Integrand size = 23, antiderivative size = 154 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {(a+b) \left (5 a^2-2 a b+5 b^2\right ) \arctan (\sinh (c+d x))}{16 d}+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 \left (a^2-b^2\right ) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {(a-b) \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d} \]
1/16*(a+b)*(5*a^2-2*a*b+5*b^2)*arctan(sinh(d*x+c))/d+1/48*(a-b)*(15*a^2+14 *a*b+15*b^2)*sech(d*x+c)*tanh(d*x+c)/d+5/24*(a^2-b^2)*sech(d*x+c)^3*(a+b*s inh(d*x+c)^2)*tanh(d*x+c)/d+1/6*(a-b)*sech(d*x+c)^5*(a+b*sinh(d*x+c)^2)^2* tanh(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.91 (sec) , antiderivative size = 1192, normalized size of antiderivative = 7.74 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx =\text {Too large to display} \]
(Csch[c + d*x]^5*(-117228825*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] - 1092656 25*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 274542345*a^2*b*A rcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 17069535*a^3*ArcTanh[Sqrt [-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 260465625*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 215549775*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^ 2]]*Sinh[c + d*x]^4 + 142065*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 41427855*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 207173295*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 58009455 *b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 210735*a^2*b*ArcTan h[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 33756345*a*b^2*ArcTanh[Sqrt[-S inh[c + d*x]^2]]*Sinh[c + d*x]^8 - 56109375*b^3*ArcTanh[Sqrt[-Sinh[c + d*x ]^2]]*Sinh[c + d*x]^8 - 174825*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[ c + d*x]^10 - 9261945*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10 - 48825*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^12 + 117228825* a^3*Sqrt[-Sinh[c + d*x]^2] + 4093425*a^3*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d* x]^2] + 168951510*a^2*b*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 215549775 *a*b^2*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 9514449*a^2*b*Sinh[c + d*x ]^6*Sqrt[-Sinh[c + d*x]^2] + 135323370*a*b^2*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 58009455*b^3*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 7808535* a*b^2*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 36772890*b^3*Sinh[c + d*...
Time = 0.35 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3669, 315, 401, 25, 298, 216}
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.
| \(\displaystyle \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \sin (i c+i d x)^2\right )^3}{\cos (i c+i d x)^7}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \frac {\left (b \sinh ^2(c+d x)+a\right )^3}{\left (\sinh ^2(c+d x)+1\right )^4}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {\left (b \sinh ^2(c+d x)+a\right ) \left (b (a+5 b) \sinh ^2(c+d x)+a (5 a+b)\right )}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}-\frac {1}{4} \int -\frac {b \left (5 a^2+4 b a+15 b^2\right ) \sinh ^2(c+d x)+a \left (15 a^2+4 b a+5 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {b \left (5 a^2+4 b a+15 b^2\right ) \sinh ^2(c+d x)+a \left (15 a^2+4 b a+5 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)+\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} (a+b) \left (5 a^2-2 a b+5 b^2\right ) \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} (a+b) \left (5 a^2-2 a b+5 b^2\right ) \arctan (\sinh (c+d x))+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\) |
(((a - b)*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(6*(1 + Sinh[c + d*x]^2 )^3) + ((5*(a^2 - b^2)*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2))/(4*(1 + Sinh [c + d*x]^2)^2) + ((3*(a + b)*(5*a^2 - 2*a*b + 5*b^2)*ArcTan[Sinh[c + d*x] ])/2 + ((a - b)*(15*a^2 + 14*a*b + 15*b^2)*Sinh[c + d*x])/(2*(1 + Sinh[c + d*x]^2)))/4)/6)/d
3.4.14.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
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. \(313\) vs. \(2(146)=292\).
Time = 0.13 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.04
\[\frac {a^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{5 \cosh \left (d x +c \right )^{6}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )}{5}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{6}}-\frac {\sinh \left (d x +c \right )}{5 \cosh \left (d x +c \right )^{6}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )}{5}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{6}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{6}}-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{6}}+\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )}{d}\]
1/d*(a^3*((1/6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x +c)+5/8*arctan(exp(d*x+c)))+3*a^2*b*(-1/5*sinh(d*x+c)/cosh(d*x+c)^6+1/5*(1 /6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x+c)+1/8*arct an(exp(d*x+c)))+3*a*b^2*(-1/3*sinh(d*x+c)^3/cosh(d*x+c)^6-1/5*sinh(d*x+c)/ cosh(d*x+c)^6+1/5*(1/6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))* tanh(d*x+c)+1/8*arctan(exp(d*x+c)))+b^3*(-sinh(d*x+c)^5/cosh(d*x+c)^6-5/3* sinh(d*x+c)^3/cosh(d*x+c)^6-sinh(d*x+c)/cosh(d*x+c)^6+(1/6*sech(d*x+c)^5+5 /24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x+c)+5/8*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 3675 vs. \(2 (146) = 292\).
Time = 0.29 (sec) , antiderivative size = 3675, normalized size of antiderivative = 23.86 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
1/24*(3*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^11 + 33*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)*sinh(d*x + c)^10 + 3*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*sinh(d*x + c)^11 + (85*a^3 + 51*a^2*b - 141*a* b^2 + 5*b^3)*cosh(d*x + c)^9 + (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3 + 16 5*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 9*(55*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^3 + (85*a^3 + 51* a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 18*(11*a^3 - 1 9*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^7 + 18*(55*(5*a^3 + 3*a^2*b + 3* a*b^2 - 11*b^3)*cosh(d*x + c)^4 + 11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3 + 2 *(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 42*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^5 + 2*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(11*a^3 - 19*a^2*b + 1 3*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 18*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^5 + 18*(77*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11 *b^3)*cosh(d*x + c)^6 + 7*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^4 - 11*a^3 + 19*a^2*b - 13*a*b^2 + 5*b^3 + 21*(11*a^3 - 19*a^2*b + 1 3*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 18*(55*(5*a^3 + 3*a^2* b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^7 + 7*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 35*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d *x + c)^3 - 5*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c))*sin...
