\(\int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx\) [20]

Optimal result

Integrand size = 31, antiderivative size = 360 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=\frac {2 e^{\frac {5 (a-d)}{3}+\frac {5}{3} (d+b x)}}{b (f+g) \left (f-g-e^{2 (d+b x)} (f+g)\right )}-\frac {5 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1-\frac {2 e^{\frac {1}{3} (d+b x)} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}}{\sqrt {3}}\right )}{\sqrt {3} b \sqrt [6]{f-g} (f+g)^{11/6}}+\frac {5 e^{\frac {5 (a-d)}{3}} \arctan \left (\frac {1+\frac {2 e^{\frac {1}{3} (d+b x)} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}}{\sqrt {3}}\right )}{\sqrt {3} b \sqrt [6]{f-g} (f+g)^{11/6}}-\frac {10 e^{\frac {5 (a-d)}{3}} \text {arctanh}\left (\frac {e^{\frac {1}{3} (d+b x)} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 b \sqrt [6]{f-g} (f+g)^{11/6}}-\frac {5 e^{\frac {5 (a-d)}{3}} \text {arctanh}\left (\frac {e^{\frac {1}{3} (d+b x)} \sqrt [6]{f-g}}{\sqrt [6]{f+g} \left (e^{\frac {2}{3} (d+b x)}+\frac {\sqrt [3]{f-g}}{\sqrt [3]{f+g}}\right )}\right )}{3 b \sqrt [6]{f-g} (f+g)^{11/6}} \] Output:

2*exp(5/3*b*x+5/3*a)/b/(f+g)/(f-g-exp(2*b*x+2*d)*(f+g))-5/3*exp(5/3*a-5/3* 
d)*arctan(1/3*(1-2*exp(1/3*b*x+1/3*d)*(f+g)^(1/6)/(f-g)^(1/6))*3^(1/2))*3^ 
(1/2)/b/(f-g)^(1/6)/(f+g)^(11/6)+5/3*exp(5/3*a-5/3*d)*arctan(1/3*(1+2*exp( 
1/3*b*x+1/3*d)*(f+g)^(1/6)/(f-g)^(1/6))*3^(1/2))*3^(1/2)/b/(f-g)^(1/6)/(f+ 
g)^(11/6)-10/3*exp(5/3*a-5/3*d)*arctanh(exp(1/3*b*x+1/3*d)*(f+g)^(1/6)/(f- 
g)^(1/6))/b/(f-g)^(1/6)/(f+g)^(11/6)-5/3*exp(5/3*a-5/3*d)*arctanh(exp(1/3* 
b*x+1/3*d)*(f-g)^(1/6)/(f+g)^(1/6)/(exp(2/3*b*x+2/3*d)+(f-g)^(1/3)/(f+g)^( 
1/3)))/b/(f-g)^(1/6)/(f+g)^(11/6)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.40 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.22 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=\frac {12 e^{\frac {5}{3} (a+b x)} \left (\frac {1}{f-g-e^{2 (d+b x)} (f+g)}-\frac {\operatorname {Hypergeometric2F1}\left (\frac {5}{6},2,\frac {11}{6},\frac {e^{2 (d+b x)} (f+g)}{f-g}\right )}{f-g}\right )}{b (f+g)} \] Input:

Integrate[E^((5*(a + b*x))/3)/(g*Cosh[d + b*x] + f*Sinh[d + b*x])^2,x]
 

Output:

(12*E^((5*(a + b*x))/3)*((f - g - E^(2*(d + b*x))*(f + g))^(-1) - Hypergeo 
metric2F1[5/6, 2, 11/6, (E^(2*(d + b*x))*(f + g))/(f - g)]/(f - g)))/(b*(f 
 + g))
 

Rubi [A] (warning: unable to verify)

Time = 0.93 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2720, 27, 817, 825, 27, 221, 1142, 25, 27, 1082, 217, 1103}

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.

