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

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

[Out]

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

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Rubi [A]  time = 0.52, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5281, 3303, 3298, 3301} \[ \frac {\cosh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Chi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cosh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Chi}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\sinh \left (a+\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Shi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}-b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\sinh \left (a-\frac {b \sqrt {-c}}{\sqrt {d}}\right ) \text {Shi}\left (x b+\frac {\sqrt {-c} b}{\sqrt {d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]/(c + d*x^2),x]

[Out]

(Cosh[a + (b*Sqrt[-c])/Sqrt[d]]*CoshIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cosh[a - (b*
Sqrt[-c])/Sqrt[d]]*CoshIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Sinh[a + (b*Sqrt[-c])/Sqr
t[d]]*SinhIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Sinh[a - (b*Sqrt[-c])/Sqrt[d]]*SinhInt
egral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])

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]

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 5281

Int[Cosh[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cosh[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*} \int \frac {\cosh (a+b x)}{c+d x^2} \, dx &=\int \left (\frac {\sqrt {-c} \cosh (a+b x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \cosh (a+b x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\cosh (a+b x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\cosh (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 {\cosh \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 {\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 {\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 {\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 ) \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 {Chi}\left (\frac {b \sqrt {-c}}{\sqrt {d}}+b x\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\sinh \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 {\sinh \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*}

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Mathematica [C]  time = 0.30, size = 180, normalized size = 0.85 \[ \frac {i \left (\cosh \left (a-\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Ci}\left (i b x-\frac {b \sqrt {c}}{\sqrt {d}}\right )-\cosh \left (a+\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Ci}\left (i x b+\frac {\sqrt {c} b}{\sqrt {d}}\right )+i \left (\sinh \left (a-\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Si}\left (\frac {b \sqrt {c}}{\sqrt {d}}-i b x\right )+\sinh \left (a+\frac {i b \sqrt {c}}{\sqrt {d}}\right ) \text {Si}\left (i x b+\frac {\sqrt {c} b}{\sqrt {d}}\right )\right )\right )}{2 \sqrt {c} \sqrt {d}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.52, size = 316, normalized size = 1.48 \[ -\frac {{\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) + \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x + \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \cosh \left (a + \sqrt {-\frac {b^{2} c}{d}}\right ) - {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) + \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x - \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \cosh \left (-a + \sqrt {-\frac {b^{2} c}{d}}\right ) + {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x - \sqrt {-\frac {b^{2} c}{d}}\right ) - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x + \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \sinh \left (a + \sqrt {-\frac {b^{2} c}{d}}\right ) + {\left (\sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (b x + \sqrt {-\frac {b^{2} c}{d}}\right ) - \sqrt {-\frac {b^{2} c}{d}} {\rm Ei}\left (-b x - \sqrt {-\frac {b^{2} c}{d}}\right )\right )} \sinh \left (-a + \sqrt {-\frac {b^{2} c}{d}}\right )}{4 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )}{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.12, size = 212, normalized size = 1.00 \[ \frac {{\mathrm e}^{-\frac {-b \sqrt {-c d}+d a}{d}} \Ei \left (1, \frac {b \sqrt {-c d}+\left (b x +a \right ) d -d a}{d}\right )}{4 \sqrt {-c d}}-\frac {{\mathrm e}^{-\frac {b \sqrt {-c d}+d a}{d}} \Ei \left (1, -\frac {b \sqrt {-c d}-\left (b x +a \right ) d +d a}{d}\right )}{4 \sqrt {-c d}}-\frac {{\mathrm e}^{\frac {b \sqrt {-c d}+d a}{d}} \Ei \left (1, \frac {b \sqrt {-c d}-\left (b x +a \right ) d +d a}{d}\right )}{4 \sqrt {-c d}}+\frac {{\mathrm e}^{\frac {-b \sqrt {-c d}+d a}{d}} \Ei \left (1, -\frac {b \sqrt {-c d}+\left (b x +a \right ) d -d a}{d}\right )}{4 \sqrt {-c d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(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]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (b x + a\right )}{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )}{d\,x^2+c} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)/(c + d*x^2),x)

[Out]

int(cosh(a + b*x)/(c + d*x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (a + b x \right )}}{c + d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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