\(\int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx\) [15]

Optimal result

Integrand size = 16, antiderivative size = 162 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {b^2}{3 d^3 (c+d x)}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \cosh ^2(a+b x)}{3 d^3 (c+d x)}+\frac {2 b^3 \text {Chi}\left (\frac {2 b c}{d}+2 b x\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )}{3 d^4}-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}+\frac {2 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4} \] Output:

1/3*b^2/d^3/(d*x+c)-1/3*cosh(b*x+a)^2/d/(d*x+c)^3-2/3*b^2*cosh(b*x+a)^2/d^ 
3/(d*x+c)+2/3*b^3*Chi(2*b*c/d+2*b*x)*sinh(2*a-2*b*c/d)/d^4-1/3*b*cosh(b*x+ 
a)*sinh(b*x+a)/d^2/(d*x+c)^2+2/3*b^3*cosh(2*a-2*b*c/d)*Shi(2*b*c/d+2*b*x)/ 
d^4
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.75 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {4 b^3 \text {Chi}\left (\frac {2 b (c+d x)}{d}\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )-\frac {d \left (\left (d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+d (d+b (c+d x) \sinh (2 (a+b x)))\right )}{(c+d x)^3}+4 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )}{6 d^4} \] Input:

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

Output:

(4*b^3*CoshIntegral[(2*b*(c + d*x))/d]*Sinh[2*a - (2*b*c)/d] - (d*((d^2 + 
2*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)] + d*(d + b*(c + d*x)*Sinh[2*(a + b*x) 
])))/(c + d*x)^3 + 4*b^3*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*(c + d*x) 
)/d])/(6*d^4)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {3042, 3795, 17, 3042, 3794, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3795

\(\displaystyle \frac {2 b^2 \int \frac {\cosh ^2(a+b x)}{(c+d x)^2}dx}{3 d^2}-\frac {b^2 \int \frac {1}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {2 b^2 \int \frac {\cosh ^2(a+b x)}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^2}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 3794

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}+\frac {2 i b \int -\frac {i \sinh (2 a+2 b x)}{2 (c+d x)}dx}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}+\frac {b \int -\frac {i \sin (2 i a+2 i b x)}{c+d x}dx}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}-\frac {i b \int \frac {\sin (2 i a+2 i b x)}{c+d x}dx}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 3784

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx+\cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {i \sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx+i \cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x+\frac {\pi }{2}\right )}{c+d x}dx+i \cosh \left (2 a-\frac {2 b c}{d}\right ) \int -\frac {i \sin \left (\frac {2 i b c}{d}+2 i b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x+\frac {\pi }{2}\right )}{c+d x}dx+\cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 3779

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {i \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

\(\Big \downarrow \) 3782

\(\displaystyle \frac {2 b^2 \left (-\frac {\cosh ^2(a+b x)}{d (c+d x)}-\frac {i b \left (\frac {i \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {i \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\cosh ^2(a+b x)}{3 d (c+d x)^3}+\frac {b^2}{3 d^3 (c+d x)}\)

Input:

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

Output:

b^2/(3*d^3*(c + d*x)) - Cosh[a + b*x]^2/(3*d*(c + d*x)^3) - (b*Cosh[a + b* 
x]*Sinh[a + b*x])/(3*d^2*(c + d*x)^2) + (2*b^2*(-(Cosh[a + b*x]^2/(d*(c + 
d*x))) - (I*b*((I*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/d 
 + (I*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d))/d))/(3*d^ 
2)
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 3042

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

rule 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 3782

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz 
}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* 
f)/d]   Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] 
&& NeQ[d*e - c*f, 0]
 

rule 3794

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si 
mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 
)))   Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n 
- 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & 
& LtQ[m, -1]
 

rule 3795

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ 
b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) 
*(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2)))   Int[(c + 
d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* 
(m + 2)))   Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, 
c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(554\) vs. \(2(150)=300\).

Time = 1.92 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.43

method	result	size
risch	\(-\frac {1}{6 \left (d x +c \right )^{3} d}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} x^{2}}{6 d \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} c x}{3 d^{2} \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} c^{2}}{6 d^{3} \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{4} {\mathrm e}^{-2 b x -2 a} x}{12 d \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{4} {\mathrm e}^{-2 b x -2 a} c}{12 d^{2} \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{3} {\mathrm e}^{-2 b x -2 a}}{12 d \left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{3} {\mathrm e}^{-\frac {2 \left (a d -c b \right )}{d}} \operatorname {expIntegral}_{1}\left (2 b x +2 a -\frac {2 \left (a d -c b \right )}{d}\right )}{3 d^{4}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{12 d^{4} \left (\frac {b c}{d}+b x \right )^{3}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{12 d^{4} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{6 d^{4} \left (\frac {b c}{d}+b x \right )}-\frac {b^{3} {\mathrm e}^{\frac {2 a d -2 c b}{d}} \operatorname {expIntegral}_{1}\left (-2 b x -2 a -\frac {2 \left (-a d +c b \right )}{d}\right )}{3 d^{4}}\)	\(555\)

