Integrand size = 23, antiderivative size = 49 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {b^2 \arctan (\sinh (c+d x))}{d}+\frac {a (a+2 b) \sinh (c+d x)}{d}+\frac {a^2 \sinh ^3(c+d x)}{3 d} \] Output:
b^2*arctan(sinh(d*x+c))/d+a*(a+2*b)*sinh(d*x+c)/d+1/3*a^2*sinh(d*x+c)^3/d
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.49 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {b^2 \cot ^{-1}(\sinh (c+d x))}{d}+\frac {2 a b \cosh (d x) \sinh (c)}{d}+\frac {2 a b \cosh (c) \sinh (d x)}{d}+\frac {a^2 \sinh (c+d x)}{d}+\frac {a^2 \sinh ^3(c+d x)}{3 d} \] Input:
Integrate[Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]
Output:
-((b^2*ArcCot[Sinh[c + d*x]])/d) + (2*a*b*Cosh[d*x]*Sinh[c])/d + (2*a*b*Co sh[c]*Sinh[d*x])/d + (a^2*Sinh[c + d*x])/d + (a^2*Sinh[c + d*x]^3)/(3*d)
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, 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.
\(\displaystyle \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sec (i c+i d x)^2\right )^2}{\sec (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 4635 |
\(\displaystyle \frac {\int \frac {\left (a \sinh ^2(c+d x)+a+b\right )^2}{\sinh ^2(c+d x)+1}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \frac {\int \left (\frac {b^2}{\sinh ^2(c+d x)+1}+a^2 \sinh ^2(c+d x)+a (a+2 b)\right )d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{3} a^2 \sinh ^3(c+d x)+a (a+2 b) \sinh (c+d x)+b^2 \arctan (\sinh (c+d x))}{d}\) |
Input:
Int[Cosh[c + d*x]^3*(a + b*Sech[c + d*x]^2)^2,x]
Output:
(b^2*ArcTan[Sinh[c + d*x]] + a*(a + 2*b)*Sinh[c + d*x] + (a^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[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 = 0.71 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.02
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+2 a b \sinh \left (d x +c \right )+2 b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{d}\) | \(50\) |
default | \(\frac {a^{2} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+2 a b \sinh \left (d x +c \right )+2 b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{d}\) | \(50\) |
parallelrisch | \(\frac {-12 i b^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+12 i b^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+a^{2} \sinh \left (3 d x +3 c \right )+9 \left (a +\frac {8 b}{3}\right ) a \sinh \left (d x +c \right )}{12 d}\) | \(72\) |
risch | \(\frac {a^{2} {\mathrm e}^{3 d x +3 c}}{24 d}+\frac {3 a^{2} {\mathrm e}^{d x +c}}{8 d}+\frac {a \,{\mathrm e}^{d x +c} b}{d}-\frac {3 a^{2} {\mathrm e}^{-d x -c}}{8 d}-\frac {a \,{\mathrm e}^{-d x -c} b}{d}-\frac {a^{2} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}-\frac {i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}\) | \(133\) |
Input:
int(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(2/3+1/3*cosh(d*x+c)^2)*sinh(d*x+c)+2*a*b*sinh(d*x+c)+2*b^2*arcta n(exp(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (47) = 94\).
