Integrand size = 14, antiderivative size = 51 \[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \] Output:
cosh(a-b*c/d)*Chi(b*c/d+b*x)/d+sinh(a-b*c/d)*Shi(b*c/d+b*x)/d
Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96 \[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=\frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )+\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d} \] Input:
Integrate[Cosh[a + b*x]/(c + d*x),x]
Output:
(Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x] + Sinh[a - (b*c)/d]*SinhInt egral[(b*c)/d + b*x])/d
Time = 0.41 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {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.
\(\displaystyle \int \frac {\cosh (a+b x)}{c+d x} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )}{c+d x}dx\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \cosh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx-i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {i \sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sinh \left (\frac {b c}{d}+b x\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\cosh \left (\frac {b c}{d}+b x\right )}{c+d x}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh \left (a-\frac {b c}{d}\right ) \int -\frac {i \sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx-i \sinh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x\right )}{c+d x}dx\) |
\(\Big \downarrow \) 3779 |
\(\displaystyle \frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}+\cosh \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {i b c}{d}+i b x+\frac {\pi }{2}\right )}{c+d x}dx\) |
\(\Big \downarrow \) 3782 |
\(\displaystyle \frac {\cosh \left (a-\frac {b c}{d}\right ) \text {Chi}\left (\frac {b c}{d}+b x\right )}{d}+\frac {\sinh \left (a-\frac {b c}{d}\right ) \text {Shi}\left (\frac {b c}{d}+b x\right )}{d}\) |
Input:
Int[Cosh[a + b*x]/(c + d*x),x]
Output:
(Cosh[a - (b*c)/d]*CoshIntegral[(b*c)/d + b*x])/d + (Sinh[a - (b*c)/d]*Sin hIntegral[(b*c)/d + b*x])/d
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[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 = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.61
method | result | size |
risch | \(-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2 d}-\frac {{\mathrm e}^{\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (-b x -a -\frac {-a d +c b}{d}\right )}{2 d}\) | \(82\) |
Input:
int(cosh(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-1/2/d*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-1/2/d*exp((a*d-b*c)/d)*Ei (1,-b*x-a-(-a*d+b*c)/d)
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.84 \[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=\frac {{\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) + {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \cosh \left (-\frac {b c - a d}{d}\right ) + {\left ({\rm Ei}\left (\frac {b d x + b c}{d}\right ) - {\rm Ei}\left (-\frac {b d x + b c}{d}\right )\right )} \sinh \left (-\frac {b c - a d}{d}\right )}{2 \, d} \] Input:
integrate(cosh(b*x+a)/(d*x+c),x, algorithm="fricas")
Output:
1/2*((Ei((b*d*x + b*c)/d) + Ei(-(b*d*x + b*c)/d))*cosh(-(b*c - a*d)/d) + ( Ei((b*d*x + b*c)/d) - Ei(-(b*d*x + b*c)/d))*sinh(-(b*c - a*d)/d))/d
\[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=\int \frac {\cosh {\left (a + b x \right )}}{c + d x}\, dx \] Input:
integrate(cosh(b*x+a)/(d*x+c),x)
Output:
Integral(cosh(a + b*x)/(c + d*x), x)
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12 \[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=-\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} - \frac {e^{\left (a - \frac {b c}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{2 \, d} \] Input:
integrate(cosh(b*x+a)/(d*x+c),x, algorithm="maxima")
Output:
-1/2*e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*b/d)/d - 1/2*e^(a - b*c/d) *exp_integral_e(1, -(d*x + c)*b/d)/d
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.10 \[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=\frac {{\rm Ei}\left (\frac {b d x + b c}{d}\right ) e^{\left (a - \frac {b c}{d}\right )} + {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )}}{2 \, d} \] Input:
integrate(cosh(b*x+a)/(d*x+c),x, algorithm="giac")
Output:
1/2*(Ei((b*d*x + b*c)/d)*e^(a - b*c/d) + Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/ d))/d
Timed out. \[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=\int \frac {\mathrm {cosh}\left (a+b\,x\right )}{c+d\,x} \,d x \] Input:
int(cosh(a + b*x)/(c + d*x),x)
Output:
int(cosh(a + b*x)/(c + d*x), x)
\[ \int \frac {\cosh (a+b x)}{c+d x} \, dx=\int \frac {\cosh \left (b x +a \right )}{d x +c}d x \] Input:
int(cosh(b*x+a)/(d*x+c),x)
Output:
int(cosh(a + b*x)/(c + d*x),x)