\(\displaystyle \int \frac {e^{\frac {5}{3} (a+b x)}}{(f \sinh (b x+d)+g \cosh (b x+d))^2} \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {3 \int \frac {4 e^{\frac {5 a}{3}+\frac {10 b x}{3}}}{\left (f-g-e^{2 b x} (f+g)\right )^2}de^{\frac {b x}{3}}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {12 e^{5 a/3} \int \frac {e^{\frac {10 b x}{3}}}{\left (f-g-e^{2 b x} (f+g)\right )^2}de^{\frac {b x}{3}}}{b}\)

\(\Big \downarrow \) 817

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \int \frac {e^{\frac {4 b x}{3}}}{f-g-e^{2 b x} (f+g)}de^{\frac {b x}{3}}}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 825

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (\frac {\int \frac {1}{\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{3 (f+g)^{2/3}}+\frac {\int -\frac {\sqrt [6]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g}}{2 \left (\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}de^{\frac {b x}{3}}}{3 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\int -\frac {\sqrt [6]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g}}{2 \left (\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}\right )}de^{\frac {b x}{3}}}{3 \sqrt [6]{f-g} (f+g)^{2/3}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (\frac {\int \frac {1}{\sqrt [3]{f-g}-e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{3 (f+g)^{2/3}}-\frac {\int \frac {\sqrt [6]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {\int \frac {\sqrt [6]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (-\frac {\int \frac {\sqrt [6]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {\int \frac {\sqrt [6]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\text {arctanh}\left (\frac {e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 \sqrt [6]{f-g} (f+g)^{5/6}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (-\frac {\frac {3}{2} \sqrt [6]{f-g} \int \frac {1}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}+\frac {\int -\frac {\sqrt [6]{f+g} \left (\sqrt [6]{f-g}-2 e^{\frac {b x}{3}} \sqrt [6]{f+g}\right )}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{2 \sqrt [6]{f+g}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{f-g} \int \frac {1}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}-\frac {\int \frac {\sqrt [6]{f+g} \left (\sqrt [6]{f-g}+2 e^{\frac {b x}{3}} \sqrt [6]{f+g}\right )}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{2 \sqrt [6]{f+g}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\text {arctanh}\left (\frac {e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 \sqrt [6]{f-g} (f+g)^{5/6}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (-\frac {\frac {3}{2} \sqrt [6]{f-g} \int \frac {1}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}-\frac {\int \frac {\sqrt [6]{f+g} \left (\sqrt [6]{f-g}-2 e^{\frac {b x}{3}} \sqrt [6]{f+g}\right )}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{2 \sqrt [6]{f+g}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{f-g} \int \frac {1}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}-\frac {\int \frac {\sqrt [6]{f+g} \left (\sqrt [6]{f-g}+2 e^{\frac {b x}{3}} \sqrt [6]{f+g}\right )}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{2 \sqrt [6]{f+g}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\text {arctanh}\left (\frac {e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 \sqrt [6]{f-g} (f+g)^{5/6}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (-\frac {\frac {3}{2} \sqrt [6]{f-g} \int \frac {1}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}-\frac {1}{2} \int \frac {\sqrt [6]{f-g}-2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {\frac {3}{2} \sqrt [6]{f-g} \int \frac {1}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}-\frac {1}{2} \int \frac {\sqrt [6]{f-g}+2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\text {arctanh}\left (\frac {e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 \sqrt [6]{f-g} (f+g)^{5/6}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (-\frac {\frac {3 \int \frac {1}{-3-e^{\frac {2 b x}{3}}}d\left (1-\frac {2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{\sqrt [6]{f+g}}-\frac {1}{2} \int \frac {\sqrt [6]{f-g}-2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {-\frac {3 \int \frac {1}{-3-e^{\frac {2 b x}{3}}}d\left (\frac {2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}+1\right )}{\sqrt [6]{f+g}}-\frac {1}{2} \int \frac {\sqrt [6]{f-g}+2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\text {arctanh}\left (\frac {e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 \sqrt [6]{f-g} (f+g)^{5/6}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (-\frac {-\frac {1}{2} \int \frac {\sqrt [6]{f-g}-2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}}{\sqrt {3}}\right )}{\sqrt [6]{f+g}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}+1}{\sqrt {3}}\right )}{\sqrt [6]{f+g}}-\frac {1}{2} \int \frac {\sqrt [6]{f-g}+2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [3]{f-g}+e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}}de^{\frac {b x}{3}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\text {arctanh}\left (\frac {e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 \sqrt [6]{f-g} (f+g)^{5/6}}\right )}{6 (f+g)}\right )}{b}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {12 e^{5 a/3} \left (\frac {e^{\frac {5 b x}{3}}}{6 (f+g) \left (-e^{2 b x} (f+g)+f-g\right )}-\frac {5 \left (-\frac {\frac {\log \left (-e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}+\sqrt [3]{f-g}\right )}{2 \sqrt [6]{f+g}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}}{\sqrt {3}}\right )}{\sqrt [6]{f+g}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}-\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}+1}{\sqrt {3}}\right )}{\sqrt [6]{f+g}}-\frac {\log \left (e^{\frac {b x}{3}} \sqrt [6]{f+g} \sqrt [6]{f-g}+e^{\frac {2 b x}{3}} \sqrt [3]{f+g}+\sqrt [3]{f-g}\right )}{2 \sqrt [6]{f+g}}}{6 \sqrt [6]{f-g} (f+g)^{2/3}}+\frac {\text {arctanh}\left (\frac {e^{\frac {b x}{3}} \sqrt [6]{f+g}}{\sqrt [6]{f-g}}\right )}{3 \sqrt [6]{f-g} (f+g)^{5/6}}\right )}{6 (f+g)}\right )}{b}\)

