3.3.30 \(\int \frac {(e+f x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [230]

Optimal. Leaf size=264 \[ -\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}+\frac {(e+f x) \cosh (c+d x)}{b d}+\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}-\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d}+\frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^2}-\frac {a^2 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \sqrt {a^2+b^2} d^2}-\frac {f \sinh (c+d x)}{b d^2} \]

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5676, 3377, 2717, 3403, 2296, 2221, 2317, 2438} \begin {gather*} \frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2 \sqrt {a^2+b^2}}-\frac {a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2 \sqrt {a^2+b^2}}+\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^2 d \sqrt {a^2+b^2}}-\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}-\frac {f \sinh (c+d x)}{b d^2}+\frac {(e+f x) \cosh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, 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 5676

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

Rubi steps

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

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Mathematica [A]
time = 1.41, size = 299, normalized size = 1.13 \begin {gather*} \frac {a (c+d x) (c f-d (2 e+f x))+2 b d (e+f x) \cosh (c+d x)+\frac {2 a^2 \left (-2 d e \tanh ^{-1}\left (\frac {a+b \cosh (c+d x)+b \sinh (c+d x)}{\sqrt {a^2+b^2}}\right )+2 c f \tanh ^{-1}\left (\frac {a+b \cosh (c+d x)+b \sinh (c+d x)}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,\frac {b (\cosh (c+d x)+\sinh (c+d x))}{-a+\sqrt {a^2+b^2}}\right )-f \text {PolyLog}\left (2,-\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-2 b f \sinh (c+d x)}{2 b^2 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x)*(c*f - d*(2*e + f*x)) + 2*b*d*(e + f*x)*Cosh[c + d*x] + (2*a^2*(-2*d*e*ArcTanh[(a + b*Cosh[c + d*
x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + 2*c*f*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]
] + f*(c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - f*(c + d*x)*Log[1 + (b*(C
osh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + f*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a +
Sqrt[a^2 + b^2])] - f*PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b
^2] - 2*b*f*Sinh[c + d*x])/(2*b^2*d^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(242)=484\).
time = 1.61, size = 510, normalized size = 1.93

method	result	size
risch	\(-\frac {a f \,x^{2}}{2 b^{2}}-\frac {a e x}{b^{2}}+\frac {\left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 d^{2} b}+\frac {\left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 d^{2} b}-\frac {2 a^{2} e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{2} \sqrt {a^{2}+b^{2}}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2} \sqrt {a^{2}+b^{2}}}+\frac {a^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2} \sqrt {a^{2}+b^{2}}}-\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2} \sqrt {a^{2}+b^{2}}}-\frac {a^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2} \sqrt {a^{2}+b^{2}}}+\frac {a^{2} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2} \sqrt {a^{2}+b^{2}}}-\frac {a^{2} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2} \sqrt {a^{2}+b^{2}}}+\frac {2 a^{2} f c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2} \sqrt {a^{2}+b^{2}}}\)	\(510\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*f*x^2/b^2-a*e*x/b^2+1/2*(d*f*x+d*e-f)/d^2/b*exp(d*x+c)+1/2*(d*f*x+d*e+f)/d^2/b*exp(-d*x-c)-2/d*a^2/b^2*
e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d*a^2/b^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d
*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/d^2*a^2/b^2*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(
1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d*a^2/b^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)
^(1/2)))*x-1/d^2*a^2/b^2*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/d^2*a^
2/b^2*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d^2*a^2/b^2*f/(a^2+b^2
)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/d^2*a^2/b^2*f*c/(a^2+b^2)^(1/2)*arctanh(
1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))

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

[Out]

