Integrand size = 21, antiderivative size = 93 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {3 b \left (8 a^2+4 a b+b^2\right ) \arctan (\sinh (c+d x))}{8 d}+\frac {a^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}+\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \] Output:
3/8*b*(8*a^2+4*a*b+b^2)*arctan(sinh(d*x+c))/d+a^3*sinh(d*x+c)/d+3/8*b^2*(4 *a+b)*sech(d*x+c)*tanh(d*x+c)/d+1/4*b^3*sech(d*x+c)^3*tanh(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 8.08 (sec) , antiderivative size = 575, normalized size of antiderivative = 6.18 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {\cosh (c+d x) \coth ^5(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (256 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^3+384 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \left (7 b+a \left (7+5 \sinh ^2(c+d x)\right )\right )+\frac {315 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (b^3 \left (16807+15000 \sinh ^2(c+d x)+2187 \sinh ^4(c+d x)-62 \sinh ^6(c+d x)\right )+a^3 \cosh ^4(c+d x) \left (16807+24604 \sinh ^2(c+d x)+11562 \sinh ^4(c+d x)+1468 \sinh ^6(c+d x)+7 \sinh ^8(c+d x)\right )+3 a b^2 \left (16807+29406 \sinh ^2(c+d x)+15312 \sinh ^4(c+d x)+1858 \sinh ^6(c+d x)+9 \sinh ^8(c+d x)\right )+3 a^2 b \left (16807+43812 \sinh ^2(c+d x)+40442 \sinh ^4(c+d x)+14956 \sinh ^6(c+d x)+1719 \sinh ^8(c+d x)+8 \sinh ^{10}(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}-21 \left (b^3 \left (252105+140965 \sinh ^2(c+d x)+8226 \sinh ^4(c+d x)\right )+3 a b^2 \left (252105+357055 \sinh ^2(c+d x)+133071 \sinh ^4(c+d x)+6393 \sinh ^6(c+d x)\right )+3 a^2 b \left (252105+573145 \sinh ^2(c+d x)+437991 \sinh ^4(c+d x)+120431 \sinh ^6(c+d x)+5640 \sinh ^8(c+d x)\right )+a^3 \left (252105+789235 \sinh ^2(c+d x)+922986 \sinh ^4(c+d x)+491574 \sinh ^6(c+d x)+107725 \sinh ^8(c+d x)+4887 \sinh ^{10}(c+d x)\right )\right )\right )}{7560 d (a+2 b+a \cosh (2 c+2 d x))^3} \] Input:
Integrate[Cosh[c + d*x]*(a + b*Sech[c + d*x]^2)^3,x]
Output:
-1/7560*(Cosh[c + d*x]*Coth[c + d*x]^5*(a + b*Sech[c + d*x]^2)^3*(256*Hype rgeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]* Sinh[c + d*x]^8*(a + b + a*Sinh[c + d*x]^2)^3 + 384*HypergeometricPFQ[{3/2 , 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + b + a*Sinh[c + d*x]^2)^2*(7*b + a*(7 + 5*Sinh[c + d*x]^2)) + (315*ArcTanh[Sqr t[-Sinh[c + d*x]^2]]*(b^3*(16807 + 15000*Sinh[c + d*x]^2 + 2187*Sinh[c + d *x]^4 - 62*Sinh[c + d*x]^6) + a^3*Cosh[c + d*x]^4*(16807 + 24604*Sinh[c + d*x]^2 + 11562*Sinh[c + d*x]^4 + 1468*Sinh[c + d*x]^6 + 7*Sinh[c + d*x]^8) + 3*a*b^2*(16807 + 29406*Sinh[c + d*x]^2 + 15312*Sinh[c + d*x]^4 + 1858*S inh[c + d*x]^6 + 9*Sinh[c + d*x]^8) + 3*a^2*b*(16807 + 43812*Sinh[c + d*x] ^2 + 40442*Sinh[c + d*x]^4 + 14956*Sinh[c + d*x]^6 + 1719*Sinh[c + d*x]^8 + 8*Sinh[c + d*x]^10)))/Sqrt[-Sinh[c + d*x]^2] - 21*(b^3*(252105 + 140965* Sinh[c + d*x]^2 + 8226*Sinh[c + d*x]^4) + 3*a*b^2*(252105 + 357055*Sinh[c + d*x]^2 + 133071*Sinh[c + d*x]^4 + 6393*Sinh[c + d*x]^6) + 3*a^2*b*(25210 5 + 573145*Sinh[c + d*x]^2 + 437991*Sinh[c + d*x]^4 + 120431*Sinh[c + d*x] ^6 + 5640*Sinh[c + d*x]^8) + a^3*(252105 + 789235*Sinh[c + d*x]^2 + 922986 *Sinh[c + d*x]^4 + 491574*Sinh[c + d*x]^6 + 107725*Sinh[c + d*x]^8 + 4887* Sinh[c + d*x]^10))))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)
Time = 0.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 4635, 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.
