3.110 \(\int \frac{(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx\)

Optimal. Leaf size=207 \[ \frac{a^2 f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{d^3}+\frac{a^2 f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{d^3}+\frac{a^2 f^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^3}+\frac{a^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^3}-\frac{4 a^2 f \sinh \left (\frac{e}{2}+\frac{f x}{2}\right ) \cosh ^3\left (\frac{e}{2}+\frac{f x}{2}\right )}{d^2 (c+d x)}-\frac{2 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)^2} \]

[Out]

(-2*a^2*Cosh[e/2 + (f*x)/2]^4)/(d*(c + d*x)^2) + (a^2*f^2*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^3 +
(a^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 - (4*a^2*f*Cosh[e/2 + (f*x)/2]^3*Sinh[e/2
+ (f*x)/2])/(d^2*(c + d*x)) + (a^2*f^2*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 + (a^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 = 0.504832, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3318, 3314, 3312, 3303, 3298, 3301} \[ \frac{a^2 f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{d^3}+\frac{a^2 f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{d^3}+\frac{a^2 f^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^3}+\frac{a^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^3}-\frac{4 a^2 f \sinh \left (\frac{e}{2}+\frac{f x}{2}\right ) \cosh ^3\left (\frac{e}{2}+\frac{f x}{2}\right )}{d^2 (c+d x)}-\frac{2 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*Cosh[e/2 + (f*x)/2]^4)/(d*(c + d*x)^2) + (a^2*f^2*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/d^3 +
(a^2*f^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d^3 - (4*a^2*f*Cosh[e/2 + (f*x)/2]^3*Sinh[e/2
+ (f*x)/2])/(d^2*(c + d*x)) + (a^2*f^2*Sinh[e - (c*f)/d]*SinhIntegral[(c*f)/d + f*x])/d^3 + (a^2*f^2*Sinh[2*e
- (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d^3

Rule 3318

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

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{(a+a \cosh (e+f x))^2}{(c+d x)^3} \, dx &=\left (4 a^2\right ) \int \frac{\sin ^4\left (\frac{1}{2} (i e+\pi )+\frac{i f x}{2}\right )}{(c+d x)^3} \, dx\\ &=-\frac{2 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)^2}-\frac{4 a^2 f \cosh ^3\left (\frac{e}{2}+\frac{f x}{2}\right ) \sinh \left (\frac{e}{2}+\frac{f x}{2}\right )}{d^2 (c+d x)}-\frac{\left (6 a^2 f^2\right ) \int \frac{\cosh ^2\left (\frac{e}{2}+\frac{f x}{2}\right )}{c+d x} \, dx}{d^2}+\frac{\left (8 a^2 f^2\right ) \int \frac{\cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{c+d x} \, dx}{d^2}\\ &=-\frac{2 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)^2}-\frac{4 a^2 f \cosh ^3\left (\frac{e}{2}+\frac{f x}{2}\right ) \sinh \left (\frac{e}{2}+\frac{f x}{2}\right )}{d^2 (c+d x)}-\frac{\left (6 a^2 f^2\right ) \int \left (\frac{1}{2 (c+d x)}+\frac{\cosh (e+f x)}{2 (c+d x)}\right ) \, dx}{d^2}+\frac{\left (8 a^2 f^2\right ) \int \left (\frac{3}{8 (c+d x)}+\frac{\cosh (e+f x)}{2 (c+d x)}+\frac{\cosh (2 e+2 f x)}{8 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac{2 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)^2}-\frac{4 a^2 f \cosh ^3\left (\frac{e}{2}+\frac{f x}{2}\right ) \sinh \left (\frac{e}{2}+\frac{f x}{2}\right )}{d^2 (c+d x)}+\frac{\left (a^2 f^2\right ) \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{d^2}-\frac{\left (3 a^2 f^2\right ) \int \frac{\cosh (e+f x)}{c+d x} \, dx}{d^2}+\frac{\left (4 a^2 f^2\right ) \int \frac{\cosh (e+f x)}{c+d x} \, dx}{d^2}\\ &=-\frac{2 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)^2}-\frac{4 a^2 f \cosh ^3\left (\frac{e}{2}+\frac{f x}{2}\right ) \sinh \left (\frac{e}{2}+\frac{f x}{2}\right )}{d^2 (c+d x)}+\frac{\left (a^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 (3 a^2 f^2 \cosh \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{\left (4 a^2 f^2 \cosh \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{\left (a^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{\left (3 a^2 f^2 \sinh \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 (4 a^2 f^2 \sinh \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{2 a^2 \cosh ^4\left (\frac{e}{2}+\frac{f x}{2}\right )}{d (c+d x)^2}+\frac{a^2 f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Chi}\left (\frac{c f}{d}+f x\right )}{d^3}+\frac{a^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{4 a^2 f \cosh ^3\left (\frac{e}{2}+\frac{f x}{2}\right ) \sinh \left (\frac{e}{2}+\frac{f x}{2}\right )}{d^2 (c+d x)}+\frac{a^2 f^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^3}+\frac{a^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 = 1.08188, size = 353, normalized size = 1.71 \[ \frac{a^2 \left (4 c^2 f^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+4 c^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+4 f^2 (c+d x)^2 \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \cosh \left (e-\frac{c f}{d}\right )+4 f^2 (c+d x)^2 \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )+4 d^2 f^2 x^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+4 d^2 f^2 x^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+8 c d f^2 x \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+8 c d f^2 x \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-4 c d f \sinh (e+f x)-2 c d f \sinh (2 (e+f x))-4 d^2 f x \sinh (e+f x)-2 d^2 f x \sinh (2 (e+f x))-4 d^2 \cosh (e+f x)-d^2 \cosh (2 (e+f x))-3 d^2\right )}{4 d^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(-3*d^2 - 4*d^2*Cosh[e + f*x] - d^2*Cosh[2*(e + f*x)] + 4*f^2*(c + d*x)^2*Cosh[e - (c*f)/d]*CoshIntegral[
f*(c/d + x)] + 4*f^2*(c + d*x)^2*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*f*(c + d*x))/d] - 4*c*d*f*Sinh[e + f*x]
- 4*d^2*f*x*Sinh[e + f*x] - 2*c*d*f*Sinh[2*(e + f*x)] - 2*d^2*f*x*Sinh[2*(e + f*x)] + 4*c^2*f^2*Sinh[e - (c*f
)/d]*SinhIntegral[f*(c/d + x)] + 8*c*d*f^2*x*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 4*d^2*f^2*x^2*Sinh[
e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 4*c^2*f^2*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] + 8*c
*d*f^2*x*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] + 4*d^2*f^2*x^2*Sinh[2*e - (2*c*f)/d]*SinhInteg
ral[(2*f*(c + d*x))/d]))/(4*d^3*(c + d*x)^2)

