Integrand size = 21, antiderivative size = 96 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\frac {3 \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b} d}+\frac {\sinh (c+d x)}{4 a d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {3 \sinh (c+d x)}{8 a^2 d \left (a+b \sinh ^2(c+d x)\right )} \]
1/4*sinh(d*x+c)/a/d/(a+b*sinh(d*x+c)^2)^2+3/8*sinh(d*x+c)/a^2/d/(a+b*sinh( d*x+c)^2)+3/8*arctan(sinh(d*x+c)*b^(1/2)/a^(1/2))/a^(5/2)/d/b^(1/2)
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.82 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\frac {\frac {3 \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}}+\frac {\sinh (c+d x) \left (5 a+3 b \sinh ^2(c+d x)\right )}{8 a^2 \left (a+b \sinh ^2(c+d x)\right )^2}}{d} \]
((3*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]) + (Sinh[c + d*x]*(5*a + 3*b*Sinh[c + d*x]^2))/(8*a^2*(a + b*Sinh[c + d*x]^2)^2))/d
Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3669, 215, 215, 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.
| \(\displaystyle \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i c+i d x)}{\left (a-b \sin (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \frac {1}{\left (b \sinh ^2(c+d x)+a\right )^3}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\left (b \sinh ^2(c+d x)+a\right )^2}d\sinh (c+d x)}{4 a}+\frac {\sinh (c+d x)}{4 a \left (a+b \sinh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {1}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{2 a}+\frac {\sinh (c+d x)}{2 a \left (a+b \sinh ^2(c+d x)\right )}\right )}{4 a}+\frac {\sinh (c+d x)}{4 a \left (a+b \sinh ^2(c+d x)\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sinh (c+d x)}{2 a \left (a+b \sinh ^2(c+d x)\right )}\right )}{4 a}+\frac {\sinh (c+d x)}{4 a \left (a+b \sinh ^2(c+d x)\right )^2}}{d}\) |
(Sinh[c + d*x]/(4*a*(a + b*Sinh[c + d*x]^2)^2) + (3*(ArcTan[(Sqrt[b]*Sinh[ c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sinh[c + d*x]/(2*a*(a + b*Sinh[c + d*x]^2))))/(4*a))/d
3.4.44.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[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 = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89
\[\frac {\sinh \left (d x +c \right )}{4 a d \left (a +b \sinh \left (d x +c \right )^{2}\right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{8 a^{2} d \left (a +b \sinh \left (d x +c \right )^{2}\right )}+\frac {3 \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{8 d \,a^{2} \sqrt {a b}}\]
1/4*sinh(d*x+c)/a/d/(a+b*sinh(d*x+c)^2)^2+3/8*sinh(d*x+c)/a^2/d/(a+b*sinh( d*x+c)^2)+3/8/d/a^2/(a*b)^(1/2)*arctan(b*sinh(d*x+c)/(a*b)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 2111 vs. \(2 (82) = 164\).
Time = 0.30 (sec) , antiderivative size = 3934, normalized size of antiderivative = 40.98 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
[1/16*(12*a*b^2*cosh(d*x + c)^7 + 84*a*b^2*cosh(d*x + c)*sinh(d*x + c)^6 + 12*a*b^2*sinh(d*x + c)^7 + 4*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^5 + 4*(63 *a*b^2*cosh(d*x + c)^2 + 20*a^2*b - 9*a*b^2)*sinh(d*x + c)^5 + 20*(21*a*b^ 2*cosh(d*x + c)^3 + (20*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - 12*a*b^2*cosh(d*x + c) - 4*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^3 + 4*(105*a *b^2*cosh(d*x + c)^4 - 20*a^2*b + 9*a*b^2 + 10*(20*a^2*b - 9*a*b^2)*cosh(d *x + c)^2)*sinh(d*x + c)^3 + 4*(63*a*b^2*cosh(d*x + c)^5 + 10*(20*a^2*b - 9*a*b^2)*cosh(d*x + c)^3 - 3*(20*a^2*b - 9*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 3*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + b^ 2*sinh(d*x + c)^8 + 4*(2*a*b - b^2)*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^6 + 8*(7*b^2*cosh(d*x + c)^3 + 3*(2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^4 + 2*(35*b^2*cosh(d*x + c)^4 + 30*(2*a*b - b^2)*cosh(d*x + c)^2 + 8 *a^2 - 8*a*b + 3*b^2)*sinh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^5 + 10*(2*a *b - b^2)*cosh(d*x + c)^3 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c))*sinh(d* x + c)^3 + 4*(2*a*b - b^2)*cosh(d*x + c)^2 + 4*(7*b^2*cosh(d*x + c)^6 + 15 *(2*a*b - b^2)*cosh(d*x + c)^4 + 3*(8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 + 3*(2*a*b - b^2)*cosh(d*x + c)^5 + (8*a^2 - 8*a*b + 3*b^2)*cosh(d*x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b)*log((b*cosh(d*x + c)^4 ...
Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (85) = 170\).
Time = 24.95 (sec) , antiderivative size = 835, normalized size of antiderivative = 8.70 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\begin {cases} \frac {\tilde {\infty } x \cosh {\left (c \right )}}{\sinh ^{6}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a^{3} d} & \text {for}\: b = 0 \\- \frac {1}{5 b^{3} d \sinh ^{5}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {x \cosh {\left (c \right )}}{\left (a + b \sinh ^{2}{\left (c \right )}\right )^{3}} & \text {for}\: d = 0 \\\frac {3 a^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} - \frac {3 a^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {10 a b \sqrt {- \frac {a}{b}} \sinh {\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {6 a b \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} - \frac {6 a b \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {6 b^{2} \sqrt {- \frac {a}{b}} \sinh ^{3}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} + \frac {3 b^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{4}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} - \frac {3 b^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{4}{\left (c + d x \right )}}{16 a^{4} b d \sqrt {- \frac {a}{b}} + 32 a^{3} b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )} + 16 a^{2} b^{3} d \sqrt {- \frac {a}{b}} \sinh ^{4}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*x*cosh(c)/sinh(c)**6, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (sin h(c + d*x)/(a**3*d), Eq(b, 0)), (-1/(5*b**3*d*sinh(c + d*x)**5), Eq(a, 0)) , (x*cosh(c)/(a + b*sinh(c)**2)**3, Eq(d, 0)), (3*a**2*log(-sqrt(-a/b) + s inh(c + d*x))/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) - 3*a**2*log(sqrt(- a/b) + sinh(c + d*x))/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)* sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) + 10*a*b*sq rt(-a/b)*sinh(c + d*x)/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b) *sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) + 6*a*b*lo g(-sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**2/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sin h(c + d*x)**4) - 6*a*b*log(sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**2/(1 6*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a* *2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) + 6*b**2*sqrt(-a/b)*sinh(c + d*x)** 3/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 1 6*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) + 3*b**2*log(-sqrt(-a/b) + sinh (c + d*x))*sinh(c + d*x)**4/(16*a**4*b*d*sqrt(-a/b) + 32*a**3*b**2*d*sqrt( -a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-a/b)*sinh(c + d*x)**4) - 3*b **2*log(sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**4/(16*a**4*b*d*sqrt(-a/ b) + 32*a**3*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2 + 16*a**2*b**3*d*sqrt(-...
\[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
1/4*((20*a*e^(5*c) - 9*b*e^(5*c))*e^(5*d*x) - (20*a*e^(3*c) - 9*b*e^(3*c)) *e^(3*d*x) + 3*b*e^(7*d*x + 7*c) - 3*b*e^(d*x + c))/(a^2*b^2*d*e^(8*d*x + 8*c) + a^2*b^2*d + 4*(2*a^3*b*d*e^(6*c) - a^2*b^2*d*e^(6*c))*e^(6*d*x) + 2 *(8*a^4*d*e^(4*c) - 8*a^3*b*d*e^(4*c) + 3*a^2*b^2*d*e^(4*c))*e^(4*d*x) + 4 *(2*a^3*b*d*e^(2*c) - a^2*b^2*d*e^(2*c))*e^(2*d*x)) + 1/2*integrate(3/2*(e ^(3*d*x + 3*c) + e^(d*x + c))/(a^2*b*e^(4*d*x + 4*c) + a^2*b + 2*(2*a^3*e^ (2*c) - a^2*b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Time = 1.86 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.91 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx=\frac {\frac {5\,\mathrm {sinh}\left (c+d\,x\right )}{8\,a}+\frac {3\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3}{8\,a^2}}{d\,a^2+2\,d\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+d\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^4}+\frac {3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {sinh}\left (c+d\,x\right )}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {b}\,d} \]