Integrand size = 14, antiderivative size = 71 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=-\frac {\cosh (a+b x)}{d (c+d x)}+\frac {b \text {Chi}\left (\frac {b c}{d}+b x\right ) \sinh \left (a-\frac {b c}{d}\right )}{d^2}+\frac {b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d^2} \]
-cosh(b*x+a)/d/(d*x+c)+b*cosh(a-b*c/d)*Shi(b*c/d+b*x)/d^2+b*Chi(b*c/d+b*x) *sinh(a-b*c/d)/d^2
Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.92 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\frac {-\frac {d \cosh (a+b x)}{c+d x}+b \text {Chi}\left (b \left (\frac {c}{d}+x\right )\right ) \sinh \left (a-\frac {b c}{d}\right )+b \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (b \left (\frac {c}{d}+x\right )\right )}{d^2} \]
(-((d*Cosh[a + b*x])/(c + d*x)) + b*CoshIntegral[b*(c/d + x)]*Sinh[a - (b* c)/d] + b*Cosh[a - (b*c)/d]*SinhIntegral[b*(c/d + x)])/d^2
Result contains complex when optimal does not.
Time = 0.49 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.18, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3042, 3778, 26, 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.
\(\displaystyle \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{(c+d x)^2}dx\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}+\frac {i b \int -\frac {i \sinh (a+b x)}{c+d x}dx}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {b \int \frac {\sinh (a+b x)}{c+d x}dx}{d}-\frac {\cosh (a+b x)}{d (c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}+\frac {b \int -\frac {i \sin (i a+i b x)}{c+d x}dx}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \int \frac {\sin (i a+i b x)}{c+d x}dx}{d}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {i \sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx+i \cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+i \cosh \left (a-\frac {b c}{d}\right ) \int -\frac {i \sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\right )}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\right )}{d}\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d}\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle -\frac {\cosh (a+b x)}{d (c+d x)}-\frac {i b \left (\frac {i \sinh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {i \cosh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\right )}{d}\) |
-(Cosh[a + b*x]/(d*(c + d*x))) - (I*b*((I*CoshIntegral[(b*c)/d + b*x]*Sinh [a - (b*c)/d])/d + (I*Cosh[a - (b*c)/d]*SinhIntegral[(b*c)/d + b*x])/d))/d
3.1.6.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
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]
Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.87
method | result | size |
risch | \(-\frac {b \,{\mathrm e}^{-b x -a}}{2 d \left (d x b +c b \right )}+\frac {b \,{\mathrm e}^{-\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {d a -c b}{d}\right )}{2 d^{2}}-\frac {b \,{\mathrm e}^{b x +a}}{2 d^{2} \left (\frac {b c}{d}+b x \right )}-\frac {b \,{\mathrm e}^{\frac {d a -c b}{d}} \operatorname {Ei}_{1}\left (-b x -a -\frac {-d a +c b}{d}\right )}{2 d^{2}}\) | \(133\) |
-1/2*b*exp(-b*x-a)/d/(b*d*x+b*c)+1/2*b/d^2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a *d-b*c)/d)-1/2*b/d^2*exp(b*x+a)/(b*c/d+b*x)-1/2*b/d^2*exp((a*d-b*c)/d)*Ei( 1,-b*x-a-(-a*d+b*c)/d)
Leaf count of result is larger than twice the leaf count of optimal. 150 vs. \(2 (71) = 142\).
Time = 0.25 (sec) , antiderivative size = 150, normalized size of antiderivative = 2.11 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=-\frac {2 \, d \cosh \left (b x + a\right ) - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) - {\left ({\left (b d x + b c\right )} {\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \]
-1/2*(2*d*cosh(b*x + a) - ((b*d*x + b*c)*Ei((b*d*x + b*c)/d) - (b*d*x + b* c)*Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) - ((b*d*x + b*c)*Ei((b*d*x + b*c)/d) + (b*d*x + b*c)*Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d))/(d^3* x + c*d^2)
Timed out. \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\frac {b {\left (\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{d} - \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{d}\right )}}{2 \, d} - \frac {\cosh \left (b x + a\right )}{{\left (d x + c\right )} d} \]
1/2*b*(e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*b/d)/d - e^(a - b*c/d)*e xp_integral_e(1, -(d*x + c)*b/d)/d)/d - cosh(b*x + a)/((d*x + c)*d)
Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (71) = 142\).
Time = 0.28 (sec) , antiderivative size = 615, normalized size of antiderivative = 8.66 \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=-\frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + b^{2} d e^{\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} + \frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} + b^{3} c {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - a b^{2} d {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) e^{\left (-\frac {b c - a d}{d}\right )} - b^{2} d e^{\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )}\right )} d^{2}}{2 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \]
-1/2*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + b^3*c *Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b* c - a*d)/d) - a*b^2*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + b^2*d*e^(-(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d))*d^2/(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*d ^4 + b*c*d^4 - a*d^5)*b) + 1/2*((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)* e^(-(b*c - a*d)/d) + b^3*c*Ei(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c )) + b*c - a*d)/d)*e^(-(b*c - a*d)/d) - a*b^2*d*Ei(((d*x + c)*(b - b*c/(d* x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^(-(b*c - a*d)/d) - b^2*d*e^((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))/d))*d^2/(((d*x + c)*(b - b*c/(d* x + c) + a*d/(d*x + c))*d^4 + b*c*d^4 - a*d^5)*b)
Timed out. \[ \int \frac {\cosh (a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]