\(\int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [282]

Optimal result

Integrand size = 23, antiderivative size = 77 \[ \int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {(a-b)^2 \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d}-\frac {(a-2 b) \sinh (c+d x)}{b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \] Output:

(a-b)^2*arctan(b^(1/2)*sinh(d*x+c)/a^(1/2))/a^(1/2)/b^(5/2)/d-(a-2*b)*sinh 
(d*x+c)/b^2/d+1/3*sinh(d*x+c)^3/b/d
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.03 \[ \int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {-\frac {12 (a-b)^2 \arctan \left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {b}}\right )}{\sqrt {a}}+3 \sqrt {b} (-4 a+7 b) \sinh (c+d x)+b^{3/2} \sinh (3 (c+d x))}{12 b^{5/2} d} \] Input:

Integrate[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]
 

Output:

((-12*(a - b)^2*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]])/Sqrt[a] + 3*Sqrt[ 
b]*(-4*a + 7*b)*Sinh[c + d*x] + b^(3/2)*Sinh[3*(c + d*x)])/(12*b^(5/2)*d)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 3669, 300, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (i c+i d x)^5}{a-b \sin (i c+i d x)^2}dx\)

\(\Big \downarrow \) 3669

\(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^2}{b \sinh ^2(c+d x)+a}d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 300

\(\displaystyle \frac {\int \left (\frac {\sinh ^2(c+d x)}{b}+\frac {a^2-2 b a+b^2}{b^2 \left (b \sinh ^2(c+d x)+a\right )}-\frac {a-2 b}{b^2}\right )d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(a-b)^2 \arctan \left (\frac {\sqrt {b} \sinh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}}-\frac {(a-2 b) \sinh (c+d x)}{b^2}+\frac {\sinh ^3(c+d x)}{3 b}}{d}\)

Input:

Int[Cosh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]
 

Output:

(((a - b)^2*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)) - ( 
(a - 2*b)*Sinh[c + d*x])/b^2 + Sinh[c + d*x]^3/(3*b))/d
 

Defintions of rubi rules used

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c 
, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3669

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]
 

Maple [A] (verified)

Time = 71.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {-\frac {-\frac {b \sinh \left (d x +c \right )^{3}}{3}+a \sinh \left (d x +c \right )-2 b \sinh \left (d x +c \right )}{b^{2}}+\frac {\left (a^{2}-2 a b +b^{2}\right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}}{d}\)	\(74\)
default	\(\frac {-\frac {-\frac {b \sinh \left (d x +c \right )^{3}}{3}+a \sinh \left (d x +c \right )-2 b \sinh \left (d x +c \right )}{b^{2}}+\frac {\left (a^{2}-2 a b +b^{2}\right ) \arctan \left (\frac {b \sinh \left (d x +c \right )}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}}{d}\)	\(74\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {{\mathrm e}^{d x +c} a}{2 b^{2} d}+\frac {7 \,{\mathrm e}^{d x +c}}{8 b d}+\frac {{\mathrm e}^{-d x -c} a}{2 b^{2} d}-\frac {7 \,{\mathrm e}^{-d x -c}}{8 b d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{\sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a^{2}}{2 \sqrt {-a b}\, d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right ) a}{\sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{\sqrt {-a b}}-1\right )}{2 \sqrt {-a b}\, d}\)	\(347\)

Input:

int(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/b^2*(-1/3*b*sinh(d*x+c)^3+a*sinh(d*x+c)-2*b*sinh(d*x+c))+(a^2-2*a* 
b+b^2)/b^2/(a*b)^(1/2)*arctan(b*sinh(d*x+c)/(a*b)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 704 vs. \(2 (67) = 134\).