Time = 0.35 (sec) , antiderivative size = 414, normalized size of antiderivative = 8.45 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {a^{2} \cosh \left (d x + c\right )^{6} + 6 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + a^{2} \sinh \left (d x + c\right )^{6} + 3 \, {\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{2} + 3 \, a^{2} + 8 \, a b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \, {\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, a^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{2} - 3 \, a^{2} - 8 \, a b\right )} \sinh \left (d x + c\right )^{2} - a^{2} + 48 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + 3 \, b^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{3}\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + 6 \, {\left (a^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )^{3} - {\left (3 \, a^{2} + 8 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/24*(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 + 3*(3*a^2 + 8*a*b)*cosh(d*x + c)^4 + 3*(5*a^2*cosh(d*x + c)^2 + 3*a^2 + 8*a*b)*sinh(d*x + c)^4 + 4*(5*a^2*cosh(d*x + c)^3 + 3*(3*a^2 + 8*a*b)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(3*a^2 + 8*a*b)*cosh(d*x + c)^2 + 3*(5*a^2*cosh(d*x + c)^4 + 6*(3*a^2 + 8*a*b)*cosh(d*x + c)^2 - 3*a^2 - 8 *a*b)*sinh(d*x + c)^2 - a^2 + 48*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c )^2*sinh(d*x + c) + 3*b^2*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c )^3)*arctan(cosh(d*x + c) + sinh(d*x + c)) + 6*(a^2*cosh(d*x + c)^5 + 2*(3 *a^2 + 8*a*b)*cosh(d*x + c)^3 - (3*a^2 + 8*a*b)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(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)
\[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \cosh ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(cosh(d*x+c)**3*(a+b*sech(d*x+c)**2)**2,x)
Output:
Integral((a + b*sech(c + d*x)**2)**2*cosh(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (47) = 94\).
Time = 0.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.14 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} - \frac {2 \, b^{2} \arctan \left (e^{\left (-d x - c\right )}\right )}{d} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/24*a^2*(e^(3*d*x + 3*c)/d + 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d - e^(-3*d *x - 3*c)/d) + a*b*(e^(d*x + c)/d - e^(-d*x - c)/d) - 2*b^2*arctan(e^(-d*x - c))/d
Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.92 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {48 \, b^{2} \arctan \left (e^{\left (d x + c\right )}\right ) + a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 9 \, a^{2} e^{\left (d x + c\right )} + 24 \, a b e^{\left (d x + c\right )} - {\left (9 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \] Input:
integrate(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/24*(48*b^2*arctan(e^(d*x + c)) + a^2*e^(3*d*x + 3*c) + 9*a^2*e^(d*x + c) + 24*a*b*e^(d*x + c) - (9*a^2*e^(2*d*x + 2*c) + 24*a*b*e^(2*d*x + 2*c) + a^2)*e^(-3*d*x - 3*c))/d
Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.33 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {2\,\mathrm {atan}\left (\frac {b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^4}}\right )\,\sqrt {b^4}}{\sqrt {d^2}}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a^2+8\,b\,a\right )}{8\,d}-\frac {a^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {a^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}+\frac {a\,{\mathrm {e}}^{c+d\,x}\,\left (3\,a+8\,b\right )}{8\,d} \] Input:
int(cosh(c + d*x)^3*(a + b/cosh(c + d*x)^2)^2,x)
Output:
(2*atan((b^2*exp(d*x)*exp(c)*(d^2)^(1/2))/(d*(b^4)^(1/2)))*(b^4)^(1/2))/(d ^2)^(1/2) - (exp(- c - d*x)*(8*a*b + 3*a^2))/(8*d) - (a^2*exp(- 3*c - 3*d* x))/(24*d) + (a^2*exp(3*c + 3*d*x))/(24*d) + (a*exp(c + d*x)*(3*a + 8*b))/ (8*d)
Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.41 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {48 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}+e^{6 d x +6 c} a^{2}+9 e^{4 d x +4 c} a^{2}+24 e^{4 d x +4 c} a b -9 e^{2 d x +2 c} a^{2}-24 e^{2 d x +2 c} a b -a^{2}}{24 e^{3 d x +3 c} d} \] Input:
int(cosh(d*x+c)^3*(a+b*sech(d*x+c)^2)^2,x)
Output:
(48*e**(3*c + 3*d*x)*atan(e**(c + d*x))*b**2 + e**(6*c + 6*d*x)*a**2 + 9*e **(4*c + 4*d*x)*a**2 + 24*e**(4*c + 4*d*x)*a*b - 9*e**(2*c + 2*d*x)*a**2 - 24*e**(2*c + 2*d*x)*a*b - a**2)/(24*e**(3*c + 3*d*x)*d)