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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5388, 3384, 3379, 3382} \[ \int \frac {\sinh (a+b x)}{c+d x^2} \, dx=-\frac {\sinh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Chi}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\sinh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Chi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cosh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Shi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cosh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Shi}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

[In]

Int[Sinh[a + b*x]/(c + d*x^2),x]

[Out]

-1/2*(CoshIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x]*Sinh[a - (b*Sqrt[-c])/Sqrt[d]])/(Sqrt[-c]*Sqrt[d]) + (CoshInteg
ral[(b*Sqrt[-c])/Sqrt[d] - b*x]*Sinh[a + (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) - (Cosh[a + (b*Sqrt[-c])/
Sqrt[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cosh[a - (b*Sqrt[-c])/Sqrt[d]]*Sinh
Integral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> 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 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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 5388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sinh[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[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])

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = -\frac {\int \frac {\sinh (a+b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\sinh (a+b x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}} \\ & = -\frac {\cosh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\sinh \left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}+\frac {\cosh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\sinh \left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\sinh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\cosh \left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\sinh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \int \frac {\cosh \left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}} \\ & = -\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}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.55 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.78 \[ \int \frac {\sinh (a+b x)}{c+d x^2} \, dx=-\frac {i e^{-a-\frac {i b \sqrt {c}}{\sqrt {d}}} \left (e^{2 a+\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (b \left (-\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )-e^{2 a} \operatorname {ExpIntegralEi}\left (b \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )\right )+e^{\frac {2 i b \sqrt {c}}{\sqrt {d}}} \operatorname {ExpIntegralEi}\left (-\frac {i b \sqrt {c}}{\sqrt {d}}-b x\right )-\operatorname {ExpIntegralEi}\left (\frac {i b \sqrt {c}}{\sqrt {d}}-b x\right )\right )}{4 \sqrt {c} \sqrt {d}} \]

[In]

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

[Out]

((-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])*ExpIntegral
Ei[((-I)*b*Sqrt[c])/Sqrt[d] - b*x] - ExpIntegralEi[(I*b*Sqrt[c])/Sqrt[d] - b*x]))/(Sqrt[c]*Sqrt[d])

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

-1/4/(-c*d)^(1/2)*exp(-(-b*(-c*d)^(1/2)+a*d)/d)*Ei(1,(b*(-c*d)^(1/2)+d*(b*x+a)-a*d)/d)-1/4/(-c*d)^(1/2)*exp((b
*(-c*d)^(1/2)+a*d)/d)*Ei(1,(b*(-c*d)^(1/2)-d*(b*x+a)+a*d)/d)+1/4/(-c*d)^(1/2)*exp((-b*(-c*d)^(1/2)+a*d)/d)*Ei(
1,-(b*(-c*d)^(1/2)+d*(b*x+a)-a*d)/d)+1/4/(-c*d)^(1/2)*exp(-(b*(-c*d)^(1/2)+a*d)/d)*Ei(1,-(b*(-c*d)^(1/2)-d*(b*
x+a)+a*d)/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 = 0.30 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.48 \[ \int \frac {\sinh (a+b x)}{c+d x^2} \, dx=-\frac {{\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x + \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \cosh \left (a + \sqrt {-\frac {b^{2} c}{d}}\right ) - {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x - \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \cosh \left (-a + \sqrt {-\frac {b^{2} c}{d}}\right ) + {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) + \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x + \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \sinh \left (a + \sqrt {-\frac {b^{2} c}{d}}\right ) + {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) + \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x - \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \sinh \left (-a + \sqrt {-\frac {b^{2} c}{d}}\right )}{4 \, b c} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sinh(a + b*x)/(c + d*x^2),x)

[Out]

int(sinh(a + b*x)/(c + d*x^2), x)