3.168 \(\int \frac{(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx\)

Optimal. Leaf size=242 \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^3}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}+\frac{b^2 f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{d^3}+\frac{b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^3}-\frac{b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2} \]

[Out]

-a^2/(2*d*(c + d*x)^2) - (a*b*f*Cosh[e + f*x])/(d^2*(c + d*x)) + (b^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(
2*c*f)/d + 2*f*x])/d^3 + (a*b*f^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^3 - (a*b*Sinh[e + f*x])/(d*
(c + d*x)^2) - (b^2*f*Cosh[e + f*x]*Sinh[e + f*x])/(d^2*(c + d*x)) - (b^2*Sinh[e + f*x]^2)/(2*d*(c + d*x)^2) +
(a*b*f^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 + (b^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*
c*f)/d + 2*f*x])/d^3

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Rubi [A]  time = 0.446764, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3317, 3297, 3303, 3298, 3301, 3314, 31, 3312} \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^3}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}+\frac{b^2 f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{d^3}+\frac{b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^3}-\frac{b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[e + f*x])^2/(c + d*x)^3,x]

[Out]

-a^2/(2*d*(c + d*x)^2) - (a*b*f*Cosh[e + f*x])/(d^2*(c + d*x)) + (b^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(
2*c*f)/d + 2*f*x])/d^3 + (a*b*f^2*CoshIntegral[(c*f)/d + f*x]*Sinh[e - (c*f)/d])/d^3 - (a*b*Sinh[e + f*x])/(d*
(c + d*x)^2) - (b^2*f*Cosh[e + f*x]*Sinh[e + f*x])/(d^2*(c + d*x)) - (b^2*Sinh[e + f*x]^2)/(2*d*(c + d*x)^2) +
(a*b*f^2*Cosh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 + (b^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*
c*f)/d + 2*f*x])/d^3

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -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]

Rule 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rubi steps

\begin{align*} \int \frac{(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx &=\int \left (\frac{a^2}{(c+d x)^3}+\frac{2 a b \sinh (e+f x)}{(c+d x)^3}+\frac{b^2 \sinh ^2(e+f x)}{(c+d x)^3}\right ) \, dx\\ &=-\frac{a^2}{2 d (c+d x)^2}+(2 a b) \int \frac{\sinh (e+f x)}{(c+d x)^3} \, dx+b^2 \int \frac{\sinh ^2(e+f x)}{(c+d x)^3} \, dx\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{(a b f) \int \frac{\cosh (e+f x)}{(c+d x)^2} \, dx}{d}+\frac{\left (b^2 f^2\right ) \int \frac{1}{c+d x} \, dx}{d^2}+\frac{\left (2 b^2 f^2\right ) \int \frac{\sinh ^2(e+f x)}{c+d x} \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac{b^2 f^2 \log (c+d x)}{d^3}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{\left (a b f^2\right ) \int \frac{\sinh (e+f x)}{c+d x} \, dx}{d^2}-\frac{\left (2 b^2 f^2\right ) \int \left (\frac{1}{2 (c+d x)}-\frac{\cosh (2 e+2 f x)}{2 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{\left (b^2 f^2\right ) \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{d^2}+\frac{\left (a b f^2 \cosh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d^2}+\frac{\left (a b f^2 \sinh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac{a b f^2 \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^3}+\frac{\left (b^2 f^2 \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d^2}+\frac{\left (b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac{b^2 f^2 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{d^3}+\frac{a b f^2 \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^3}+\frac{b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{d^3}\\ \end{align*}

Mathematica [A]  time = 0.960048, size = 395, normalized size = 1.63 \[ \frac{-2 a^2 d^2+4 a b c^2 f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+4 a b f^2 (c+d x)^2 \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+4 a b d^2 f^2 x^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+8 a b c d f^2 x \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )-4 a b c d f \cosh (e+f x)-4 a b d^2 \sinh (e+f x)-4 a b d^2 f x \cosh (e+f x)+4 b^2 c^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+4 b^2 f^2 (c+d x)^2 \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )+4 b^2 d^2 f^2 x^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+8 b^2 c d f^2 x \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-2 b^2 c d f \sinh (2 (e+f x))-2 b^2 d^2 f x \sinh (2 (e+f x))-b^2 d^2 \cosh (2 (e+f x))+b^2 d^2}{4 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sinh[e + f*x])^2/(c + d*x)^3,x]

[Out]

