Integrand size = 23, antiderivative size = 49 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx=\frac {2 \log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2 d}+\frac {2 a}{b^2 d \left (a+b \sqrt {\sinh (c+d x)}\right )} \]
Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx=\frac {2 \left (\log \left (a+b \sqrt {\sinh (c+d x)}\right )+\frac {a}{a+b \sqrt {\sinh (c+d x)}}\right )}{b^2 d} \]
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 3702, 774, 49, 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.
\(\displaystyle \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (i c+i d x)}{\left (a+b \sqrt {-i \sin (i c+i d x)}\right )^2}dx\) |
\(\Big \downarrow \) 3702 |
\(\displaystyle \frac {\int \frac {1}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 774 |
\(\displaystyle \frac {2 \int \frac {\sqrt {\sinh (c+d x)}}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2}d\sqrt {\sinh (c+d x)}}{d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {2 \int \left (\frac {1}{b \left (a+b \sqrt {\sinh (c+d x)}\right )}-\frac {a}{b \left (a+b \sqrt {\sinh (c+d x)}\right )^2}\right )d\sqrt {\sinh (c+d x)}}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (\frac {a}{b^2 \left (a+b \sqrt {\sinh (c+d x)}\right )}+\frac {\log \left (a+b \sqrt {\sinh (c+d x)}\right )}{b^2}\right )}{d}\) |
3.5.17.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x _)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Si mp[ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0] || IGtQ[p, 0] || IntegersQ[m, p])
Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs. \(2(45)=90\).
Time = 0.14 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.59
method | result | size |
derivativedivides | \(\frac {\frac {2 a^{2}}{\left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right ) b^{2}}+\frac {\ln \left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right )}{b^{2}}-\frac {a}{b^{2} \left (-b \sqrt {\sinh \left (d x +c \right )}+a \right )}-\frac {\ln \left (-b \sqrt {\sinh \left (d x +c \right )}+a \right )}{b^{2}}+\frac {a}{b^{2} \left (a +b \sqrt {\sinh \left (d x +c \right )}\right )}+\frac {\ln \left (a +b \sqrt {\sinh \left (d x +c \right )}\right )}{b^{2}}}{d}\) | \(127\) |
default | \(\frac {\frac {2 a^{2}}{\left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right ) b^{2}}+\frac {\ln \left (-b^{2} \sinh \left (d x +c \right )+a^{2}\right )}{b^{2}}-\frac {a}{b^{2} \left (-b \sqrt {\sinh \left (d x +c \right )}+a \right )}-\frac {\ln \left (-b \sqrt {\sinh \left (d x +c \right )}+a \right )}{b^{2}}+\frac {a}{b^{2} \left (a +b \sqrt {\sinh \left (d x +c \right )}\right )}+\frac {\ln \left (a +b \sqrt {\sinh \left (d x +c \right )}\right )}{b^{2}}}{d}\) | \(127\) |
1/d*(2*a^2/(-b^2*sinh(d*x+c)+a^2)/b^2+1/b^2*ln(-b^2*sinh(d*x+c)+a^2)-a/b^2 /(-b*sinh(d*x+c)^(1/2)+a)-1/b^2*ln(-b*sinh(d*x+c)^(1/2)+a)+a/b^2/(a+b*sinh (d*x+c)^(1/2))+1/b^2*ln(a+b*sinh(d*x+c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (45) = 90\).