Input:

Int[E^((5*(a + b*x))/3)/(g*Cosh[d + b*x] + f*Sinh[d + b*x])^2,x]
 

Output:

(12*E^((5*a)/3)*(E^((5*b*x)/3)/(6*(f + g)*(f - g - E^(2*b*x)*(f + g))) - ( 
5*(ArcTanh[(E^((b*x)/3)*(f + g)^(1/6))/(f - g)^(1/6)]/(3*(f - g)^(1/6)*(f 
+ g)^(5/6)) - (-((Sqrt[3]*ArcTan[(1 - (2*E^((b*x)/3)*(f + g)^(1/6))/(f - g 
)^(1/6))/Sqrt[3]])/(f + g)^(1/6)) + Log[(f - g)^(1/3) - E^((b*x)/3)*(f - g 
)^(1/6)*(f + g)^(1/6) + E^((2*b*x)/3)*(f + g)^(1/3)]/(2*(f + g)^(1/6)))/(6 
*(f - g)^(1/6)*(f + g)^(2/3)) - ((Sqrt[3]*ArcTan[(1 + (2*E^((b*x)/3)*(f + 
g)^(1/6))/(f - g)^(1/6))/Sqrt[3]])/(f + g)^(1/6) - Log[(f - g)^(1/3) + E^( 
(b*x)/3)*(f - g)^(1/6)*(f + g)^(1/6) + E^((2*b*x)/3)*(f + g)^(1/3)]/(2*(f 
+ g)^(1/6)))/(6*(f - g)^(1/6)*(f + g)^(2/3))))/(6*(f + g))))/b
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 817

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( 
n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n 
*((m - n + 1)/(b*n*(p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x 
] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  ! 
ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 825