Time = 0.12 (sec) , antiderivative size = 1493, normalized size of antiderivative = 19.39 \[ \int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/24*(a*b^2*cosh(d*x + c)^6 + 6*a*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + a*b 
^2*sinh(d*x + c)^6 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^4 + 3*(5*a*b^2*co 
sh(d*x + c)^2 - 4*a^2*b + 7*a*b^2)*sinh(d*x + c)^4 + 4*(5*a*b^2*cosh(d*x + 
 c)^3 - 3*(4*a^2*b - 7*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - a*b^2 + 3*( 
4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2 + 3*(5*a*b^2*cosh(d*x + c)^4 + 4*a^2*b 
- 7*a*b^2 - 6*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 12*(( 
a^2 - 2*a*b + b^2)*cosh(d*x + c)^3 + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)^2 
*sinh(d*x + c) + 3*(a^2 - 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (a^ 
2 - 2*a*b + b^2)*sinh(d*x + c)^3)*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*c 
osh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*s 
inh(d*x + c) - cosh(d*x + c))*sqrt(-a*b) + b)/(b*cosh(d*x + c)^4 + 4*b*cos 
h(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)) + 6*(a*b^2*cosh(d*x + c 
)^5 - 2*(4*a^2*b - 7*a*b^2)*cosh(d*x + c)^3 + (4*a^2*b - 7*a*b^2)*cosh(d*x 
 + c))*sinh(d*x + c))/(a*b^3*d*cosh(d*x + c)^3 + 3*a*b^3*d*cosh(d*x + c)^2 
*sinh(d*x + c) + 3*a*b^3*d*cosh(d*x + c)*sinh(d*x + c)^2 + a*b^3*d*sinh...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cosh(d*x+c)**5/(a+b*sinh(d*x+c)**2),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/24*(3*(4*a*e^(4*c) - 7*b*e^(4*c))*e^(4*d*x) - 3*(4*a*e^(2*c) - 7*b*e^(2 
*c))*e^(2*d*x) - b*e^(6*d*x + 6*c) + b)*e^(-3*d*x - 3*c)/(b^2*d) + 1/32*in 
tegrate(64*((a^2*e^(3*c) - 2*a*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) + (a^2*e 
^c - 2*a*b*e^c + b^2*e^c)*e^(d*x))/(b^3*e^(4*d*x + 4*c) + b^3 + 2*(2*a*b^2 
*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)
 

Giac [F(-2)]

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

integrate(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^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
 

Mupad [B] (verification not implemented)

Time = 2.17 (sec) , antiderivative size = 668, normalized size of antiderivative = 8.68 \[ \int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,{\left (a-b\right )}^2\,\sqrt {a\,b^5\,d^2}}{2\,a\,b^2\,d\,\sqrt {{\left (a-b\right )}^4}}\right )-2\,\mathrm {atan}\left (\frac {a\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a\,b^5\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}-6\,a^2\,b^4\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}+6\,a^3\,b^3\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}-2\,a^4\,b^2\,d\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}\right )}{a^2\,b^{11}\,d^2\,{\left (a-b\right )}^2}+\frac {2\,\left (a^5\,\sqrt {a\,b^5\,d^2}-b^5\,\sqrt {a\,b^5\,d^2}+5\,a\,b^4\,\sqrt {a\,b^5\,d^2}-5\,a^4\,b\,\sqrt {a\,b^5\,d^2}-10\,a^2\,b^3\,\sqrt {a\,b^5\,d^2}+10\,a^3\,b^2\,\sqrt {a\,b^5\,d^2}\right )}{a^2\,b^8\,d\,\sqrt {{\left (a-b\right )}^4}\,\sqrt {a\,b^5\,d^2}}\right )\,\sqrt {a\,b^5\,d^2}}{4\,a^3-12\,a^2\,b+12\,a\,b^2-4\,b^3}-\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (a^5\,\sqrt {a\,b^5\,d^2}-b^5\,\sqrt {a\,b^5\,d^2}+5\,a\,b^4\,\sqrt {a\,b^5\,d^2}-5\,a^4\,b\,\sqrt {a\,b^5\,d^2}-10\,a^2\,b^3\,\sqrt {a\,b^5\,d^2}+10\,a^3\,b^2\,\sqrt {a\,b^5\,d^2}\right )}{a\,b^2\,d\,\sqrt {{\left (a-b\right )}^4}\,\left (4\,a^3-12\,a^2\,b+12\,a\,b^2-4\,b^3\right )}\right )\right )\,\sqrt {a^4-4\,a^3\,b+6\,a^2\,b^2-4\,a\,b^3+b^4}}{2\,\sqrt {a\,b^5\,d^2}}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a-7\,b\right )}{8\,b^2\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a-7\,b\right )}{8\,b^2\,d} \] Input:

int(cosh(c + d*x)^5/(a + b*sinh(c + d*x)^2),x)
 

