3.2.70 \(\int \frac {(c+d x)^2}{a+b \sinh (e+f x)} \, dx\) [170]

Optimal. Leaf size=296 \[ \frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {2 d (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {2 d (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {2 d^2 \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {2 d^2 \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3} \]

[Out]

(d*x+c)^2*ln(1+b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/f/(a^2+b^2)^(1/2)-(d*x+c)^2*ln(1+b*exp(f*x+e)/(a+(a^2+b^2)^(1
/2)))/f/(a^2+b^2)^(1/2)+2*d*(d*x+c)*polylog(2,-b*exp(f*x+e)/(a-(a^2+b^2)^(1/2)))/f^2/(a^2+b^2)^(1/2)-2*d*(d*x+
c)*polylog(2,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/f^2/(a^2+b^2)^(1/2)-2*d^2*polylog(3,-b*exp(f*x+e)/(a-(a^2+b^2)
^(1/2)))/f^3/(a^2+b^2)^(1/2)+2*d^2*polylog(3,-b*exp(f*x+e)/(a+(a^2+b^2)^(1/2)))/f^3/(a^2+b^2)^(1/2)

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Rubi [A]
time = 0.47, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3403, 2296, 2221, 2611, 2320, 6724} \begin {gather*} \frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \sqrt {a^2+b^2}}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^2 \sqrt {a^2+b^2}}+\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \sqrt {a^2+b^2}}-\frac {(c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \sqrt {a^2+b^2}}-\frac {2 d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^3 \sqrt {a^2+b^2}}+\frac {2 d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^3 \sqrt {a^2+b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) - ((c + d*x)^2*Log[1 + (b*E^(
e + f*x))/(a + Sqrt[a^2 + b^2])])/(Sqrt[a^2 + b^2]*f) + (2*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a - Sqrt[
a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^2) - (2*d*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(S
qrt[a^2 + b^2]*f^2) - (2*d^2*PolyLog[3, -((b*E^(e + f*x))/(a - Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^3) + (2*
d^2*PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))])/(Sqrt[a^2 + b^2]*f^3)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {(c+d x)^2}{a+b \sinh (e+f x)} \, dx &=2 \int \frac {e^{e+f x} (c+d x)^2}{-b+2 a e^{e+f x}+b e^{2 (e+f x)}} \, dx\\ &=\frac {(2 b) \int \frac {e^{e+f x} (c+d x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\sqrt {a^2+b^2}}-\frac {(2 b) \int \frac {e^{e+f x} (c+d x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\sqrt {a^2+b^2}}\\ &=\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(2 d) \int (c+d x) \log \left (1+\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f}+\frac {(2 d) \int (c+d x) \log \left (1+\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f}\\ &=\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {\left (2 d^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f^2}+\frac {\left (2 d^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f^2}\\ &=\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\sqrt {a^2+b^2} f^3}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\sqrt {a^2+b^2} f^3}\\ &=\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {2 d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {2 d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {2 d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 233, normalized size = 0.79 \begin {gather*} \frac {(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )-(c+d x)^2 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+\frac {2 d \left (f (c+d x) \text {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-d \text {PolyLog}\left (3,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )\right )}{f^2}-\frac {2 d \left (f (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-d \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{f^2}}{\sqrt {a^2+b^2} f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a - Sqrt[a^2 + b^2])] - (c + d*x)^2*Log[1 + (b*E^(e + f*x))/(a + Sqrt[a^
2 + b^2])] + (2*d*(f*(c + d*x)*PolyLog[2, (b*E^(e + f*x))/(-a + Sqrt[a^2 + b^2])] - d*PolyLog[3, (b*E^(e + f*x
))/(-a + Sqrt[a^2 + b^2])]))/f^2 - (2*d*(f*(c + d*x)*PolyLog[2, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))] - d*
PolyLog[3, -((b*E^(e + f*x))/(a + Sqrt[a^2 + b^2]))]))/f^2)/(Sqrt[a^2 + b^2]*f)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{2}}{a +b \sinh \left (f x +e \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

c^2*log((b*e^(-f*x - e) - a - sqrt(a^2 + b^2))/(b*e^(-f*x - e) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*f) + i
ntegrate(2*d^2*x^2/(b*(e^(f*x + e) - e^(-f*x - e)) + 2*a) + 4*c*d*x/(b*(e^(f*x + e) - e^(-f*x - e)) + 2*a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (270) = 540\).
time = 0.40, size = 910, normalized size = 3.07 \begin {gather*} -\frac {2 \, b d^{2} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm polylog}\left (3, \frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}}}{b}\right ) - 2 \, b d^{2} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm polylog}\left (3, \frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}}}{b}\right ) - 2 \, {\left (b d^{2} f x + b c d f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, {\left (b d^{2} f x + b c d f\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + {\left (b c^{2} f^{2} - 2 \, b c d f \cosh \left (1\right ) + b d^{2} \cosh \left (1\right )^{2} + b d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - {\left (b c^{2} f^{2} - 2 \, b c d f \cosh \left (1\right ) + b d^{2} \cosh \left (1\right )^{2} + b d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 2 \, b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + 2 \, b c d f \cosh \left (1\right ) - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} + 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) + {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + 2 \, b c d f \cosh \left (1\right ) - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} + 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + a \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - {\left (b \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + b \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right )}{{\left (a^{2} + b^{2}\right )} f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*d^2*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1))
+ (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2))/b) - 2*b*d^2*sqrt
((a^2 + b^2)/b^2)*polylog(3, (a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) - (b*cosh(f*x
+ cosh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2))/b) - 2*(b*d^2*f*x + b*c*d*f)*sq
rt((a^2 + b^2)/b^2)*dilog((a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) + (b*cosh(f*x + c
osh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(b*d^2*f*x + b*c*d*
f)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(1) + sinh(1)) - (b*cosh(f*
x + cosh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + (b*c^2*f^2 - 2*b
*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*sqrt((a^2 + b^2)/b^2
)*log(2*b*cosh(f*x + cosh(1) + sinh(1)) + 2*b*sinh(f*x + cosh(1) + sinh(1)) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a)
- (b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*s
qrt((a^2 + b^2)/b^2)*log(2*b*cosh(f*x + cosh(1) + sinh(1)) + 2*b*sinh(f*x + cosh(1) + sinh(1)) - 2*b*sqrt((a^2
+ b^2)/b^2) + 2*a) - (b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + 2*b*c*d*f*cosh(1) - b*d^2*cosh(1)^2 - b*d^2*sinh(1)^2 +
2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x
+ cosh(1) + sinh(1)) + (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b
^2) - b)/b) + (b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + 2*b*c*d*f*cosh(1) - b*d^2*cosh(1)^2 - b*d^2*sinh(1)^2 + 2*(b*c*
d*f - b*d^2*cosh(1))*sinh(1))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(f*x + cosh(1) + sinh(1)) + a*sinh(f*x + cosh(
1) + sinh(1)) - (b*cosh(f*x + cosh(1) + sinh(1)) + b*sinh(f*x + cosh(1) + sinh(1)))*sqrt((a^2 + b^2)/b^2) - b)
/b))/((a^2 + b^2)*f^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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