 a+689 b+61 b \cosh (2 (c+d x))) \sinh ^2(c+d x)+303 b^2 \sinh ^4(c+d x)+61 b^2 \sinh ^6(c+d x)\right )+\frac {105 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (125 a^2+2 a (27 a+125 b) \sinh ^2(c+d x)+\left (9 a^2+124 a b+101 b^2\right ) \sinh ^4(c+d x)+2 b (a+27 b) \sinh ^6(c+d x)+b^2 \sinh ^8(c+d x)\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{1680 d} \]
(Csch[c + d*x]^3*(-64*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh [c + d*x]^2]*Sinh[c + d*x]^6*(a + b*Sinh[c + d*x]^2)^2 - 35*(375*a^2 + a*( 37*a + 689*b + 61*b*Cosh[2*(c + d*x)])*Sinh[c + d*x]^2 + 303*b^2*Sinh[c + d*x]^4 + 61*b^2*Sinh[c + d*x]^6) + (105*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(1 25*a^2 + 2*a*(27*a + 125*b)*Sinh[c + d*x]^2 + (9*a^2 + 124*a*b + 101*b^2)* Sinh[c + d*x]^4 + 2*b*(a + 27*b)*Sinh[c + d*x]^6 + b^2*Sinh[c + d*x]^8))/S qrt[-Sinh[c + d*x]^2]))/(1680*d)
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, 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.
| \(\displaystyle \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \sin (i c+i d x)^2\right )^2}{\cos (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \frac {\left (b \sinh ^2(c+d x)+a\right )^2}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (b^2+\frac {a^2-b^2+2 (a-b) b \sinh ^2(c+d x)}{\left (\sinh ^2(c+d x)+1\right )^2}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{2} (a+3 b) (a-b) \arctan (\sinh (c+d x))+\frac {(a-b)^2 \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}+b^2 \sinh (c+d x)}{d}\) |
(((a - b)*(a + 3*b)*ArcTan[Sinh[c + d*x]])/2 + b^2*Sinh[c + d*x] + ((a - b )^2*Sinh[c + d*x])/(2*(1 + Sinh[c + d*x]^2)))/d
3.3.99.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(60)=120\).
Time = 37.89 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.11
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(135\) |
| default | \(\frac {a^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(135\) |
| risch | \(\frac {{\mathrm e}^{d x +c} b^{2}}{2 d}-\frac {{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {{\mathrm e}^{d x +c} \left (a^{2}-2 a b +b^{2}\right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a b}{d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{2}}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a b}{d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{2}}{2 d}\) | \(190\) |
1/d*(a^2*(1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+2*a*b*(-sinh(d*x +c)/cosh(d*x+c)^2+1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+b^2*(sin h(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. 759 vs. \(2 (60) = 120\).
Time = 0.27 (sec) , antiderivative size = 759, normalized size of antiderivative = 11.86 \[ \int \text {sech}^3(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} + {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - 2 \, a^{2} + 4 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - b^{2} + 2 \, {\left ({\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right ) + {\left (5 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 6 \, {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (2 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + d \sinh \left (d x + c\right )^{5} + 2 \, d \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + d \cosh \left (d x + c\right ) + {\left (5 \, d \cosh \left (d x + c\right )^{4} + 6 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )\right )}} \]
1/2*(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 + (2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c)^4 + (15*b^2*cosh(d*x + c)^2 + 2*a^2 - 4*a*b + 3*b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + (2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - (2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c)^2 + (15*b^2*cosh(d*x + c)^4 + 6*(2*a^2 - 4*a*b + 3*b ^2)*cosh(d*x + c)^2 - 2*a^2 + 4*a*b - 3*b^2)*sinh(d*x + c)^2 - b^2 + 2*((a ^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^5 + 5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c )*sinh(d*x + c)^4 + (a^2 + 2*a*b - 3*b^2)*sinh(d*x + c)^5 + 2*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^3 + 2*(5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^2 + a ^2 + 2*a*b - 3*b^2)*sinh(d*x + c)^3 + 2*(5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + (a^2 + 2 *a*b - 3*b^2)*cosh(d*x + c) + (5*(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^4 + 6 *(a^2 + 2*a*b - 3*b^2)*cosh(d*x + c)^2 + a^2 + 2*a*b - 3*b^2)*sinh(d*x + c ))*arctan(cosh(d*x + c) + sinh(d*x + c)) + 2*(3*b^2*cosh(d*x + c)^5 + 2*(2 *a^2 - 4*a*b + 3*b^2)*cosh(d*x + c)^3 - (2*a^2 - 4*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(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)*s inh(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)*sin h(d*x + c))
\[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (60) = 120\).
Time = 0.28 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.66 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {1}{2} \, b^{2} {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} - 2 \, a b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - a^{2} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \]
1/2*b^2*(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)))) - 2*a*b*(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))) - a^2*(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)))
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (60) = 120\).
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.55 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {2 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{2} + 2 \, a b - 3 \, b^{2}\right )} + \frac {4 \, {\left (a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{4 \, d} \]
1/4*(2*b^2*(e^(d*x + c) - e^(-d*x - c)) + (pi + 2*arctan(1/2*(e^(2*d*x + 2 *c) - 1)*e^(-d*x - c)))*(a^2 + 2*a*b - 3*b^2) + 4*(a^2*(e^(d*x + c) - e^(- d*x - c)) - 2*a*b*(e^(d*x + c) - e^(-d*x - c)) + b^2*(e^(d*x + c) - e^(-d* x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4))/d
Time = 1.66 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.44 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^2\,{\mathrm {e}}^{c+d\,x}}{2\,d}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^2\,\sqrt {d^2}-3\,b^2\,\sqrt {d^2}+2\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {a^4+4\,a^3\,b-2\,a^2\,b^2-12\,a\,b^3+9\,b^4}}\right )\,\sqrt {a^4+4\,a^3\,b-2\,a^2\,b^2-12\,a\,b^3+9\,b^4}}{\sqrt {d^2}}-\frac {b^2\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(b^2*exp(c + d*x))/(2*d) + (atan((exp(d*x)*exp(c)*(a^2*(d^2)^(1/2) - 3*b^2 *(d^2)^(1/2) + 2*a*b*(d^2)^(1/2)))/(d*(4*a^3*b - 12*a*b^3 + a^4 + 9*b^4 - 2*a^2*b^2)^(1/2)))*(4*a^3*b - 12*a*b^3 + a^4 + 9*b^4 - 2*a^2*b^2)^(1/2))/( d^2)^(1/2) - (b^2*exp(- c - d*x))/(2*d) + (exp(c + d*x)*(a^2 - 2*a*b + b^2 ))/(d*(exp(2*c + 2*d*x) + 1)) - (2*exp(c + d*x)*(a^2 - 2*a*b + b^2))/(d*(2 *exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))