(-2*a^2*d^2 + b^2*d^2 - 4*a*b*c*d*f*Cosh[e + f*x] - 4*a*b*d^2*f*x*Cosh[e + f*x] - b^2*d^2*Cosh[2*(e + f*x)] +
4*b^2*f^2*(c + d*x)^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*f*(c + d*x))/d] + 4*a*b*f^2*(c + d*x)^2*CoshIntegr
al[f*(c/d + x)]*Sinh[e - (c*f)/d] - 4*a*b*d^2*Sinh[e + f*x] - 2*b^2*c*d*f*Sinh[2*(e + f*x)] - 2*b^2*d^2*f*x*Si
nh[2*(e + f*x)] + 4*a*b*c^2*f^2*Cosh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 8*a*b*c*d*f^2*x*Cosh[e - (c*f)/d
]*SinhIntegral[f*(c/d + x)] + 4*a*b*d^2*f^2*x^2*Cosh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 4*b^2*c^2*f^2*Si
nh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] + 8*b^2*c*d*f^2*x*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*
(c + d*x))/d] + 4*b^2*d^2*f^2*x^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d])/(4*d^3*(c + d*x)^2)

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Maple [B]  time = 0.135, size = 626, normalized size = 2.6 \begin{align*} -{\frac{ab{f}^{2}{{\rm e}^{fx+e}}}{2\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{ab{f}^{2}{{\rm e}^{fx+e}}}{2\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{ab{f}^{2}}{2\,{d}^{3}}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) }-{\frac{{a}^{2}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{2}}{4\,d \left ( dx+c \right ) ^{2}}}+{\frac{{f}^{3}{b}^{2}{{\rm e}^{-2\,fx-2\,e}}x}{4\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{3}{b}^{2}{{\rm e}^{-2\,fx-2\,e}}c}{4\,{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{2}{b}^{2}{{\rm e}^{-2\,fx-2\,e}}}{8\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{2}{b}^{2}}{2\,{d}^{3}}{{\rm e}^{2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) }-{\frac{{f}^{2}{b}^{2}{{\rm e}^{2\,fx+2\,e}}}{8\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{{f}^{2}{b}^{2}{{\rm e}^{2\,fx+2\,e}}}{4\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{{f}^{2}{b}^{2}}{2\,{d}^{3}}{{\rm e}^{-2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-2\,fx-2\,e-2\,{\frac{cf-de}{d}} \right ) }-{\frac{ab{f}^{3}{{\rm e}^{-fx-e}}x}{2\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{ab{f}^{3}{{\rm e}^{-fx-e}}c}{2\,{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{ab{f}^{2}{{\rm e}^{-fx-e}}}{2\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{ab{f}^{2}}{2\,{d}^{3}}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sinh(f*x+e))^2/(d*x+c)^3,x)

[Out]

-1/2*a*b*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)^2-1/2*a*b*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)-1/2*a*b*f^2/d^3*exp(-(c*f-d*e
)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)-1/2*a^2/d/(d*x+c)^2+1/4*b^2/d/(d*x+c)^2+1/4*b^2*f^3*exp(-2*f*x-2*e)/d/(d^2*f^2*x
^2+2*c*d*f^2*x+c^2*f^2)*x+1/4*b^2*f^3*exp(-2*f*x-2*e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c-1/8*b^2*f^2*exp(
-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)-1/2*b^2*f^2/d^3*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/
d)-1/8*f^2*b^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)^2-1/4*f^2*b^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)-1/2*f^2*b^2/d^3*exp
(-2*(c*f-d*e)/d)*Ei(1,-2*f*x-2*e-2*(c*f-d*e)/d)-1/2*a*b*f^3*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*x-
1/2*a*b*f^3*exp(-f*x-e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c+1/2*a*b*f^2*exp(-f*x-e)/d/(d^2*f^2*x^2+2*c*d*f
^2*x+c^2*f^2)+1/2*a*b*f^2/d^3*exp((c*f-d*e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)

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Maxima [A]  time = 1.58655, size = 274, normalized size = 1.13 \begin{align*} \frac{1}{4} \, b^{2}{\left (\frac{1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} - \frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{3}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac{e^{\left (2 \, e - \frac{2 \, c f}{d}\right )} E_{3}\left (-\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} + a b{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{3}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac{e^{\left (e - \frac{c f}{d}\right )} E_{3}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac{a^{2}}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

