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

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

[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)

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Rubi [A]
time = 0.41, 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 = {5388, 3384, 3379, 3382} \begin {gather*} -\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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 180, normalized size = 0.85 \begin {gather*} \frac {i \left (\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 ) \sinh \left (a+\frac {i b \sqrt {c}}{\sqrt {d}}\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 (\frac {b \sqrt {c}}{\sqrt {d}}+i b x\right )\right )\right )}{2 \sqrt {c} \sqrt {d}} \end {gather*}

Antiderivative was successfully verified.

[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 = 0.50, size = 212, normalized size = 1.00

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (157) = 314\).
time = 0.37, size = 316, normalized size = 1.48 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (a + b x \right )}}{c + d x^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {sinh}\left (a+b\,x\right )}{d\,x^2+c} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)/(c + d*x^2),x)

[Out]

int(sinh(a + b*x)/(c + d*x^2), x)

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