| \(\displaystyle \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sec (i c+i d x)^2\right )^3}{\sec (i c+i d x)}dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\left (a \sinh ^2(c+d x)+a+b\right )^3}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (a^3+\frac {3 a^2 b \sinh ^4(c+d x)+3 a b (2 a+b) \sinh ^2(c+d x)+b \left (3 a^2+3 b a+b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^3}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \sinh (c+d x)+\frac {3}{8} b \left (8 a^2+4 a b+b^2\right ) \arctan (\sinh (c+d x))+\frac {3 b^2 (4 a+b) \sinh (c+d x)}{8 \left (\sinh ^2(c+d x)+1\right )}+\frac {b^3 \sinh (c+d x)}{4 \left (\sinh ^2(c+d x)+1\right )^2}}{d}\) |
Input:
Int[Cosh[c + d*x]*(a + b*Sech[c + d*x]^2)^3,x]
Output:
((3*b*(8*a^2 + 4*a*b + b^2)*ArcTan[Sinh[c + d*x]])/8 + a^3*Sinh[c + d*x] + (b^3*Sinh[c + d*x])/(4*(1 + Sinh[c + d*x]^2)^2) + (3*b^2*(4*a + b)*Sinh[c + d*x])/(8*(1 + Sinh[c + d*x]^2)))/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[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && In tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Time = 1.79 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.04
| method | result | size |
| derivativedivides | \(\frac {a^{3} \sinh \left (d x +c \right )+6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(97\) |
| default | \(\frac {a^{3} \sinh \left (d x +c \right )+6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )+3 a \,b^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )+\frac {3 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(97\) |
| parallelrisch | \(\frac {-48 i \left (\frac {3}{4}+\frac {\cosh \left (4 d x +4 c \right )}{4}+\cosh \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {1}{2} a b +\frac {1}{8} b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+48 i \left (\frac {3}{4}+\frac {\cosh \left (4 d x +4 c \right )}{4}+\cosh \left (2 d x +2 c \right )\right ) b \left (a^{2}+\frac {1}{2} a b +\frac {1}{8} b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+3 \left (2 a^{3}+4 a \,b^{2}+b^{3}\right ) \sinh \left (3 d x +3 c \right )+2 a^{3} \sinh \left (5 d x +5 c \right )+4 \sinh \left (d x +c \right ) \left (a^{3}+3 a \,b^{2}+\frac {11}{4} b^{3}\right )}{4 d \left (\cosh \left (4 d x +4 c \right )+4 \cosh \left (2 d x +2 c \right )+3\right )}\) | \(198\) |
| risch | \(\frac {a^{3} {\mathrm e}^{d x +c}}{2 d}-\frac {a^{3} {\mathrm e}^{-d x -c}}{2 d}+\frac {b^{2} {\mathrm e}^{d x +c} \left (12 \,{\mathrm e}^{6 d x +6 c} a +3 \,{\mathrm e}^{6 d x +6 c} b +12 \,{\mathrm e}^{4 d x +4 c} a +11 \,{\mathrm e}^{4 d x +4 c} b -12 a \,{\mathrm e}^{2 d x +2 c}-11 b \,{\mathrm e}^{2 d x +2 c}-12 a -3 b \right )}{4 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{4}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{d}+\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 d}+\frac {3 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{d}-\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 d}-\frac {3 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 d}\) | \(257\) |
Input:
int(cosh(d*x+c)*(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*sinh(d*x+c)+6*a^2*b*arctan(exp(d*x+c))+3*a*b^2*(1/2*sech(d*x+c)*t anh(d*x+c)+arctan(exp(d*x+c)))+b^3*((1/4*sech(d*x+c)^3+3/8*sech(d*x+c))*ta nh(d*x+c)+3/4*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 1992 vs. \(2 (87) = 174\).
