\(\int \frac {\sinh (a+b x)}{c+d x^2} \, dx\) [368]

Optimal result

Integrand size = 16, antiderivative size = 213 \[ \int \frac {\sinh (a+b x)}{c+d x^2} \, dx=-\frac {\text {Chi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right ) \sinh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Chi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right ) \sinh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cosh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Shi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cosh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Shi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.49 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.78 \[ \int \frac {\sinh (a+b x)}{c+d x^2} \, dx=-\frac {i e^{-a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{2 a+\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )+e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {i b \sqrt {c}}{\sqrt {d}}-b x\right )-\operatorname {ExpIntegralEi}\left (\frac {i b \sqrt {c}}{\sqrt {d}}-b x\right )\right )}{4 \sqrt {c} \sqrt {d}} \] Input:

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

Output:

((-1/4*I)*E^(-a - (I*b*Sqrt[c])/Sqrt[d])*(E^(2*a + ((2*I)*b*Sqrt[c])/Sqrt[ 
d])*ExpIntegralEi[b*(((-I)*Sqrt[c])/Sqrt[d] + x)] - E^(2*a)*ExpIntegralEi[ 
b*((I*Sqrt[c])/Sqrt[d] + x)] + E^(((2*I)*b*Sqrt[c])/Sqrt[d])*ExpIntegralEi 
[((-I)*b*Sqrt[c])/Sqrt[d] - b*x] - ExpIntegralEi[(I*b*Sqrt[c])/Sqrt[d] - b 
*x]))/(Sqrt[c]*Sqrt[d])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5803, 2009}

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.

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

\(\Big \downarrow \) 5803

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(CoshIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x]*Sinh[a - (b*Sqrt[-c])/Sqrt[ 
d]])/(Sqrt[-c]*Sqrt[d]) + (CoshIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x]*Sinh[a 
 + (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) - (Cosh[a + (b*Sqrt[-c])/Sq 
rt[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (C 
osh[a - (b*Sqrt[-c])/Sqrt[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2 
*Sqrt[-c]*Sqrt[d])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5803

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> In 
t[ExpandIntegrand[Sinh[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d 
}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00

method	result	size
risch	\(-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}-\left (b x +a \right ) d +d a}{d}\right )}{4 \sqrt {-c d}}+\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}+\left (b x +a \right ) d -d a}{d}\right )}{4 \sqrt {-c d}}+\frac {{\mathrm e}^{-\frac {b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (-\frac {b \sqrt {-c d}-\left (b x +a \right ) d +d a}{d}\right )}{4 \sqrt {-c d}}-\frac {{\mathrm e}^{-\frac {-b \sqrt {-c d}+d a}{d}} \operatorname {expIntegral}_{1}\left (\frac {b \sqrt {-c d}+\left (b x +a \right ) d -d a}{d}\right )}{4 \sqrt {-c d}}\)	\(212\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (157) = 314\).

Time = 0.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.48 \[ \int \frac {\sinh (a+b x)}{c+d x^2} \, dx=-\frac {{\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x + \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \cosh \left (a + \sqrt {-\frac {b^{2} c}{d}}\right ) - {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x - \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \cosh \left (-a + \sqrt {-\frac {b^{2} c}{d}}\right ) + {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) + \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x + \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \sinh \left (a + \sqrt {-\frac {b^{2} c}{d}}\right ) + {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) + \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x - \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \sinh \left (-a + \sqrt {-\frac {b^{2} c}{d}}\right )}{4 \, b c} \] Input:

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

Output:

-1/4*((sqrt(-b^2*c/d)*Ei(b*x - sqrt(-b^2*c/d)) - sqrt(-b^2*c/d)*Ei(-b*x + 
sqrt(-b^2*c/d)))*cosh(a + sqrt(-b^2*c/d)) - (sqrt(-b^2*c/d)*Ei(b*x + sqrt( 
-b^2*c/d)) - sqrt(-b^2*c/d)*Ei(-b*x - sqrt(-b^2*c/d)))*cosh(-a + sqrt(-b^2 
*c/d)) + (sqrt(-b^2*c/d)*Ei(b*x - sqrt(-b^2*c/d)) + sqrt(-b^2*c/d)*Ei(-b*x 
 + sqrt(-b^2*c/d)))*sinh(a + sqrt(-b^2*c/d)) + (sqrt(-b^2*c/d)*Ei(b*x + sq 
rt(-b^2*c/d)) + sqrt(-b^2*c/d)*Ei(-b*x - sqrt(-b^2*c/d)))*sinh(-a + sqrt(- 
b^2*c/d)))/(b*c)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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