Integrand size = 21, antiderivative size = 66 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} d}+\frac {\sinh (c+d x)}{2 a d \left (a+b \sinh ^2(c+d x)\right )} \] Output:
1/2*arctan(b^(1/2)*sinh(d*x+c)/a^(1/2))/a^(3/2)/b^(1/2)/d+1/2*sinh(d*x+c)/ a/d/(a+b*sinh(d*x+c)^2)
Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\frac {\arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sinh (c+d x)}{2 a \left (a+b \sinh ^2(c+d x)\right )}}{d} \] Input:
Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(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)))/d
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3669, 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.
\(\displaystyle \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i c+i d x)}{\left (a-b \sin (i c+i d x)^2\right )^2}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {1}{\left (b \sinh ^2(c+d x)+a\right )^2}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle \frac {\frac {\int \frac {1}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{2 a}+\frac {\sinh (c+d x)}{2 a \left (a+b \sinh ^2(c+d x)\right )}}{d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\sinh (c+d x)}{2 a \left (a+b \sinh ^2(c+d x)\right )}}{d}\) |
Input:
Int[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]
Output:
(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)))/d
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 = 59.60 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {\sinh \left (d x +c \right )}{2 a \left (a +b \sinh \left (d x +c \right )^{2}\right )}+\frac {\arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{d}\) | \(55\) |
default | \(\frac {\frac {\sinh \left (d x +c \right )}{2 a \left (a +b \sinh \left (d x +c \right )^{2}\right )}+\frac {\arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{d}\) | \(55\) |
risch | \(\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a d \left (b \,{\mathrm e}^{4 d x +4 c}+4 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +b \right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{4 \sqrt {-a b}\, d a}\) | \(147\) |
Input:
int(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/2*sinh(d*x+c)/a/(a+b*sinh(d*x+c)^2)+1/2/a/(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. 630 vs. \(2 (54) = 108\).
Time = 0.10 (sec) , antiderivative size = 1324, normalized size of antiderivative = 20.06 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")
Output:
[1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a*b *sinh(d*x + c)^3 - 4*a*b*cosh(d*x + c) - (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)*sqrt(-a*b)*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)*co sh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*c osh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 4*(3*a*b*c osh(d*x + c)^2 - a*b)*sinh(d*x + c))/(a^2*b^2*d*cosh(d*x + c)^4 + 4*a^2*b^ 2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*b^2*d*sinh(d*x + c)^4 + a^2*b^2*d + 2*(2*a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*a^2*b^2*d*cosh(d*x + c)^2 + (2*a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 + 4*(a^2*b^2*d*cosh(d*x + c)^3 + (2*a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), 1/2*(2*a*b*cosh(d*x + c)^3 + 6*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*b*sinh(d*x + c)^3 - 2*a *b*cosh(d*x + c) - (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)...
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (54) = 108\).
Time = 6.13 (sec) , antiderivative size = 377, normalized size of antiderivative = 5.71 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\begin {cases} \frac {\tilde {\infty } x \cosh {\left (c \right )}}{\sinh ^{4}{\left (c \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a^{2} d} & \text {for}\: b = 0 \\- \frac {1}{3 b^{2} d \sinh ^{3}{\left (c + d x \right )}} & \text {for}\: a = 0 \\\frac {x \cosh {\left (c \right )}}{\left (a + b \sinh ^{2}{\left (c \right )}\right )^{2}} & \text {for}\: d = 0 \\\frac {a \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} - \frac {a \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} + \frac {2 b \sqrt {- \frac {a}{b}} \sinh {\left (c + d x \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} + \frac {b \log {\left (- \sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} - \frac {b \log {\left (\sqrt {- \frac {a}{b}} + \sinh {\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{4 a^{2} b d \sqrt {- \frac {a}{b}} + 4 a b^{2} d \sqrt {- \frac {a}{b}} \sinh ^{2}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)**2)**2,x)
Output:
Piecewise((zoo*x*cosh(c)/sinh(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (sin h(c + d*x)/(a**2*d), Eq(b, 0)), (-1/(3*b**2*d*sinh(c + d*x)**3), Eq(a, 0)) , (x*cosh(c)/(a + b*sinh(c)**2)**2, Eq(d, 0)), (a*log(-sqrt(-a/b) + sinh(c + d*x))/(4*a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2) - a*log(sqrt(-a/b) + sinh(c + d*x))/(4*a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sq rt(-a/b)*sinh(c + d*x)**2) + 2*b*sqrt(-a/b)*sinh(c + d*x)/(4*a**2*b*d*sqrt (-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2) + b*log(-sqrt(-a/b) + sin h(c + d*x))*sinh(c + d*x)**2/(4*a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b )*sinh(c + d*x)**2) - b*log(sqrt(-a/b) + sinh(c + d*x))*sinh(c + d*x)**2/( 4*a**2*b*d*sqrt(-a/b) + 4*a*b**2*d*sqrt(-a/b)*sinh(c + d*x)**2), True))
\[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \] Input:
integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")
Output:
(e^(3*d*x + 3*c) - e^(d*x + c))/(a*b*d*e^(4*d*x + 4*c) + a*b*d + 2*(2*a^2* d*e^(2*c) - a*b*d*e^(2*c))*e^(2*d*x)) + 1/2*integrate(2*(e^(3*d*x + 3*c) + e^(d*x + c))/(a*b*e^(4*d*x + 4*c) + a*b + 2*(2*a^2*e^(2*c) - a*b*e^(2*c)) *e^(2*d*x)), x)
Exception generated. \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 1.76 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\mathrm {sinh}\left (c+d\,x\right )}{2\,a\,\left (b\,d\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\,d\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {sinh}\left (c+d\,x\right )}{\sqrt {a}}\right )}{2\,a^{3/2}\,\sqrt {b}\,d} \] Input:
int(cosh(c + d*x)/(a + b*sinh(c + d*x)^2)^2,x)
Output:
sinh(c + d*x)/(2*a*(a*d + b*d*sinh(c + d*x)^2)) + atan((b^(1/2)*sinh(c + d *x))/a^(1/2))/(2*a^(3/2)*b^(1/2)*d)
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.35 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sinh \left (d x +c \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) \sinh \left (d x +c \right )^{2} b +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {\sinh \left (d x +c \right ) b}{\sqrt {b}\, \sqrt {a}}\right ) a +\sinh \left (d x +c \right ) a b}{2 a^{2} b d \left (\sinh \left (d x +c \right )^{2} b +a \right )} \] Input:
int(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x)
Output:
(sqrt(b)*sqrt(a)*atan((sinh(c + d*x)*b)/(sqrt(b)*sqrt(a)))*sinh(c + d*x)** 2*b + sqrt(b)*sqrt(a)*atan((sinh(c + d*x)*b)/(sqrt(b)*sqrt(a)))*a + sinh(c + d*x)*a*b)/(2*a**2*b*d*(sinh(c + d*x)**2*b + a))