1/4*b^2*(1/(d^3*x^2 + 2*c*d^2*x + c^2*d) - e^(-2*e + 2*c*f/d)*exp_integral_e(3, 2*(d*x + c)*f/d)/((d*x + c)^2*
d) - e^(2*e - 2*c*f/d)*exp_integral_e(3, -2*(d*x + c)*f/d)/((d*x + c)^2*d)) + a*b*(e^(-e + c*f/d)*exp_integral
_e(3, (d*x + c)*f/d)/((d*x + c)^2*d) - e^(e - c*f/d)*exp_integral_e(3, -(d*x + c)*f/d)/((d*x + c)^2*d)) - 1/2*
a^2/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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Fricas [B]  time = 2.47013, size = 1237, normalized size = 5.11 \begin{align*} -\frac{b^{2} d^{2} \cosh \left (f x + e\right )^{2} + b^{2} d^{2} \sinh \left (f x + e\right )^{2} +{\left (2 \, a^{2} - b^{2}\right )} d^{2} + 4 \,{\left (a b d^{2} f x + a b c d f\right )} \cosh \left (f x + e\right ) - 2 \,{\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \cosh \left (-\frac{d e - c f}{d}\right ) - 2 \,{\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) +{\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + 4 \,{\left (a b d^{2} +{\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \,{\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{d}\right ) + 2 \,{\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) -{\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right )}{4 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/4*(b^2*d^2*cosh(f*x + e)^2 + b^2*d^2*sinh(f*x + e)^2 + (2*a^2 - b^2)*d^2 + 4*(a*b*d^2*f*x + a*b*c*d*f)*cosh
(f*x + e) - 2*((a*b*d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^2*f^2)*Ei((d*f*x + c*f)/d) - (a*b*d^2*f^2*x^2 + 2*a*
b*c*d*f^2*x + a*b*c^2*f^2)*Ei(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) - 2*((b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x
+ b^2*c^2*f^2)*Ei(2*(d*f*x + c*f)/d) + (b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x + b^2*c^2*f^2)*Ei(-2*(d*f*x + c*f)/d
))*cosh(-2*(d*e - c*f)/d) + 4*(a*b*d^2 + (b^2*d^2*f*x + b^2*c*d*f)*cosh(f*x + e))*sinh(f*x + e) + 2*((a*b*d^2*
f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^2*f^2)*Ei((d*f*x + c*f)/d) + (a*b*d^2*f^2*x^2 + 2*a*b*c*d*f^2*x + a*b*c^2*f^
2)*Ei(-(d*f*x + c*f)/d))*sinh(-(d*e - c*f)/d) + 2*((b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x + b^2*c^2*f^2)*Ei(2*(d*f
*x + c*f)/d) - (b^2*d^2*f^2*x^2 + 2*b^2*c*d*f^2*x + b^2*c^2*f^2)*Ei(-2*(d*f*x + c*f)/d))*sinh(-2*(d*e - c*f)/d
))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sinh{\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e))**2/(d*x+c)**3,x)

[Out]

Integral((a + b*sinh(e + f*x))**2/(c + d*x)**3, x)

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Giac [B]  time = 1.30969, size = 948, normalized size = 3.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sinh(f*x+e))^2/(d*x+c)^3,x, algorithm="giac")

[Out]

1/8*(4*b^2*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) - 4*a*b*d^2*f^2*x^2*Ei(-(d*f*x + c*f)/d)*e^(c*
f/d - e) + 4*a*b*d^2*f^2*x^2*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 4*b^2*d^2*f^2*x^2*Ei(2*(d*f*x + c*f)/d)*e^(-
2*c*f/d + 2*e) + 8*b^2*c*d*f^2*x*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) - 8*a*b*c*d*f^2*x*Ei(-(d*f*x + c*f)/
d)*e^(c*f/d - e) + 8*a*b*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 8*b^2*c*d*f^2*x*Ei(2*(d*f*x + c*f)/d)*
e^(-2*c*f/d + 2*e) + 4*b^2*c^2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) - 4*a*b*c^2*f^2*Ei(-(d*f*x + c*f)/
d)*e^(c*f/d - e) + 4*a*b*c^2*f^2*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 4*b^2*c^2*f^2*Ei(2*(d*f*x + c*f)/d)*e^(-
2*c*f/d + 2*e) - 2*b^2*d^2*f*x*e^(2*f*x + 2*e) - 4*a*b*d^2*f*x*e^(f*x + e) - 4*a*b*d^2*f*x*e^(-f*x - e) + 2*b^
2*d^2*f*x*e^(-2*f*x - 2*e) - 2*b^2*c*d*f*e^(2*f*x + 2*e) - 4*a*b*c*d*f*e^(f*x + e) - 4*a*b*c*d*f*e^(-f*x - e)
+ 2*b^2*c*d*f*e^(-2*f*x - 2*e) - b^2*d^2*e^(2*f*x + 2*e) - 4*a*b*d^2*e^(f*x + e) + 4*a*b*d^2*e^(-f*x - e) - b^
2*d^2*e^(-2*f*x - 2*e) - 4*a^2*d^2 + 2*b^2*d^2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)