Integrand size = 21, antiderivative size = 56 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {b (4 a+b) \arctan (\sinh (c+d x))}{2 d}+\frac {a^2 \sinh (c+d x)}{d}+\frac {b^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.43 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {2 a b \arctan (\sinh (c+d x))}{d}+\frac {b^2 \arctan (\sinh (c+d x))}{2 d}+\frac {a^2 \cosh (d x) \sinh (c)}{d}+\frac {a^2 \cosh (c) \sinh (d x)}{d}+\frac {b^2 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \]
(2*a*b*ArcTan[Sinh[c + d*x]])/d + (b^2*ArcTan[Sinh[c + d*x]])/(2*d) + (a^2 *Cosh[d*x]*Sinh[c])/d + (a^2*Cosh[c]*Sinh[d*x])/d + (b^2*Sech[c + d*x]*Tan h[c + d*x])/(2*d)
Time = 0.27 (sec) , antiderivative size = 57, 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.
\(\displaystyle \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sec (i c+i d x)^2\right )^2}{\sec (i c+i d x)}dx\) |
\(\Big \downarrow \) 4635 |
\(\displaystyle \frac {\int \frac {\left (a \sinh ^2(c+d x)+a+b\right )^2}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \frac {\int \left (a^2+\frac {2 a b \sinh ^2(c+d x)+b (2 a+b)}{\left (\sinh ^2(c+d x)+1\right )^2}\right )d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \sinh (c+d x)+\frac {1}{2} b (4 a+b) \arctan (\sinh (c+d x))+\frac {b^2 \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}}{d}\) |
((b*(4*a + b)*ArcTan[Sinh[c + d*x]])/2 + a^2*Sinh[c + d*x] + (b^2*Sinh[c + d*x])/(2*(1 + Sinh[c + d*x]^2)))/d
3.1.60.3.1 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[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 = 1.00 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(\frac {a^{2} \sinh \left (d x +c \right )+4 a b \arctan \left ({\mathrm e}^{d x +c}\right )+b^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(53\) |
default | \(\frac {a^{2} \sinh \left (d x +c \right )+4 a b \arctan \left ({\mathrm e}^{d x +c}\right )+b^{2} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(53\) |
parallelrisch | \(\frac {-4 i \left (a +\frac {b}{4}\right ) \left (1+\cosh \left (2 d x +2 c \right )\right ) b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+4 i \left (a +\frac {b}{4}\right ) \left (1+\cosh \left (2 d x +2 c \right )\right ) b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+a^{2} \sinh \left (3 d x +3 c \right )+\sinh \left (d x +c \right ) \left (a^{2}+2 b^{2}\right )}{2 d \left (1+\cosh \left (2 d x +2 c \right )\right )}\) | \(115\) |
risch | \(\frac {a^{2} {\mathrm e}^{d x +c}}{2 d}-\frac {a^{2} {\mathrm e}^{-d x -c}}{2 d}+\frac {b^{2} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {2 i b a \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}+\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {2 i b a \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}-\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\) | \(144\) |
1/d*(a^2*sinh(d*x+c)+4*a*b*arctan(exp(d*x+c))+b^2*(1/2*sech(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. 653 vs. \(2 (52) = 104\).
Time = 0.28 (sec) , antiderivative size = 653, normalized size of antiderivative = 11.66 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {a^{2} \cosh \left (d x + c\right )^{6} + 6 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + a^{2} \sinh \left (d x + c\right )^{6} + {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, a^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 2 \, {\left ({\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + {\left (4 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{5} + 2 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 4 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3} + 2 \, {\left (5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) + {\left (5 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 6 \, {\left (4 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 4 \, a b + 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 \, a^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (a^{2} + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + 2 \, 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*(a^2*cosh(d*x + c)^6 + 6*a^2*cosh(d*x + c)*sinh(d*x + c)^5 + a^2*sinh( d*x + c)^6 + (a^2 + 2*b^2)*cosh(d*x + c)^4 + (15*a^2*cosh(d*x + c)^2 + a^2 + 2*b^2)*sinh(d*x + c)^4 + 4*(5*a^2*cosh(d*x + c)^3 + (a^2 + 2*b^2)*cosh( d*x + c))*sinh(d*x + c)^3 - (a^2 + 2*b^2)*cosh(d*x + c)^2 + (15*a^2*cosh(d *x + c)^4 + 6*(a^2 + 2*b^2)*cosh(d*x + c)^2 - a^2 - 2*b^2)*sinh(d*x + c)^2 - a^2 + 2*((4*a*b + b^2)*cosh(d*x + c)^5 + 5*(4*a*b + b^2)*cosh(d*x + c)* sinh(d*x + c)^4 + (4*a*b + b^2)*sinh(d*x + c)^5 + 2*(4*a*b + b^2)*cosh(d*x + c)^3 + 2*(5*(4*a*b + b^2)*cosh(d*x + c)^2 + 4*a*b + b^2)*sinh(d*x + c)^ 3 + 2*(5*(4*a*b + b^2)*cosh(d*x + c)^3 + 3*(4*a*b + b^2)*cosh(d*x + c))*si nh(d*x + c)^2 + (4*a*b + b^2)*cosh(d*x + c) + (5*(4*a*b + b^2)*cosh(d*x + c)^4 + 6*(4*a*b + b^2)*cosh(d*x + c)^2 + 4*a*b + b^2)*sinh(d*x + c))*arcta n(cosh(d*x + c) + sinh(d*x + c)) + 2*(3*a^2*cosh(d*x + c)^5 + 2*(a^2 + 2*b ^2)*cosh(d*x + c)^3 - (a^2 + 2*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*c osh(d*x + c)^3 + 2*(5*d*cosh(d*x + c)^2 + d)*sinh(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)*sinh(d*x + c))
\[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \cosh {\left (c + d x \right )}\, dx \]
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.80 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-b^{2} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - \frac {4 \, a b \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {a^{2} \sinh \left (d x + c\right )}{d} \]
-b^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))) - 4*a*b*arctan(e^(-d*x - c))/d + a ^2*sinh(d*x + c)/d
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (52) = 104\).
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.00 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {2 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (4 \, a b + b^{2}\right )} + \frac {4 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{4 \, d} \]
1/4*(2*a^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)))*(4*a*b + b^2) + 4*b^2*(e^(d*x + c) - e^(-d*x - c)) /((e^(d*x + c) - e^(-d*x - c))^2 + 4))/d
Time = 2.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 3.07 \[ \int \cosh (c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^2\,\sqrt {d^2}+4\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {16\,a^2\,b^2+8\,a\,b^3+b^4}}\right )\,\sqrt {16\,a^2\,b^2+8\,a\,b^3+b^4}}{\sqrt {d^2}}+\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{2\,d}-\frac {a^2\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {b^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]
(atan((exp(d*x)*exp(c)*(b^2*(d^2)^(1/2) + 4*a*b*(d^2)^(1/2)))/(d*(8*a*b^3 + b^4 + 16*a^2*b^2)^(1/2)))*(8*a*b^3 + b^4 + 16*a^2*b^2)^(1/2))/(d^2)^(1/2 ) + (a^2*exp(c + d*x))/(2*d) - (a^2*exp(- c - d*x))/(2*d) + (b^2*exp(c + d *x))/(d*(exp(2*c + 2*d*x) + 1)) - (2*b^2*exp(c + d*x))/(d*(2*exp(2*c + 2*d *x) + exp(4*c + 4*d*x) + 1))