\(\int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [283]

Optimal result

Integrand size = 23, antiderivative size = 81 \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {(2 a-3 b) x}{2 b^2}+\frac {(a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99 \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {4 (a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} (-2 (2 a-3 b) (c+d x)+b \sinh (2 (c+d x)))}{4 \sqrt {a} b^2 d} \] Input:

Integrate[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]
 

Output:

(4*(a - b)^(3/2)*ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]] + Sqrt[a]*(- 
2*(2*a - 3*b)*(c + d*x) + b*Sinh[2*(c + d*x)]))/(4*Sqrt[a]*b^2*d)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3670, 316, 25, 397, 219, 221}

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.

\(\displaystyle \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (i c+i d x)^4}{a-b \sin (i c+i d x)^2}dx\)

\(\Big \downarrow \) 3670

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 397

\(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {(2 a-3 b) \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{b}-\frac {2 (a-b)^2 \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{2 b}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {(2 a-3 b) \text {arctanh}(\tanh (c+d x))}{b}-\frac {2 (a-b)^2 \int \frac {1}{a-(a-b) \tanh ^2(c+d x)}d\tanh (c+d x)}{b}}{2 b}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\tanh (c+d x)}{2 b \left (1-\tanh ^2(c+d x)\right )}-\frac {\frac {(2 a-3 b) \text {arctanh}(\tanh (c+d x))}{b}-\frac {2 (a-b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b}}{2 b}}{d}\)

Input:

Int[Cosh[c + d*x]^4/(a + b*Sinh[c + d*x]^2),x]
 

Output:

(-1/2*(((2*a - 3*b)*ArcTanh[Tanh[c + d*x]])/b - (2*(a - b)^(3/2)*ArcTanh[( 
Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]])/(Sqrt[a]*b))/b + Tanh[c + d*x]/(2*b*( 
1 - Tanh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 25

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[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 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

rule 397

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

rule 3042

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

rule 3670

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( 
p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f   Su 
bst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e 
 + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(252\) vs. \(2(69)=138\).

Time = 29.17 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.12

method	result	size
risch	\(-\frac {a x}{b^{2}}+\frac {3 x}{2 b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {a \left (a -b \right )}+b}{b}\right )}{2 d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {-2 a +2 \sqrt {a \left (a -b \right )}+b}{b}\right )}{2 a d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {a \left (a -b \right )}-b}{b}\right )}{2 d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a +2 \sqrt {a \left (a -b \right )}-b}{b}\right )}{2 a d b}\)	\(253\)
derivativedivides	\(\frac {\frac {2 \left (a^{2}-2 a b +b^{2}\right ) a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{2}}}{d}\)	\(316\)
default	\(\frac {\frac {2 \left (a^{2}-2 a b +b^{2}\right ) a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \operatorname {arctanh}\left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{2}}}{d}\)	\(316\)

Input:

int(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)
 

Output:

-a*x/b^2+3/2*x/b+1/8/b/d*exp(2*d*x+2*c)-1/8/b/d*exp(-2*d*x-2*c)+1/2*(a*(a- 
b))^(1/2)/d/b^2*ln(exp(2*d*x+2*c)-(-2*a+2*(a*(a-b))^(1/2)+b)/b)-1/2/a*(a*( 
a-b))^(1/2)/d/b*ln(exp(2*d*x+2*c)-(-2*a+2*(a*(a-b))^(1/2)+b)/b)-1/2*(a*(a- 
b))^(1/2)/d/b^2*ln(exp(2*d*x+2*c)+(2*a+2*(a*(a-b))^(1/2)-b)/b)+1/2/a*(a*(a 
-b))^(1/2)/d/b*ln(exp(2*d*x+2*c)+(2*a+2*(a*(a-b))^(1/2)-b)/b)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (69) = 138\).

Time = 0.11 (sec) , antiderivative size = 875, normalized size of antiderivative = 10.80 \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \] Input:

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
 

Output:

[-1/8*(4*(2*a - 3*b)*d*x*cosh(d*x + c)^2 - 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*(2*a - 3*b)*d*x - 3*b*co 
sh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a - b)*cosh(d*x + c)^2 + 2*(a - b)*co 
sh(d*x + c)*sinh(d*x + c) + (a - b)*sinh(d*x + c)^2)*sqrt((a - b)/a)*log(( 
b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + 
 c)^4 + 2*(2*a*b - b^2)*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b 
 - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*x + c)^3 + ( 
2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) + 4*(a*b*cosh(d*x + c)^2 + 2*a*b 
*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 + 2*a^2 - a*b)*sqrt((a 
- b)/a))/(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)) + 4*(2*(2*a - 3*b)*d*x*cosh(d*x + c) - b*cosh(d*x + c)^3)*si 
nh(d*x + c) + b)/(b^2*d*cosh(d*x + c)^2 + 2*b^2*d*cosh(d*x + c)*sinh(d*x + 
 c) + b^2*d*sinh(d*x + c)^2), -1/8*(4*(2*a - 3*b)*d*x*cosh(d*x + c)^2 - 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*(2*a - 3*b)*d*x - 3*b*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a - b)*c 
osh(d*x + c)^2 + 2*(a - b)*cosh(d*x + c)*sinh(d*x + c) + (a - b)*sinh(d*x 
+ c)^2)*sqrt(-(a - b)/a)*arctan(-1/2*(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c 
)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-(a - b)/a)/(a - b)...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Timed out} \] Input:

integrate(cosh(d*x+c)**4/(a+b*sinh(d*x+c)**2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^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(b-a>0)', see `assume?` for more 
details)Is
 