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[-a/b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r*Cos[2*k 
*m*(Pi/n)] - s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[2*k*(Pi/n)]*x + 
s^2*x^2), x] + Int[(r*Cos[2*k*m*(Pi/n)] + s*Cos[2*k*(m + 1)*(Pi/n)]*x)/(r^2 
 + 2*r*s*Cos[2*k*(Pi/n)]*x + s^2*x^2), x]; 2*(r^(m + 2)/(a*n*s^m))   Int[1/ 
(r^2 - s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], 
x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1 
] && NegQ[a/b]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 92.47 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.86

method	result	size
risch	\(-\frac {2 \,{\mathrm e}^{\frac {5 b x}{3}+\frac {5 a}{3}}}{\left (f \,{\mathrm e}^{2 b x +2 d}+{\mathrm e}^{2 b x +2 d} g -f +g \right ) b \left (f +g \right )}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-15625+\left (729 b^{6} f^{12}+7290 b^{6} f^{11} g +32076 b^{6} f^{10} g^{2}+80190 b^{6} f^{9} g^{3}+120285 b^{6} f^{8} g^{4}+96228 b^{6} f^{7} g^{5}-96228 b^{6} f^{5} g^{7}-120285 b^{6} f^{4} g^{8}-80190 b^{6} f^{3} g^{9}-32076 b^{6} f^{2} g^{10}-7290 b^{6} f \,g^{11}-729 b^{6} g^{12}\right ) \textit {\_Z}^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{\frac {b x}{3}+\frac {d}{3}}+\left (-\frac {243}{3125} b^{5} f^{10}-\frac {1944}{3125} b^{5} f^{9} g -\frac {6561}{3125} b^{5} f^{8} g^{2}-\frac {11664}{3125} b^{5} f^{7} g^{3}-\frac {10206}{3125} b^{5} f^{6} g^{4}+\frac {10206}{3125} b^{5} f^{4} g^{6}+\frac {11664}{3125} b^{5} f^{3} g^{7}+\frac {6561}{3125} b^{5} f^{2} g^{8}+\frac {1944}{3125} b^{5} f \,g^{9}+\frac {243}{3125} b^{5} g^{10}\right ) \textit {\_R}^{5}\right )\right ) {\mathrm e}^{\frac {5 a}{3}-\frac {5 d}{3}}\)	\(310\)

Input:

int(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^2,x,method=_RETURNVER 
BOSE)
                                                                                    
                                                                                    
 

Output:

-2/(f*exp(2*b*x+2*d)+exp(2*b*x+2*d)*g-f+g)/b/(f+g)*exp(5/3*b*x+5/3*a)+sum( 
_R*ln(exp(1/3*b*x+1/3*d)+(-243/3125*b^5*f^10-1944/3125*b^5*f^9*g-6561/3125 
*b^5*f^8*g^2-11664/3125*b^5*f^7*g^3-10206/3125*b^5*f^6*g^4+10206/3125*b^5* 
f^4*g^6+11664/3125*b^5*f^3*g^7+6561/3125*b^5*f^2*g^8+1944/3125*b^5*f*g^9+2 
43/3125*b^5*g^10)*_R^5),_R=RootOf(-15625+(729*b^6*f^12+7290*b^6*f^11*g+320 
76*b^6*f^10*g^2+80190*b^6*f^9*g^3+120285*b^6*f^8*g^4+96228*b^6*f^7*g^5-962 
28*b^6*f^5*g^7-120285*b^6*f^4*g^8-80190*b^6*f^3*g^9-32076*b^6*f^2*g^10-729 
0*b^6*f*g^11-729*b^6*g^12)*_Z^6))*exp(5/3*a-5/3*d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 12943 vs. \(2 (279) = 558\).

Time = 0.23 (sec) , antiderivative size = 12943, normalized size of antiderivative = 35.95 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=\text {Too large to display} \] Input:

integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^2,x, algorithm= 
"fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=e^{\frac {5 a}{3}} \int \frac {e^{\frac {5 b x}{3}}}{f^{2} \sinh ^{2}{\left (b x + d \right )} + 2 f g \sinh {\left (b x + d \right )} \cosh {\left (b x + d \right )} + g^{2} \cosh ^{2}{\left (b x + d \right )}}\, dx \] Input:

integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))**2,x)
 

Output:

exp(5*a/3)*Integral(exp(5*b*x/3)/(f**2*sinh(b*x + d)**2 + 2*f*g*sinh(b*x + 
 d)*cosh(b*x + d) + g**2*cosh(b*x + d)**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^2,x, algorithm= 
"maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(g-f>0)', see `assume?` for more 
details)Is
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1060 vs. \(2 (279) = 558\).