Time = 0.29 (sec) , antiderivative size = 564, normalized size of antiderivative = 11.51 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx=\frac {b^{2} d x + b^{2} c - {\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )^{2} - {\left (b^{2} d x + b^{2} c\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} d x + a^{2} c - 2 \, a^{2}\right )} \cosh \left (d x + c\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a b \cosh \left (d x + c\right ) + a b \sinh \left (d x + c\right )\right )} \sqrt {\sinh \left (d x + c\right )}}{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )}\right ) + {\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (b^{2} \sinh \left (d x + c\right ) - a^{2}\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (a^{2} d x + a^{2} c - 2 \, a^{2} - {\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a b \cosh \left (d x + c\right ) + a b \sinh \left (d x + c\right )\right )} \sqrt {\sinh \left (d x + c\right )}}{b^{4} d \cosh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{2} - 2 \, a^{2} b^{2} d \cosh \left (d x + c\right ) - b^{4} d + 2 \, {\left (b^{4} d \cosh \left (d x + c\right ) - a^{2} b^{2} d\right )} \sinh \left (d x + c\right )} \]
(b^2*d*x + b^2*c - (b^2*d*x + b^2*c)*cosh(d*x + c)^2 - (b^2*d*x + b^2*c)*s inh(d*x + c)^2 + 2*(a^2*d*x + a^2*c - 2*a^2)*cosh(d*x + c) + (b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 - 2*a^2*cosh(d*x + c) - b^2 + 2*(b^2*cosh(d* x + c) - a^2)*sinh(d*x + c))*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^ 2 + 2*a^2*cosh(d*x + c) - b^2 + 2*(b^2*cosh(d*x + c) + a^2)*sinh(d*x + c) + 4*(a*b*cosh(d*x + c) + a*b*sinh(d*x + c))*sqrt(sinh(d*x + c)))/(b^2*cosh (d*x + c)^2 + b^2*sinh(d*x + c)^2 - 2*a^2*cosh(d*x + c) - b^2 + 2*(b^2*cos h(d*x + c) - a^2)*sinh(d*x + c))) + (b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 - 2*a^2*cosh(d*x + c) - b^2 + 2*(b^2*cosh(d*x + c) - a^2)*sinh(d*x + c))*log(2*(b^2*sinh(d*x + c) - a^2)/(cosh(d*x + c) - sinh(d*x + c))) + 2*( a^2*d*x + a^2*c - 2*a^2 - (b^2*d*x + b^2*c)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*b*cosh(d*x + c) + a*b*sinh(d*x + c))*sqrt(sinh(d*x + c)))/(b^4*d*cos h(d*x + c)^2 + b^4*d*sinh(d*x + c)^2 - 2*a^2*b^2*d*cosh(d*x + c) - b^4*d + 2*(b^4*d*cosh(d*x + c) - a^2*b^2*d)*sinh(d*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (42) = 84\).
Time = 3.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.08 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx=\begin {cases} \frac {x \cosh {\left (c \right )}}{a^{2}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sinh {\left (c + d x \right )}}{a^{2} d} & \text {for}\: b = 0 \\\frac {x \cosh {\left (c \right )}}{\left (a + b \sqrt {\sinh {\left (c \right )}}\right )^{2}} & \text {for}\: d = 0 \\\frac {2 a \log {\left (\frac {a}{b} + \sqrt {\sinh {\left (c + d x \right )}} \right )}}{a b^{2} d + b^{3} d \sqrt {\sinh {\left (c + d x \right )}}} + \frac {2 a}{a b^{2} d + b^{3} d \sqrt {\sinh {\left (c + d x \right )}}} + \frac {2 b \log {\left (\frac {a}{b} + \sqrt {\sinh {\left (c + d x \right )}} \right )} \sqrt {\sinh {\left (c + d x \right )}}}{a b^{2} d + b^{3} d \sqrt {\sinh {\left (c + d x \right )}}} & \text {otherwise} \end {cases} \]
Piecewise((x*cosh(c)/a**2, Eq(b, 0) & Eq(d, 0)), (sinh(c + d*x)/(a**2*d), Eq(b, 0)), (x*cosh(c)/(a + b*sqrt(sinh(c)))**2, Eq(d, 0)), (2*a*log(a/b + sqrt(sinh(c + d*x)))/(a*b**2*d + b**3*d*sqrt(sinh(c + d*x))) + 2*a/(a*b**2 *d + b**3*d*sqrt(sinh(c + d*x))) + 2*b*log(a/b + sqrt(sinh(c + d*x)))*sqrt (sinh(c + d*x))/(a*b**2*d + b**3*d*sqrt(sinh(c + d*x))), True))
\[ \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \sqrt {\sinh \left (d x + c\right )} + a\right )}^{2}} \,d x } \]
\[ \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )}{{\left (b \sqrt {\sinh \left (d x + c\right )} + a\right )}^{2}} \,d x } \]
Time = 2.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh (c+d x)}{\left (a+b \sqrt {\sinh (c+d x)}\right )^2} \, dx=\frac {2\,a}{b^2\,\left (a\,d+b\,d\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\right )}+\frac {2\,\ln \left (a+b\,\sqrt {\mathrm {sinh}\left (c+d\,x\right )}\right )}{b^2\,d} \]