Giac [A] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.70 \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=-\frac {\frac {4 \, {\left (d x + c\right )} {\left (2 \, a - 3 \, b\right )}}{b^{2}} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{b} - \frac {{\left (4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2}} - \frac {8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} b^{2}}}{8 \, d} \] Input:

integrate(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x, algorithm="giac")
 

Output:

-1/8*(4*(d*x + c)*(2*a - 3*b)/b^2 - e^(2*d*x + 2*c)/b - (4*a*e^(2*d*x + 2* 
c) - 6*b*e^(2*d*x + 2*c) - b)*e^(-2*d*x - 2*c)/b^2 - 8*(a^2 - 2*a*b + b^2) 
*arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a 
*b)*b^2))/d
 

Mupad [B] (verification not implemented)

Time = 2.50 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.70 \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}-\frac {x\,\left (2\,a-3\,b\right )}{2\,b^2}-\frac {\ln \left (\frac {4\,{\left (a-b\right )}^3\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^6}-\frac {8\,{\left (a-b\right )}^{7/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^6}\right )\,{\left (a-b\right )}^{3/2}}{2\,\sqrt {a}\,b^2\,d}+\frac {\ln \left (\frac {4\,{\left (a-b\right )}^3\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^6}+\frac {8\,{\left (a-b\right )}^{7/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^6}\right )\,{\left (a-b\right )}^{3/2}}{2\,\sqrt {a}\,b^2\,d} \] Input:

int(cosh(c + d*x)^4/(a + b*sinh(c + d*x)^2),x)
 

Output:

exp(2*c + 2*d*x)/(8*b*d) - exp(- 2*c - 2*d*x)/(8*b*d) - (x*(2*a - 3*b))/(2 
*b^2) - (log((4*(a - b)^3*(2*a*b - b^2 + 8*a^2*exp(2*c + 2*d*x) + b^2*exp( 
2*c + 2*d*x) - 8*a*b*exp(2*c + 2*d*x)))/(a*b^6) - (8*(a - b)^(7/2)*(b + 4* 
a*exp(2*c + 2*d*x) - 2*b*exp(2*c + 2*d*x)))/(a^(1/2)*b^6))*(a - b)^(3/2))/ 
(2*a^(1/2)*b^2*d) + (log((4*(a - b)^3*(2*a*b - b^2 + 8*a^2*exp(2*c + 2*d*x 
) + b^2*exp(2*c + 2*d*x) - 8*a*b*exp(2*c + 2*d*x)))/(a*b^6) + (8*(a - b)^( 
7/2)*(b + 4*a*exp(2*c + 2*d*x) - 2*b*exp(2*c + 2*d*x)))/(a^(1/2)*b^6))*(a 
- b)^(3/2))/(2*a^(1/2)*b^2*d)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.64 \[ \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {4 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a -4 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) b +4 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) a -4 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a -b}-2 a +b}+e^{d x +c} \sqrt {b}\right ) b -4 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a -b}+e^{2 d x +2 c} b +2 a -b \right ) a +4 e^{2 d x +2 c} \sqrt {a}\, \sqrt {a -b}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {a -b}+e^{2 d x +2 c} b +2 a -b \right ) b +e^{4 d x +4 c} a b -8 e^{2 d x +2 c} a^{2} d x +12 e^{2 d x +2 c} a b d x -a b}{8 e^{2 d x +2 c} a \,b^{2} d} \] Input:

int(cosh(d*x+c)^4/(a+b*sinh(d*x+c)^2),x)
 

Output:

(4*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) 
- 2*a + b) + e**(c + d*x)*sqrt(b))*a - 4*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - 
 b)*log( - sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*b 
 + 4*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 
 2*a + b) + e**(c + d*x)*sqrt(b))*a - 4*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - 
b)*log(sqrt(2*sqrt(a)*sqrt(a - b) - 2*a + b) + e**(c + d*x)*sqrt(b))*b - 4 
*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log(2*sqrt(a)*sqrt(a - b) + e**(2*c 
+ 2*d*x)*b + 2*a - b)*a + 4*e**(2*c + 2*d*x)*sqrt(a)*sqrt(a - b)*log(2*sqr 
t(a)*sqrt(a - b) + e**(2*c + 2*d*x)*b + 2*a - b)*b + e**(4*c + 4*d*x)*a*b 
- 8*e**(2*c + 2*d*x)*a**2*d*x + 12*e**(2*c + 2*d*x)*a*b*d*x - a*b)/(8*e**( 
2*c + 2*d*x)*a*b**2*d)