Time = 3.33 (sec) , antiderivative size = 1060, normalized size of antiderivative = 2.94 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=\text {Too large to display} \] Input:

integrate(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^2,x, algorithm= 
"giac")
 

Output:

-1/6*(10*(-f^6*e^(4*d) - 4*f^5*g*e^(4*d) - 5*f^4*g^2*e^(4*d) + 5*f^2*g^4*e 
^(4*d) + 4*f*g^5*e^(4*d) + g^6*e^(4*d))^(5/6)*arctan((sqrt(3)*(-(f - g)/(f 
*e^(2*d) + g*e^(2*d)))^(1/6) + 2*e^(1/3*b*x))/(-(f - g)/(f*e^(2*d) + g*e^( 
2*d)))^(1/6))/(f^7*e^(7*d) + 5*f^6*g*e^(7*d) + 9*f^5*g^2*e^(7*d) + 5*f^4*g 
^3*e^(7*d) - 5*f^3*g^4*e^(7*d) - 9*f^2*g^5*e^(7*d) - 5*f*g^6*e^(7*d) - g^7 
*e^(7*d)) + 10*(-f^6*e^(4*d) - 4*f^5*g*e^(4*d) - 5*f^4*g^2*e^(4*d) + 5*f^2 
*g^4*e^(4*d) + 4*f*g^5*e^(4*d) + g^6*e^(4*d))^(5/6)*arctan(-(sqrt(3)*(-(f 
- g)/(f*e^(2*d) + g*e^(2*d)))^(1/6) - 2*e^(1/3*b*x))/(-(f - g)/(f*e^(2*d) 
+ g*e^(2*d)))^(1/6))/(f^7*e^(7*d) + 5*f^6*g*e^(7*d) + 9*f^5*g^2*e^(7*d) + 
5*f^4*g^3*e^(7*d) - 5*f^3*g^4*e^(7*d) - 9*f^2*g^5*e^(7*d) - 5*f*g^6*e^(7*d 
) - g^7*e^(7*d)) + 20*(-(f - g)/(f*e^(2*d) + g*e^(2*d)))^(5/6)*arctan(e^(1 
/3*b*x)/(-(f - g)/(f*e^(2*d) + g*e^(2*d)))^(1/6))/(f^2*e^(2*d) - g^2*e^(2* 
d)) - 15*(-f^6*e^(4*d) - 4*f^5*g*e^(4*d) - 5*f^4*g^2*e^(4*d) + 5*f^2*g^4*e 
^(4*d) + 4*f*g^5*e^(4*d) + g^6*e^(4*d))^(5/6)*log(sqrt(3)*(-(f - g)/(f*e^( 
2*d) + g*e^(2*d)))^(1/6)*e^(1/3*b*x) + (-(f - g)/(f*e^(2*d) + g*e^(2*d)))^ 
(1/3) + e^(2/3*b*x))/(sqrt(3)*f^7*e^(7*d) + 5*sqrt(3)*f^6*g*e^(7*d) + 9*sq 
rt(3)*f^5*g^2*e^(7*d) + 5*sqrt(3)*f^4*g^3*e^(7*d) - 5*sqrt(3)*f^3*g^4*e^(7 
*d) - 9*sqrt(3)*f^2*g^5*e^(7*d) - 5*sqrt(3)*f*g^6*e^(7*d) - sqrt(3)*g^7*e^ 
(7*d)) + 15*(-f^6*e^(4*d) - 4*f^5*g*e^(4*d) - 5*f^4*g^2*e^(4*d) + 5*f^2*g^ 
4*e^(4*d) + 4*f*g^5*e^(4*d) + g^6*e^(4*d))^(5/6)*log(-sqrt(3)*(-(f - g)...
 