Timed out. \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (146) = 292\).
Time = 0.33 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.19 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=-\frac {1}{24} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {33 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 90 \, e^{\left (-5 \, d x - 5 \, c\right )} - 90 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} - 33 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {1}{24} \, a^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {1}{8} \, a^{2} b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 17 \, e^{\left (-3 \, d x - 3 \, c\right )} - 114 \, e^{\left (-5 \, d x - 5 \, c\right )} + 114 \, e^{\left (-7 \, d x - 7 \, c\right )} - 17 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} - 47 \, e^{\left (-3 \, d x - 3 \, c\right )} + 78 \, e^{\left (-5 \, d x - 5 \, c\right )} - 78 \, e^{\left (-7 \, d x - 7 \, c\right )} + 47 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} \]
-1/24*b^3*(15*arctan(e^(-d*x - c))/d + (33*e^(-d*x - c) - 5*e^(-3*d*x - 3* c) + 90*e^(-5*d*x - 5*c) - 90*e^(-7*d*x - 7*c) + 5*e^(-9*d*x - 9*c) - 33*e ^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6 *d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12 *c) + 1))) - 1/24*a^3*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) + 85*e ^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) - 198*e^(-7*d*x - 7*c) - 85*e^(-9*d *x - 9*c) - 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) - 1/8*a^2*b*(3*arctan(e^(-d*x - c))/d - (3*e^(- d*x - c) + 17*e^(-3*d*x - 3*c) - 114*e^(-5*d*x - 5*c) + 114*e^(-7*d*x - 7* c) - 17*e^(-9*d*x - 9*c) - 3*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10 *d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) - 1/8*a*b^2*(3*arctan(e^(-d*x - c ))/d - (3*e^(-d*x - c) - 47*e^(-3*d*x - 3*c) + 78*e^(-5*d*x - 5*c) - 78*e^ (-7*d*x - 7*c) + 47*e^(-9*d*x - 9*c) - 3*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d *x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c ) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (146) = 292\).
Time = 0.34 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.49 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (5 \, a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (15 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} - 33 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 160 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 96 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 96 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 160 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 240 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3}}}{96 \, d} \]
1/96*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(5*a^3 + 3 *a^2*b + 3*a*b^2 + 5*b^3) + 4*(15*a^3*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a ^2*b*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a*b^2*(e^(d*x + c) - e^(-d*x - c)) ^5 - 33*b^3*(e^(d*x + c) - e^(-d*x - c))^5 + 160*a^3*(e^(d*x + c) - e^(-d* x - c))^3 + 96*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3 - 96*a*b^2*(e^(d*x + c ) - e^(-d*x - c))^3 - 160*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 528*a^3*(e^ (d*x + c) - e^(-d*x - c)) - 144*a^2*b*(e^(d*x + c) - e^(-d*x - c)) - 144*a *b^2*(e^(d*x + c) - e^(-d*x - c)) - 240*b^3*(e^(d*x + c) - e^(-d*x - c)))/ ((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3)/d
Time = 0.20 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.90 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^3\,\sqrt {d^2}+5\,b^3\,\sqrt {d^2}+3\,a\,b^2\,\sqrt {d^2}+3\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {25\,a^6+30\,a^5\,b+39\,a^4\,b^2+68\,a^3\,b^3+39\,a^2\,b^4+30\,a\,b^5+25\,b^6}}\right )\,\sqrt {25\,a^6+30\,a^5\,b+39\,a^4\,b^2+68\,a^3\,b^3+39\,a^2\,b^4+30\,a\,b^5+25\,b^6}}{8\,\sqrt {d^2}}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (3\,a^3-11\,a^2\,b+13\,a\,b^2-5\,b^3\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3-57\,a^2\,b+111\,a\,b^2-55\,b^3\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+3\,a^2\,b+3\,a\,b^2-11\,b^3\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+3\,a^2\,b-93\,a\,b^2+85\,b^3\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {80\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {32\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )} \]
(atan((exp(d*x)*exp(c)*(5*a^3*(d^2)^(1/2) + 5*b^3*(d^2)^(1/2) + 3*a*b^2*(d ^2)^(1/2) + 3*a^2*b*(d^2)^(1/2)))/(d*(30*a*b^5 + 30*a^5*b + 25*a^6 + 25*b^ 6 + 39*a^2*b^4 + 68*a^3*b^3 + 39*a^4*b^2)^(1/2)))*(30*a*b^5 + 30*a^5*b + 2 5*a^6 + 25*b^6 + 39*a^2*b^4 + 68*a^3*b^3 + 39*a^4*b^2)^(1/2))/(8*(d^2)^(1/ 2)) - (6*exp(c + d*x)*(13*a*b^2 - 11*a^2*b + 3*a^3 - 5*b^3))/(d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1 )) + (exp(c + d*x)*(111*a*b^2 - 57*a^2*b + a^3 - 55*b^3))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (exp(c + d*x)*(3* a*b^2 + 3*a^2*b + 5*a^3 - 11*b^3))/(8*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(3*a^2*b - 93*a*b^2 + 5*a^3 + 85*b^3))/(12*d*(2*exp(2*c + 2*d*x) + e xp(4*c + 4*d*x) + 1)) + (80*exp(c + d*x)*(3*a*b^2 - 3*a^2*b + a^3 - b^3))/ (3*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*e xp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (32*exp(c + d*x)*(3*a*b^2 - 3 *a^2*b + a^3 - b^3))/(3*d*(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*e xp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1))