Integrand size = 23, antiderivative size = 52 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}+\frac {\sinh (c+d x)}{b d} \]
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}+\frac {\sinh (c+d x)}{b}}{d} \]
Time = 0.24 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3669, 299, 218}
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.
| \(\displaystyle \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i c+i d x)^3}{a-b \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \frac {\sinh ^2(c+d x)+1}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{b}-\frac {(a-b) \int \frac {1}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{b}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{b}-\frac {(a-b) \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}}}{d}\) |
3.4.20.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 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]
Time = 5.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {\frac {\sinh \left (d x +c \right )}{b}+\frac {\left (-a +b \right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}}{d}\) | \(44\) |
| default | \(\frac {\frac {\sinh \left (d x +c \right )}{b}+\frac {\left (-a +b \right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{b \sqrt {a b}}}{d}\) | \(44\) |
| risch | \(\frac {{\mathrm e}^{d x +c}}{2 b d}-\frac {{\mathrm e}^{-d x -c}}{2 b d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{2 \sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}\) | \(193\) |
Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (44) = 88\).
Time = 0.29 (sec) , antiderivative size = 659, normalized size of antiderivative = 12.67 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\left [\frac {a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + \sqrt {-a b} {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 1\right )} \sinh \left (d x + c\right ) - \cosh \left (d x + c\right )\right )} \sqrt {-a b} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right ) - a b}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )\right )}}, \frac {a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} - 2 \, \sqrt {a b} {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\frac {\sqrt {a b} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, a}\right ) - 2 \, \sqrt {a b} {\left ({\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )\right )} \arctan \left (\frac {{\left (b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - b\right )} \sinh \left (d x + c\right )\right )} \sqrt {a b}}{2 \, a b}\right ) - a b}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )\right )}}\right ] \]
[1/2*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d *x + c)^2 + sqrt(-a*b)*((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c))*log ((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^ 4 - 2*(2*a + b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a - b)*sinh(d *x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a*b) + b)/(b *cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b )) - a*b)/(a*b^2*d*cosh(d*x + c) + a*b^2*d*sinh(d*x + c)), 1/2*(a*b*cosh(d *x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 - 2*sq rt(a*b)*((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c))*arctan(1/2*sqrt(a* b)*(cosh(d*x + c) + sinh(d*x + c))/a) - 2*sqrt(a*b)*((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c))*arctan(1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c )*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - b)*cosh(d*x + c) + (3*b*cos h(d*x + c)^2 + 4*a - b)*sinh(d*x + c))*sqrt(a*b)/(a*b)) - a*b)/(a*b^2*d*co sh(d*x + c) + a*b^2*d*sinh(d*x + c))]
Timed out. \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)/(b*d) - 1/8*integrate(16*((a*e^(3*c ) - b*e^(3*c))*e^(3*d*x) + (a*e^c - b*e^c)*e^(d*x))/(b^2*e^(4*d*x + 4*c) + b^2 + 2*(2*a*b*e^(2*c) - b^2*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 1.93 (sec) , antiderivative size = 426, normalized size of antiderivative = 8.19 \[ \int \frac {\cosh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {a\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a\,b^3\,d\,\sqrt {a^2-2\,a\,b+b^2}+2\,a^3\,b\,d\,\sqrt {a^2-2\,a\,b+b^2}-4\,a^2\,b^2\,d\,\sqrt {a^2-2\,a\,b+b^2}\right )}{a^2\,b^7\,d^2\,\left (a-b\right )}-\frac {2\,\left (a^3\,\sqrt {a\,b^3\,d^2}-b^3\,\sqrt {a\,b^3\,d^2}+3\,a\,b^2\,\sqrt {a\,b^3\,d^2}-3\,a^2\,b\,\sqrt {a\,b^3\,d^2}\right )}{a^2\,b^5\,d\,\sqrt {{\left (a-b\right )}^2}\,\sqrt {a\,b^3\,d^2}}\right )\,\sqrt {a\,b^3\,d^2}}{4\,a^2-8\,a\,b+4\,b^2}+\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (a^3\,\sqrt {a\,b^3\,d^2}-b^3\,\sqrt {a\,b^3\,d^2}+3\,a\,b^2\,\sqrt {a\,b^3\,d^2}-3\,a^2\,b\,\sqrt {a\,b^3\,d^2}\right )}{a\,b\,d\,\sqrt {{\left (a-b\right )}^2}\,\left (4\,a^2-8\,a\,b+4\,b^2\right )}\right )+2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a-b\right )\,\sqrt {a\,b^3\,d^2}}{2\,a\,b\,d\,\sqrt {{\left (a-b\right )}^2}}\right )\right )\,\sqrt {a^2-2\,a\,b+b^2}}{2\,\sqrt {a\,b^3\,d^2}}-\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d} \]
exp(c + d*x)/(2*b*d) - ((2*atan((a*b^4*exp(d*x)*exp(c)*((4*(2*a*b^3*d*(a^2 - 2*a*b + b^2)^(1/2) + 2*a^3*b*d*(a^2 - 2*a*b + b^2)^(1/2) - 4*a^2*b^2*d* (a^2 - 2*a*b + b^2)^(1/2)))/(a^2*b^7*d^2*(a - b)) - (2*(a^3*(a*b^3*d^2)^(1 /2) - b^3*(a*b^3*d^2)^(1/2) + 3*a*b^2*(a*b^3*d^2)^(1/2) - 3*a^2*b*(a*b^3*d ^2)^(1/2)))/(a^2*b^5*d*((a - b)^2)^(1/2)*(a*b^3*d^2)^(1/2)))*(a*b^3*d^2)^( 1/2))/(4*a^2 - 8*a*b + 4*b^2) + (2*exp(3*c)*exp(3*d*x)*(a^3*(a*b^3*d^2)^(1 /2) - b^3*(a*b^3*d^2)^(1/2) + 3*a*b^2*(a*b^3*d^2)^(1/2) - 3*a^2*b*(a*b^3*d ^2)^(1/2)))/(a*b*d*((a - b)^2)^(1/2)*(4*a^2 - 8*a*b + 4*b^2))) + 2*atan((e xp(d*x)*exp(c)*(a - b)*(a*b^3*d^2)^(1/2))/(2*a*b*d*((a - b)^2)^(1/2))))*(a ^2 - 2*a*b + b^2)^(1/2))/(2*(a*b^3*d^2)^(1/2)) - exp(- c - d*x)/(2*b*d)