Input:

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

Output:

-1/6/(d*x+c)^3/d-1/6*b^5*exp(-2*b*x-2*a)/d/(b^3*d^3*x^3+3*b^3*c*d^2*x^2+3* 
b^3*c^2*d*x+b^3*c^3)*x^2-1/3*b^5*exp(-2*b*x-2*a)/d^2/(b^3*d^3*x^3+3*b^3*c* 
d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*c*x-1/6*b^5*exp(-2*b*x-2*a)/d^3/(b^3*d^3*x^ 
3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*c^2+1/12*b^4*exp(-2*b*x-2*a)/d/(b 
^3*d^3*x^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*x+1/12*b^4*exp(-2*b*x-2* 
a)/d^2/(b^3*d^3*x^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*c-1/12*b^3*exp( 
-2*b*x-2*a)/d/(b^3*d^3*x^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)+1/3*b^3/ 
d^4*exp(-2*(a*d-b*c)/d)*Ei(1,2*b*x+2*a-2*(a*d-b*c)/d)-1/12*b^3/d^4*exp(2*b 
*x+2*a)/(b*c/d+b*x)^3-1/12*b^3/d^4*exp(2*b*x+2*a)/(b*c/d+b*x)^2-1/6*b^3/d^ 
4*exp(2*b*x+2*a)/(b*c/d+b*x)-1/3*b^3/d^4*exp(2*(a*d-b*c)/d)*Ei(1,-2*b*x-2* 
a-2*(-a*d+b*c)/d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (150) = 300\).

Time = 0.09 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.52 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {d^{3} + {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b d^{3} x + b c d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2} - 2 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \] Input:

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

Output:

-1/6*(d^3 + (2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + d^3)*cosh(b*x + 
 a)^2 + 2*(b*d^3*x + b*c*d^2)*cosh(b*x + a)*sinh(b*x + a) + (2*b^2*d^3*x^2 
 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + d^3)*sinh(b*x + a)^2 - 2*((b^3*d^3*x^3 + 
3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*Ei(2*(b*d*x + b*c)/d) - (b^3*d^ 
3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*Ei(-2*(b*d*x + b*c)/d)) 
*cosh(-2*(b*c - a*d)/d) - 2*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d* 
x + b^3*c^3)*Ei(2*(b*d*x + b*c)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^ 
3*c^2*d*x + b^3*c^3)*Ei(-2*(b*d*x + b*c)/d))*sinh(-2*(b*c - a*d)/d))/(d^7* 
x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
 

Sympy [F]

\[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {\cosh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.68 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {1}{6 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} - \frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{4}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{3} d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{4}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{3} d} \] Input:

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

Output:

-1/6/(d^4*x^3 + 3*c*d^3*x^2 + 3*c^2*d^2*x + c^3*d) - 1/4*e^(-2*a + 2*b*c/d 
)*exp_integral_e(4, 2*(d*x + c)*b/d)/((d*x + c)^3*d) - 1/4*e^(2*a - 2*b*c/ 
d)*exp_integral_e(4, -2*(d*x + c)*b/d)/((d*x + c)^3*d)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (150) = 300\).

Time = 0.13 (sec) , antiderivative size = 537, normalized size of antiderivative = 3.31 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {4 \, b^{3} d^{3} x^{3} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 4 \, b^{3} d^{3} x^{3} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 12 \, b^{3} c d^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 12 \, b^{3} c d^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 12 \, b^{3} c^{2} d x {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 12 \, b^{3} c^{2} d x {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 2 \, b^{2} d^{3} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b^{2} d^{3} x^{2} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, b^{3} c^{3} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 4 \, b^{3} c^{3} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 4 \, b^{2} c d^{2} x e^{\left (2 \, b x + 2 \, a\right )} - 4 \, b^{2} c d^{2} x e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, b^{2} c^{2} d e^{\left (2 \, b x + 2 \, a\right )} - b d^{3} x e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b^{2} c^{2} d e^{\left (-2 \, b x - 2 \, a\right )} + b d^{3} x e^{\left (-2 \, b x - 2 \, a\right )} - b c d^{2} e^{\left (2 \, b x + 2 \, a\right )} + b c d^{2} e^{\left (-2 \, b x - 2 \, a\right )} - d^{3} e^{\left (2 \, b x + 2 \, a\right )} - d^{3} e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, d^{3}}{12 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \] Input:

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

Output:

1/12*(4*b^3*d^3*x^3*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) - 4*b^3*d^3*x^ 
3*Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2*b*c/d) + 12*b^3*c*d^2*x^2*Ei(2*(b*d*x 
 + b*c)/d)*e^(2*a - 2*b*c/d) - 12*b^3*c*d^2*x^2*Ei(-2*(b*d*x + b*c)/d)*e^( 
-2*a + 2*b*c/d) + 12*b^3*c^2*d*x*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) - 
 12*b^3*c^2*d*x*Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2*b*c/d) - 2*b^2*d^3*x^2* 
e^(2*b*x + 2*a) - 2*b^2*d^3*x^2*e^(-2*b*x - 2*a) + 4*b^3*c^3*Ei(2*(b*d*x + 
 b*c)/d)*e^(2*a - 2*b*c/d) - 4*b^3*c^3*Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2* 
b*c/d) - 4*b^2*c*d^2*x*e^(2*b*x + 2*a) - 4*b^2*c*d^2*x*e^(-2*b*x - 2*a) - 
2*b^2*c^2*d*e^(2*b*x + 2*a) - b*d^3*x*e^(2*b*x + 2*a) - 2*b^2*c^2*d*e^(-2* 
b*x - 2*a) + b*d^3*x*e^(-2*b*x - 2*a) - b*c*d^2*e^(2*b*x + 2*a) + b*c*d^2* 
e^(-2*b*x - 2*a) - d^3*e^(2*b*x + 2*a) - d^3*e^(-2*b*x - 2*a) - 2*d^3)/(d^ 
7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {3 e^{2 a} \left (\int \frac {e^{2 b x}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) c^{3} d +9 e^{2 a} \left (\int \frac {e^{2 b x}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) c^{2} d^{2} x +9 e^{2 a} \left (\int \frac {e^{2 b x}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) c \,d^{3} x^{2}+3 e^{2 a} \left (\int \frac {e^{2 b x}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}d x \right ) d^{4} x^{3}+3 \left (\int \frac {1}{e^{2 b x +2 a} c^{4}+4 e^{2 b x +2 a} c^{3} d x +6 e^{2 b x +2 a} c^{2} d^{2} x^{2}+4 e^{2 b x +2 a} c \,d^{3} x^{3}+e^{2 b x +2 a} d^{4} x^{4}}d x \right ) c^{3} d +9 \left (\int \frac {1}{e^{2 b x +2 a} c^{4}+4 e^{2 b x +2 a} c^{3} d x +6 e^{2 b x +2 a} c^{2} d^{2} x^{2}+4 e^{2 b x +2 a} c \,d^{3} x^{3}+e^{2 b x +2 a} d^{4} x^{4}}d x \right ) c^{2} d^{2} x +9 \left (\int \frac {1}{e^{2 b x +2 a} c^{4}+4 e^{2 b x +2 a} c^{3} d x +6 e^{2 b x +2 a} c^{2} d^{2} x^{2}+4 e^{2 b x +2 a} c \,d^{3} x^{3}+e^{2 b x +2 a} d^{4} x^{4}}d x \right ) c \,d^{3} x^{2}+3 \left (\int \frac {1}{e^{2 b x +2 a} c^{4}+4 e^{2 b x +2 a} c^{3} d x +6 e^{2 b x +2 a} c^{2} d^{2} x^{2}+4 e^{2 b x +2 a} c \,d^{3} x^{3}+e^{2 b x +2 a} d^{4} x^{4}}d x \right ) d^{4} x^{3}-2}{12 d \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )} \] Input:

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

Output:

(3*e**(2*a)*int(e**(2*b*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d** 
3*x**3 + d**4*x**4),x)*c**3*d + 9*e**(2*a)*int(e**(2*b*x)/(c**4 + 4*c**3*d 
*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*c**2*d**2*x + 9*e**( 
2*a)*int(e**(2*b*x)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 
+ d**4*x**4),x)*c*d**3*x**2 + 3*e**(2*a)*int(e**(2*b*x)/(c**4 + 4*c**3*d*x 
 + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4),x)*d**4*x**3 + 3*int(1/(e 
**(2*a + 2*b*x)*c**4 + 4*e**(2*a + 2*b*x)*c**3*d*x + 6*e**(2*a + 2*b*x)*c* 
*2*d**2*x**2 + 4*e**(2*a + 2*b*x)*c*d**3*x**3 + e**(2*a + 2*b*x)*d**4*x**4 
),x)*c**3*d + 9*int(1/(e**(2*a + 2*b*x)*c**4 + 4*e**(2*a + 2*b*x)*c**3*d*x 
 + 6*e**(2*a + 2*b*x)*c**2*d**2*x**2 + 4*e**(2*a + 2*b*x)*c*d**3*x**3 + e* 
*(2*a + 2*b*x)*d**4*x**4),x)*c**2*d**2*x + 9*int(1/(e**(2*a + 2*b*x)*c**4 
+ 4*e**(2*a + 2*b*x)*c**3*d*x + 6*e**(2*a + 2*b*x)*c**2*d**2*x**2 + 4*e**( 
2*a + 2*b*x)*c*d**3*x**3 + e**(2*a + 2*b*x)*d**4*x**4),x)*c*d**3*x**2 + 3* 
int(1/(e**(2*a + 2*b*x)*c**4 + 4*e**(2*a + 2*b*x)*c**3*d*x + 6*e**(2*a + 2 
*b*x)*c**2*d**2*x**2 + 4*e**(2*a + 2*b*x)*c*d**3*x**3 + e**(2*a + 2*b*x)*d 
**4*x**4),x)*d**4*x**3 - 2)/(12*d*(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d** 
3*x**3))