Time = 0.39 (sec) , antiderivative size = 1992, normalized size of antiderivative = 21.42 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
Output:
1/4*(2*a^3*cosh(d*x + c)^10 + 20*a^3*cosh(d*x + c)*sinh(d*x + c)^9 + 2*a^3 *sinh(d*x + c)^10 + 3*(2*a^3 + 4*a*b^2 + b^3)*cosh(d*x + c)^8 + 3*(30*a^3* cosh(d*x + c)^2 + 2*a^3 + 4*a*b^2 + b^3)*sinh(d*x + c)^8 + 24*(10*a^3*cosh (d*x + c)^3 + (2*a^3 + 4*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + (4* a^3 + 12*a*b^2 + 11*b^3)*cosh(d*x + c)^6 + (420*a^3*cosh(d*x + c)^4 + 4*a^ 3 + 12*a*b^2 + 11*b^3 + 84*(2*a^3 + 4*a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d *x + c)^6 + 6*(84*a^3*cosh(d*x + c)^5 + 28*(2*a^3 + 4*a*b^2 + b^3)*cosh(d* x + c)^3 + (4*a^3 + 12*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - (4 *a^3 + 12*a*b^2 + 11*b^3)*cosh(d*x + c)^4 + (420*a^3*cosh(d*x + c)^6 + 210 *(2*a^3 + 4*a*b^2 + b^3)*cosh(d*x + c)^4 - 4*a^3 - 12*a*b^2 - 11*b^3 + 15* (4*a^3 + 12*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(60*a^3*c osh(d*x + c)^7 + 42*(2*a^3 + 4*a*b^2 + b^3)*cosh(d*x + c)^5 + 5*(4*a^3 + 1 2*a*b^2 + 11*b^3)*cosh(d*x + c)^3 - (4*a^3 + 12*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*a^3 - 3*(2*a^3 + 4*a*b^2 + b^3)*cosh(d*x + c)^2 + 3*(30*a^3*cosh(d*x + c)^8 + 28*(2*a^3 + 4*a*b^2 + b^3)*cosh(d*x + c)^6 + 5*(4*a^3 + 12*a*b^2 + 11*b^3)*cosh(d*x + c)^4 - 2*a^3 - 4*a*b^2 - b^3 - 2* (4*a^3 + 12*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((8*a^2*b + 4*a*b^2 + b^3)*cosh(d*x + c)^9 + 9*(8*a^2*b + 4*a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^8 + (8*a^2*b + 4*a*b^2 + b^3)*sinh(d*x + c)^9 + 4*(8*a^2 *b + 4*a*b^2 + b^3)*cosh(d*x + c)^7 + 4*(8*a^2*b + 4*a*b^2 + b^3 + 9*(8...
\[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \cosh {\left (c + d x \right )}\, dx \] Input:
integrate(cosh(d*x+c)*(a+b*sech(d*x+c)**2)**3,x)
Output:
Integral((a + b*sech(c + d*x)**2)**3*cosh(c + d*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (87) = 174\).
Time = 0.11 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.38 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {1}{4} \, b^{3} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - 3 \, a b^{2} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - \frac {6 \, a^{2} b \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {a^{3} \sinh \left (d x + c\right )}{d} \] Input:
integrate(cosh(d*x+c)*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
Output:
-1/4*b^3*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6*e^ (-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - 3*a*b^2*(a rctan(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))) - 6*a^2*b*arctan(e^(-d*x - c))/d + a^3*si nh(d*x + c)/d
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (87) = 174\).
Time = 0.15 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.14 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {8 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (8 \, a^{2} b + 4 \, a b^{2} + b^{3}\right )} + \frac {4 \, {\left (12 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 3 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 48 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 20 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \] Input:
integrate(cosh(d*x+c)*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
Output:
1/16*(8*a^3*(e^(d*x + c) - e^(-d*x - c)) + 3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(8*a^2*b + 4*a*b^2 + b^3) + 4*(12*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 3*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 48*a*b^2* (e^(d*x + c) - e^(-d*x - c)) + 20*b^3*(e^(d*x + c) - e^(-d*x - c)))/((e^(d *x + c) - e^(-d*x - c))^2 + 4)^2)/d
