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

Optimal. Leaf size=213 \[ -\frac{\sinh \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Chi}\left (x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sinh \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Chi}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Shi}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Shi}\left (x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]

[Out]

-(CoshIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x]*Sinh[a - (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) + (CoshIntegra
l[(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]) - (Cosh[a - (b*Sqrt[-c])/Sqrt[d]]*SinhIn
tegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])

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Rubi [A]  time = 0.556159, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5280, 3303, 3298, 3301} \[ -\frac{\sinh \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Chi}\left (x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}}+\frac{\sinh \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Chi}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh \left (a+\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Shi}\left (\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right )}{2 \sqrt{-c} \sqrt{d}}-\frac{\cosh \left (a-\frac{b \sqrt{-c}}{\sqrt{d}}\right ) \text{Shi}\left (x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right )}{2 \sqrt{-c} \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]/(c + d*x^2),x]

[Out]

-(CoshIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x]*Sinh[a - (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) + (CoshIntegra
l[(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]) - (Cosh[a - (b*Sqrt[-c])/Sqrt[d]]*SinhIn
tegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])

Rule 5280

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])

Rule 3303

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 3298

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 3301

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]

Rubi steps

\begin{align*} \int \frac{\sinh (a+b x)}{c+d x^2} \, dx &=\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]  time = 0.313823, size = 180, normalized size = 0.85 \[ \frac{i \left (\sinh \left (a-\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (-\frac{b \sqrt{c}}{\sqrt{d}}+i b x\right )-\sinh \left (a+\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{CosIntegral}\left (\frac{b \sqrt{c}}{\sqrt{d}}+i b x\right )+i \left (\cosh \left (a-\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{Si}\left (\frac{b \sqrt{c}}{\sqrt{d}}-i b x\right )+\cosh \left (a+\frac{i b \sqrt{c}}{\sqrt{d}}\right ) \text{Si}\left (i x b+\frac{\sqrt{c} b}{\sqrt{d}}\right )\right )\right )}{2 \sqrt{c} \sqrt{d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/2)*(CosIntegral[-((b*Sqrt[c])/Sqrt[d]) + I*b*x]*Sinh[a - (I*b*Sqrt[c])/Sqrt[d]] - CosIntegral[(b*Sqrt[c])/
Sqrt[d] + I*b*x]*Sinh[a + (I*b*Sqrt[c])/Sqrt[d]] + I*(Cosh[a - (I*b*Sqrt[c])/Sqrt[d]]*SinIntegral[(b*Sqrt[c])/
Sqrt[d] - I*b*x] + Cosh[a + (I*b*Sqrt[c])/Sqrt[d]]*SinIntegral[(b*Sqrt[c])/Sqrt[d] + I*b*x])))/(Sqrt[c]*Sqrt[d
])

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Maple [A]  time = 0.039, size = 212, normalized size = 1. \begin{align*}{\frac{1}{4}{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}+da \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}- \left ( bx+a \right ) d+da \right ) } \right ){\frac{1}{\sqrt{-cd}}}}-{\frac{1}{4}{{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}-da \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}+ \left ( bx+a \right ) d-da \right ) } \right ){\frac{1}{\sqrt{-cd}}}}-{\frac{1}{4}{{\rm e}^{{\frac{1}{d} \left ( b\sqrt{-cd}+da \right ) }}}{\it Ei} \left ( 1,{\frac{1}{d} \left ( b\sqrt{-cd}- \left ( bx+a \right ) d+da \right ) } \right ){\frac{1}{\sqrt{-cd}}}}+{\frac{1}{4}{{\rm e}^{-{\frac{1}{d} \left ( b\sqrt{-cd}-da \right ) }}}{\it Ei} \left ( 1,-{\frac{1}{d} \left ( b\sqrt{-cd}+ \left ( bx+a \right ) d-da \right ) } \right ){\frac{1}{\sqrt{-cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.18255, size = 624, normalized size = 2.93 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b x \right )}}{c + d x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (b x + a\right )}{d x^{2} + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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