3.2.67 \(\int \frac {(a+b \sinh (e+f x))^2}{(c+d x)^2} \, dx\) [167]

Optimal. Leaf size=183 \[ -\frac {a^2}{d (c+d x)}+\frac {2 a b f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}-\frac {2 a b \sinh (e+f x)}{d (c+d x)}-\frac {b^2 \sinh ^2(e+f x)}{d (c+d x)}+\frac {2 a b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {b^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3398, 3378, 3384, 3379, 3382, 3394, 12} \begin {gather*} -\frac {a^2}{d (c+d x)}+\frac {2 a b f \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a b f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {2 a b \sinh (e+f x)}{d (c+d x)}+\frac {b^2 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {b^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d^2}-\frac {b^2 \sinh ^2(e+f x)}{d (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3378

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 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 3394

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

Rule 3398

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])

Rubi steps

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

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Mathematica [A]
time = 0.40, size = 232, normalized size = 1.27 \begin {gather*} \frac {-2 a^2 d+b^2 d-b^2 d \cosh (2 (e+f x))+4 a b f (c+d x) \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )+2 b^2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )-4 a b d \sinh (e+f x)+4 a b c f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 a b d f x \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+2 b^2 c f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+2 b^2 d f x \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )}{2 d^2 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 6.30, size = 319, normalized size = 1.74

method	result	size
risch	\(-\frac {f a b \,{\mathrm e}^{f x +e}}{d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f a b \,{\mathrm e}^{-\frac {c f -d e}{d}} \expIntegral \left (1, -f x -e -\frac {c f -d e}{d}\right )}{d^{2}}-\frac {a^{2}}{d \left (d x +c \right )}+\frac {b^{2}}{2 \left (d x +c \right ) d}-\frac {f \,b^{2} {\mathrm e}^{-2 f x -2 e}}{4 d \left (d x f +c f \right )}+\frac {f \,b^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \expIntegral \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{2}}-\frac {f \,b^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,b^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \expIntegral \left (1, -2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{2}}+\frac {f a b \,{\mathrm e}^{-f x -e}}{d \left (d x f +c f \right )}-\frac {f a b \,{\mathrm e}^{\frac {c f -d e}{d}} \expIntegral \left (1, f x +e +\frac {c f -d e}{d}\right )}{d^{2}}\)	\(319\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 185, normalized size = 1.01 \begin {gather*} -\frac {1}{4} \, b^{2} {\left (\frac {e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (-\frac {2 \, c f}{d} + 2 \, e\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {2}{d^{2} x + c d}\right )} + a b {\left (\frac {e^{\left (\frac {c f}{d} - e\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (-\frac {c f}{d} + e\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a^{2}}{d^{2} x + c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 542 vs. \(2 (192) = 384\).
time = 0.37, size = 542, normalized size = 2.96 \begin {gather*} -\frac {b^{2} d \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 4 \, a b d \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (b^{2} d + {\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right )\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + {\left (2 \, a^{2} - b^{2}\right )} d - 2 \, {\left ({\left (a b d f x + a b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a b d f x + a b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right ) - {\left ({\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} - {\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right ) - 2 \, {\left ({\left (a b d f x + a b c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a b d f x + a b c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right ) - {\left ({\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} - {\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + {\left (b^{2} d f x + b^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (c f - d \cosh \left (1\right ) - d \sinh \left (1\right )\right )}}{d}\right )}{2 \, {\left ({\left (d^{3} x + c d^{2}\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} - {\left (d^{3} x + c d^{2}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1135 vs. \(2 (186) = 372\).
time = 0.49, size = 1135, normalized size = 6.20 \begin {gather*} \frac {{\left (2 \, {\left (d x + c\right )} b^{2} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 2 \, b^{2} d e f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, b^{2} c f^{3} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 4 \, {\left (d x + c\right )} a b {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - 4 \, a b d e f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + 4 \, a b c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + 4 \, {\left (d x + c\right )} a b {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - 4 \, a b d e f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + 4 \, a b c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - 2 \, {\left (d x + c\right )} b^{2} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, b^{2} d e f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 2 \, b^{2} c f^{3} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - b^{2} d f^{2} e^{\left (\frac {2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - 4 \, a b d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} + 4 \, a b d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - b^{2} d f^{2} e^{\left (-\frac {2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - 4 \, a^{2} d f^{2} + 2 \, b^{2} d f^{2}\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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