Time = 0.17 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.70 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {a^3\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+4\,a\,b^2\,\sqrt {d^2}+8\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {64\,a^4\,b^2+64\,a^3\,b^3+32\,a^2\,b^4+8\,a\,b^5+b^6}}\right )\,\sqrt {64\,a^4\,b^2+64\,a^3\,b^3+32\,a^2\,b^4+8\,a\,b^5+b^6}}{4\,\sqrt {d^2}}-\frac {6\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (12\,a\,b^2-b^3\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {4\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (b^3+4\,a\,b^2\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int(cosh(c + d*x)*(a + b/cosh(c + d*x)^2)^3,x)
Output:
(a^3*exp(c + d*x))/(2*d) - (a^3*exp(- c - d*x))/(2*d) + (3*atan((exp(d*x)* exp(c)*(b^3*(d^2)^(1/2) + 4*a*b^2*(d^2)^(1/2) + 8*a^2*b*(d^2)^(1/2)))/(d*( 8*a*b^5 + b^6 + 32*a^2*b^4 + 64*a^3*b^3 + 64*a^4*b^2)^(1/2)))*(8*a*b^5 + b ^6 + 32*a^2*b^4 + 64*a^3*b^3 + 64*a^4*b^2)^(1/2))/(4*(d^2)^(1/2)) - (6*b^3 *exp(c + d*x))/(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)*(12*a*b^2 - b^3))/(2*d*(2*exp(2*c + 2*d*x) + exp (4*c + 4*d*x) + 1)) + (4*b^3*exp(c + d*x))/(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)) + (3*exp(c + d* x)*(4*a*b^2 + b^3))/(4*d*(exp(2*c + 2*d*x) + 1))
Time = 0.24 (sec) , antiderivative size = 615, normalized size of antiderivative = 6.61 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {-12 e^{4 d x +4 c} a \,b^{2}+3 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+3 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+12 e^{8 d x +8 c} a \,b^{2}+12 e^{6 d x +6 c} a \,b^{2}-11 e^{4 d x +4 c} b^{3}+2 e^{10 d x +10 c} a^{3}-12 e^{2 d x +2 c} a \,b^{2}-2 a^{3}-3 e^{2 d x +2 c} b^{3}+48 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+72 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+48 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+6 e^{8 d x +8 c} a^{3}+3 e^{8 d x +8 c} b^{3}+4 e^{6 d x +6 c} a^{3}+11 e^{6 d x +6 c} b^{3}-4 e^{4 d x +4 c} a^{3}-6 e^{2 d x +2 c} a^{3}+24 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +12 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+96 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +144 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +96 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +24 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +12 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+12 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+18 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+12 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}}{4 e^{d x +c} d \left (e^{8 d x +8 c}+4 e^{6 d x +6 c}+6 e^{4 d x +4 c}+4 e^{2 d x +2 c}+1\right )} \] Input:
int(cosh(d*x+c)*(a+b*sech(d*x+c)^2)^3,x)
Output:
(24*e**(9*c + 9*d*x)*atan(e**(c + d*x))*a**2*b + 12*e**(9*c + 9*d*x)*atan( e**(c + d*x))*a*b**2 + 3*e**(9*c + 9*d*x)*atan(e**(c + d*x))*b**3 + 96*e** (7*c + 7*d*x)*atan(e**(c + d*x))*a**2*b + 48*e**(7*c + 7*d*x)*atan(e**(c + d*x))*a*b**2 + 12*e**(7*c + 7*d*x)*atan(e**(c + d*x))*b**3 + 144*e**(5*c + 5*d*x)*atan(e**(c + d*x))*a**2*b + 72*e**(5*c + 5*d*x)*atan(e**(c + d*x) )*a*b**2 + 18*e**(5*c + 5*d*x)*atan(e**(c + d*x))*b**3 + 96*e**(3*c + 3*d* x)*atan(e**(c + d*x))*a**2*b + 48*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a*b* *2 + 12*e**(3*c + 3*d*x)*atan(e**(c + d*x))*b**3 + 24*e**(c + d*x)*atan(e* *(c + d*x))*a**2*b + 12*e**(c + d*x)*atan(e**(c + d*x))*a*b**2 + 3*e**(c + d*x)*atan(e**(c + d*x))*b**3 + 2*e**(10*c + 10*d*x)*a**3 + 6*e**(8*c + 8* d*x)*a**3 + 12*e**(8*c + 8*d*x)*a*b**2 + 3*e**(8*c + 8*d*x)*b**3 + 4*e**(6 *c + 6*d*x)*a**3 + 12*e**(6*c + 6*d*x)*a*b**2 + 11*e**(6*c + 6*d*x)*b**3 - 4*e**(4*c + 4*d*x)*a**3 - 12*e**(4*c + 4*d*x)*a*b**2 - 11*e**(4*c + 4*d*x )*b**3 - 6*e**(2*c + 2*d*x)*a**3 - 12*e**(2*c + 2*d*x)*a*b**2 - 3*e**(2*c + 2*d*x)*b**3 - 2*a**3)/(4*e**(c + d*x)*d*(e**(8*c + 8*d*x) + 4*e**(6*c + 6*d*x) + 6*e**(4*c + 4*d*x) + 4*e**(2*c + 2*d*x) + 1))