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Maple [B]  time = 0.134, size = 618, normalized size = 3. \begin{align*}{\frac{{f}^{3}{a}^{2}{{\rm e}^{-fx-e}}x}{2\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{3}{a}^{2}{{\rm e}^{-fx-e}}c}{2\,{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{a}^{2}{f}^{2}{{\rm e}^{-fx-e}}}{2\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{a}^{2}{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}{f}^{2}{{\rm e}^{fx+e}}}{2\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{{a}^{2}{f}^{2}{{\rm e}^{fx+e}}}{2\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{{a}^{2}{f}^{2}}{2\,{d}^{3}}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) }-{\frac{3\,{a}^{2}}{4\,d \left ( dx+c \right ) ^{2}}}+{\frac{{f}^{3}{a}^{2}{{\rm e}^{-2\,fx-2\,e}}x}{4\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{3}{a}^{2}{{\rm e}^{-2\,fx-2\,e}}c}{4\,{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{a}^{2}{f}^{2}{{\rm e}^{-2\,fx-2\,e}}}{8\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{a}^{2}{f}^{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{{a}^{2}{f}^{2}{{\rm e}^{2\,fx+2\,e}}}{8\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{{a}^{2}{f}^{2}{{\rm e}^{2\,fx+2\,e}}}{4\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{{a}^{2}{f}^{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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*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^2*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^2*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^2*f^2/d^3*exp((c*f-d*e)/d)*E
i(1,f*x+e+(c*f-d*e)/d)-1/2*a^2*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)^2-1/2*a^2*f^2/d^3*exp(f*x+e)/(c*f/d+f*x)-1/2*a^2
*f^2/d^3*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)-3/4*a^2/d/(d*x+c)^2+1/4*a^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*a^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*a^2*f^2*ex
p(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)-1/2*a^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*a^2*f^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)^2-1/4*a^2*f^2/d^3*exp(2*f*x+2*e)/(c*f/d+f*x)-1/2*a^2*f^2/d^3*e
xp(-2*(c*f-d*e)/d)*Ei(1,-2*f*x-2*e-2*(c*f-d*e)/d)