Output:

((2*atan((exp(d*x)*exp(c)*(a - b)^2*(a*b^5*d^2)^(1/2))/(2*a*b^2*d*((a - b) 
^4)^(1/2))) - 2*atan((a*b^6*exp(d*x)*exp(c)*((4*(2*a*b^5*d*(a^4 - 4*a^3*b 
- 4*a*b^3 + b^4 + 6*a^2*b^2)^(1/2) - 6*a^2*b^4*d*(a^4 - 4*a^3*b - 4*a*b^3 
+ b^4 + 6*a^2*b^2)^(1/2) + 6*a^3*b^3*d*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6* 
a^2*b^2)^(1/2) - 2*a^4*b^2*d*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)^( 
1/2)))/(a^2*b^11*d^2*(a - b)^2) + (2*(a^5*(a*b^5*d^2)^(1/2) - b^5*(a*b^5*d 
^2)^(1/2) + 5*a*b^4*(a*b^5*d^2)^(1/2) - 5*a^4*b*(a*b^5*d^2)^(1/2) - 10*a^2 
*b^3*(a*b^5*d^2)^(1/2) + 10*a^3*b^2*(a*b^5*d^2)^(1/2)))/(a^2*b^8*d*((a - b 
)^4)^(1/2)*(a*b^5*d^2)^(1/2)))*(a*b^5*d^2)^(1/2))/(12*a*b^2 - 12*a^2*b + 4 
*a^3 - 4*b^3) - (2*exp(3*c)*exp(3*d*x)*(a^5*(a*b^5*d^2)^(1/2) - b^5*(a*b^5 
*d^2)^(1/2) + 5*a*b^4*(a*b^5*d^2)^(1/2) - 5*a^4*b*(a*b^5*d^2)^(1/2) - 10*a 
^2*b^3*(a*b^5*d^2)^(1/2) + 10*a^3*b^2*(a*b^5*d^2)^(1/2)))/(a*b^2*d*((a - b 
)^4)^(1/2)*(12*a*b^2 - 12*a^2*b + 4*a^3 - 4*b^3))))*(a^4 - 4*a^3*b - 4*a*b 
^3 + b^4 + 6*a^2*b^2)^(1/2))/(2*(a*b^5*d^2)^(1/2)) - exp(- 3*c - 3*d*x)/(2 
4*b*d) + exp(3*c + 3*d*x)/(24*b*d) - (exp(c + d*x)*(4*a - 7*b))/(8*b^2*d) 
+ (exp(- c - d*x)*(4*a - 7*b))/(8*b^2*d)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1346, normalized size of antiderivative = 17.48 \[ \int \frac {\cosh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx =\text {Too large to display} \] Input:

int(cosh(d*x+c)^5/(a+b*sinh(d*x+c)^2),x)
 

Output:

( - 24*e**(3*c + 3*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a 
- b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a - b) 
+ 2*a - b)))*a**2 + 48*e**(3*c + 3*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2 
*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2*sqrt 
(a)*sqrt(a - b) + 2*a - b)))*a*b - 24*e**(3*c + 3*d*x)*sqrt(b)*sqrt(a)*sqr 
t(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt 
(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*b**2 + 24*e**(3*c + 3*d*x)*sqr 
t(b)*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)* 
sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a**3 - 48*e**(3*c + 3*d*x)*sqrt(b) 
*sqrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt 
(2*sqrt(a)*sqrt(a - b) + 2*a - b)))*a**2*b + 24*e**(3*c + 3*d*x)*sqrt(b)*s 
qrt(2*sqrt(a)*sqrt(a - b) + 2*a - b)*atan((e**(c + d*x)*b)/(sqrt(b)*sqrt(2 
*sqrt(a)*sqrt(a - b) + 2*a - b)))*a*b**2 - 12*e**(3*c + 3*d*x)*sqrt(b)*sqr 
t(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log( - sqrt(2*sqrt( 
a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a**2 + 24*e**(3*c + 3*d* 
x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b)*log( 
- sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*a*b - 12*e 
**(3*c + 3*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a)*sqrt(a - b) - 2 
*a + b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b 
))*b**2 + 12*e**(3*c + 3*d*x)*sqrt(b)*sqrt(a)*sqrt(a - b)*sqrt(2*sqrt(a...