\(\int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx\) [7]

Optimal result

Integrand size = 14, antiderivative size = 104 \[ \int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx=-\frac {\cosh (a+b x)}{2 d (c+d x)^2}+\frac {b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{2 d^3}-\frac {b \sinh (a+b x)}{2 d^2 (c+d x)}+\frac {b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{2 d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx=\frac {b^2 \cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right )-\frac {d (d \cosh (a+b x)+b (c+d x) \sinh (a+b x))}{(c+d x)^2}+b^2 \sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )}{2 d^3} \] Input:

Integrate[Cosh[a + b*x]/(c + d*x)^3,x]
 

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.67 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {3042, 3778, 26, 3042, 26, 3778, 3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3778

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3784

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 3782

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

Input:

Int[Cosh[a + b*x]/(c + d*x)^3,x]
 

Output:

-1/2*Cosh[a + b*x]/(d*(c + d*x)^2) - ((I/2)*b*(((-I)*Sinh[a + b*x])/(d*(c 
+ d*x)) + (I*b*((Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/d + (Sinh[ 
a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d))/d))/d
 

Defintions of rubi rules used

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 3042

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

rule 3778

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

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(96)=192\).

Time = 0.53 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.66

method	result	size
risch	\(\frac {b^{3} {\mathrm e}^{-b x -a} x}{4 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{3} {\mathrm e}^{-b x -a} c}{4 d^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {b^{2} {\mathrm e}^{-b x -a}}{4 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{4 d^{3}}-\frac {b^{2} {\mathrm e}^{b x +a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{2} {\mathrm e}^{b x +a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {b^{2} {\mathrm e}^{\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-a d +c b}{d}\right )}{4 d^{3}}\)	\(277\)

Input:

int(cosh(b*x+a)/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (96) = 192\).

Time = 0.09 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.44 \[ \int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx=-\frac {2 \, d^{2} \cosh \left (b x + a\right ) - {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + 2 \, {\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right ) - {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

integrate(cosh(b*x+a)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx=\text {Timed out} \] Input:

integrate(cosh(b*x+a)/(d*x+c)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.91 \[ \int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx=\frac {b {\left (\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (a - \frac {b c}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d}\right )}}{4 \, d} - \frac {\cosh \left (b x + a\right )}{2 \, {\left (d x + c\right )}^{2} d} \] Input:

integrate(cosh(b*x+a)/(d*x+c)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (96) = 192\).

Time = 0.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.87 \[ \int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx=\frac {b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 2 \, b^{2} c d x {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + 2 \, b^{2} c d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + b^{2} c^{2} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + b^{2} c^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - b d^{2} x e^{\left (b x + a\right )} + b d^{2} x e^{\left (-b x - a\right )} - b c d e^{\left (b x + a\right )} + b c d e^{\left (-b x - a\right )} - d^{2} e^{\left (b x + a\right )} - d^{2} e^{\left (-b x - a\right )}}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:

integrate(cosh(b*x+a)/(d*x+c)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\cosh (a+b x)}{(c+d x)^3} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \] Input:

int(cosh(a + b*x)/(c + d*x)^3,x)
 

Output:

int(cosh(a + b*x)/(c + d*x)^3, x)
 

Reduce [F]

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

int(cosh(b*x+a)/(d*x+c)^3,x)
 

Output:

(e**(2*a)*int(e**(b*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x) 
+ int(1/(e**(b*x)*c**3 + 3*e**(b*x)*c**2*d*x + 3*e**(b*x)*c*d**2*x**2 + e* 
*(b*x)*d**3*x**3),x))/(2*e**a)