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Maxima [A]  time = 1.40956, size = 273, normalized size = 1.32 \begin{align*} -\frac{1}{4} \, a^{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^{2}{\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+a*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

-1/4*a^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^2*(e^(-e + c*f/d)*exp_integra
l_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 = 2.1955, size = 1223, normalized size = 5.91 \begin{align*} -\frac{a^{2} d^{2} \cosh \left (f x + e\right )^{2} + a^{2} d^{2} \sinh \left (f x + e\right )^{2} + 4 \, a^{2} d^{2} \cosh \left (f x + e\right ) + 3 \, a^{2} d^{2} - 2 \,{\left ({\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} 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 (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) +{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{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^{2} d^{2} f x + a^{2} c d f +{\left (a^{2} d^{2} f x + a^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \,{\left ({\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} 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 (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) -{\left (a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} c d f^{2} x + a^{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+a*cosh(f*x+e))^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/4*(a^2*d^2*cosh(f*x + e)^2 + a^2*d^2*sinh(f*x + e)^2 + 4*a^2*d^2*cosh(f*x + e) + 3*a^2*d^2 - 2*((a^2*d^2*f^
2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei((d*f*x + c*f)/d) + (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)
*Ei(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) - 2*((a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei(2*(d*f*x
+ c*f)/d) + (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei(-2*(d*f*x + c*f)/d))*cosh(-2*(d*e - c*f)/d)
+ 4*(a^2*d^2*f*x + a^2*c*d*f + (a^2*d^2*f*x + a^2*c*d*f)*cosh(f*x + e))*sinh(f*x + e) + 2*((a^2*d^2*f^2*x^2 +
2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei((d*f*x + c*f)/d) - (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei(-(d*
f*x + c*f)/d))*sinh(-(d*e - c*f)/d) + 2*((a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^2*c^2*f^2)*Ei(2*(d*f*x + c*f)/
d) - (a^2*d^2*f^2*x^2 + 2*a^2*c*d*f^2*x + a^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*} a^{2} \left (\int \frac{2 \cosh{\left (e + f x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac{\cosh ^{2}{\left (e + f x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac{1}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(Integral(2*cosh(e + f*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(cosh(e + f*x)**2
/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(1/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x*
*3), x))

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Giac [B]  time = 1.27554, size = 953, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*a^2*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 4*a^2*d^2*f^2*x^2*Ei(-(d*f*x + c*f)/d)*e^(c*
f/d - e) + 4*a^2*d^2*f^2*x^2*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 4*a^2*d^2*f^2*x^2*Ei(2*(d*f*x + c*f)/d)*e^(-
2*c*f/d + 2*e) + 8*a^2*c*d*f^2*x*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 8*a^2*c*d*f^2*x*Ei(-(d*f*x + c*f)/
d)*e^(c*f/d - e) + 8*a^2*c*d*f^2*x*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 8*a^2*c*d*f^2*x*Ei(2*(d*f*x + c*f)/d)*
e^(-2*c*f/d + 2*e) + 4*a^2*c^2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d - 2*e) + 4*a^2*c^2*f^2*Ei(-(d*f*x + c*f)/
d)*e^(c*f/d - e) + 4*a^2*c^2*f^2*Ei((d*f*x + c*f)/d)*e^(-c*f/d + e) + 4*a^2*c^2*f^2*Ei(2*(d*f*x + c*f)/d)*e^(-
2*c*f/d + 2*e) - 2*a^2*d^2*f*x*e^(2*f*x + 2*e) - 4*a^2*d^2*f*x*e^(f*x + e) + 4*a^2*d^2*f*x*e^(-f*x - e) + 2*a^
2*d^2*f*x*e^(-2*f*x - 2*e) - 2*a^2*c*d*f*e^(2*f*x + 2*e) - 4*a^2*c*d*f*e^(f*x + e) + 4*a^2*c*d*f*e^(-f*x - e)
+ 2*a^2*c*d*f*e^(-2*f*x - 2*e) - a^2*d^2*e^(2*f*x + 2*e) - 4*a^2*d^2*e^(f*x + e) - 4*a^2*d^2*e^(-f*x - e) - a^
2*d^2*e^(-2*f*x - 2*e) - 6*a^2*d^2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)