Mupad [B] (verification not implemented)

Time = 10.03 (sec) , antiderivative size = 1125, normalized size of antiderivative = 3.12 \[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=\text {Too large to display} \] Input:

int(exp((5*a)/3 + (5*b*x)/3)/(g*cosh(d + b*x) + f*sinh(d + b*x))^2,x)
 

Output:

(5*exp(10*a - 10*d)^(1/6)*log(f^2*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/3 
) - exp((5*a)/3 - (5*d)/3)*(f + g)^(11/6)*(f - g)^(1/6) + g^2*exp(10*a - 1 
0*d)^(1/6)*exp(d/3 + (b*x)/3) + 2*f*g*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b* 
x)/3)))/(3*b*(f + g)^(11/6)*(f - g)^(1/6)) - (5*exp(10*a - 10*d)^(1/6)*log 
(exp((5*a)/3 - (5*d)/3)*(f + g)^(11/6)*(f - g)^(1/6) + f^2*exp(10*a - 10*d 
)^(1/6)*exp(d/3 + (b*x)/3) + g^2*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/3) 
 + 2*f*g*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/3)))/(3*b*(f + g)^(11/6)*( 
f - g)^(1/6)) - (2*exp((5*a)/3 + (5*b*x)/3))/(b*(f + g)*(g - f + exp(2*d + 
 2*b*x)*(f + g))) + (5*exp(10*a - 10*d)^(1/6)*log(f^2*exp(10*a - 10*d)^(1/ 
6)*exp(d/3 + (b*x)/3) - 2*exp((5*a)/3 - (5*d)/3)*(f + g)^(11/6)*(f - g)^(1 
/6) + g^2*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/3) + 2*f*g*exp(10*a - 10* 
d)^(1/6)*exp(d/3 + (b*x)/3) + 3^(1/2)*f^2*exp(10*a - 10*d)^(1/6)*exp(d/3 + 
 (b*x)/3)*1i + 3^(1/2)*g^2*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/3)*1i + 
3^(1/2)*f*g*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/3)*2i)*((3^(1/2)*1i)/2 
+ 1/2))/(3*b*(f + g)^(11/6)*(f - g)^(1/6)) + (5*exp(10*a - 10*d)^(1/6)*log 
(2*exp((5*a)/3 - (5*d)/3)*(f + g)^(11/6)*(f - g)^(1/6) + f^2*exp(10*a - 10 
*d)^(1/6)*exp(d/3 + (b*x)/3) + g^2*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/ 
3) + 2*f*g*exp(10*a - 10*d)^(1/6)*exp(d/3 + (b*x)/3) - 3^(1/2)*f^2*exp(10* 
a - 10*d)^(1/6)*exp(d/3 + (b*x)/3)*1i - 3^(1/2)*g^2*exp(10*a - 10*d)^(1/6) 
*exp(d/3 + (b*x)/3)*1i - 3^(1/2)*f*g*exp(10*a - 10*d)^(1/6)*exp(d/3 + (...
 

Reduce [F]

\[ \int \frac {e^{\frac {5}{3} (a+b x)}}{(g \cosh (d+b x)+f \sinh (d+b x))^2} \, dx=\int \frac {e^{\frac {5 b x}{3}+\frac {5 a}{3}}}{\cosh \left (b x +d \right )^{2} g^{2}+2 \cosh \left (b x +d \right ) \sinh \left (b x +d \right ) f g +\sinh \left (b x +d \right )^{2} f^{2}}d x \] Input:

int(exp(5/3*b*x+5/3*a)/(g*cosh(b*x+d)+f*sinh(b*x+d))^2,x)
 

Output:

int(e**((5*a + 5*b*x)/3)/(cosh(b*x + d)**2*g**2 + 2*cosh(b*x + d)*sinh(b*x 
 + d)*f*g + sinh(b*x + d)**2*f**2),x)