1/2*(4*a^2*integrate(x*e^(d*x + c)/(b^3*e^(2*d*x + 2*c) + 2*a*b^2*e^(d*x + c) - b^3), x) - (a*d^2*x^2*e^c - (b
*d*x*e^(2*c) - b*e^(2*c))*e^(d*x) - (b*d*x + b)*e^(-d*x))*e^(-c)/(b^2*d^2))*f + 1/2*(2*a^2*log((b*e^(-d*x - c)
- a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^2*d) - 2*(d*x + c)*a/(b^2*d
) + e^(d*x + c)/(b*d) + e^(-d*x - c)/(b*d))*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1078 vs. \(2 (244) = 488\).
time = 0.37, size = 1078, normalized size = 4.08 \begin {gather*} \frac {{\left (a^{2} b + b^{3}\right )} d f x + {\left (a^{2} b + b^{3}\right )} d \cosh \left (1\right ) + {\left ({\left (a^{2} b + b^{3}\right )} d f x + {\left (a^{2} b + b^{3}\right )} d \cosh \left (1\right ) + {\left (a^{2} b + b^{3}\right )} d \sinh \left (1\right ) - {\left (a^{2} b + b^{3}\right )} f\right )} \cosh \left (d x + c\right )^{2} + {\left (a^{2} b + b^{3}\right )} d \sinh \left (1\right ) + {\left ({\left (a^{2} b + b^{3}\right )} d f x + {\left (a^{2} b + b^{3}\right )} d \cosh \left (1\right ) + {\left (a^{2} b + b^{3}\right )} d \sinh \left (1\right ) - {\left (a^{2} b + b^{3}\right )} f\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} b f \cosh \left (d x + c\right ) + a^{2} b f \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) - 2 \, {\left (a^{2} b f \cosh \left (d x + c\right ) + a^{2} b f \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} {\rm Li}_2\left (\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b} + 1\right ) + 2 \, {\left ({\left (a^{2} b c f - a^{2} b d \cosh \left (1\right ) - a^{2} b d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left (a^{2} b c f - a^{2} b d \cosh \left (1\right ) - a^{2} b d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - 2 \, {\left ({\left (a^{2} b c f - a^{2} b d \cosh \left (1\right ) - a^{2} b d \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) + {\left (a^{2} b c f - a^{2} b d \cosh \left (1\right ) - a^{2} b d \sinh \left (1\right )\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (2 \, b \cosh \left (d x + c\right ) + 2 \, b \sinh \left (d x + c\right ) - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + 2 \, {\left ({\left (a^{2} b d f x + a^{2} b c f\right )} \cosh \left (d x + c\right ) + {\left (a^{2} b d f x + a^{2} b c f\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) + {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) - 2 \, {\left ({\left (a^{2} b d f x + a^{2} b c f\right )} \cosh \left (d x + c\right ) + {\left (a^{2} b d f x + a^{2} b c f\right )} \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} \log \left (-\frac {a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right ) - {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} - b}{b}\right ) + {\left (a^{2} b + b^{3}\right )} f - {\left ({\left (a^{3} + a b^{2}\right )} d^{2} f x^{2} + 2 \, {\left (a^{3} + a b^{2}\right )} d^{2} x \cosh \left (1\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} d^{2} x \sinh \left (1\right )\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{3} + a b^{2}\right )} d^{2} f x^{2} + 2 \, {\left (a^{3} + a b^{2}\right )} d^{2} x \cosh \left (1\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} d^{2} x \sinh \left (1\right ) - 2 \, {\left ({\left (a^{2} b + b^{3}\right )} d f x + {\left (a^{2} b + b^{3}\right )} d \cosh \left (1\right ) + {\left (a^{2} b + b^{3}\right )} d \sinh \left (1\right ) - {\left (a^{2} b + b^{3}\right )} f\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{2} + b^{4}\right )} d^{2} \cosh \left (d x + c\right ) + {\left (a^{2} b^{2} + b^{4}\right )} d^{2} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b +
b^3)*d*sinh(1) - (a^2*b + b^3)*f)*cosh(d*x + c)^2 + (a^2*b + b^3)*d*sinh(1) + ((a^2*b + b^3)*d*f*x + (a^2*b +
b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) - (a^2*b + b^3)*f)*sinh(d*x + c)^2 + 2*(a^2*b*f*cosh(d*x + c) + a^2*
b*f*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(
d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*(a^2*b*f*cosh(d*x + c) + a^2*b*f*sinh(d*x + c))*sqrt((a^2 + b^
2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) -
b)/b + 1) + 2*((a^2*b*c*f - a^2*b*d*cosh(1) - a^2*b*d*sinh(1))*cosh(d*x + c) + (a^2*b*c*f - a^2*b*d*cosh(1) -
a^2*b*d*sinh(1))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a
^2 + b^2)/b^2) + 2*a) - 2*((a^2*b*c*f - a^2*b*d*cosh(1) - a^2*b*d*sinh(1))*cosh(d*x + c) + (a^2*b*c*f - a^2*b*
d*cosh(1) - a^2*b*d*sinh(1))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) -
2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*((a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c) + (a^2*b*d*f*x + a^2*b*c*f)*sinh
(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))
*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*((a^2*b*d*f*x + a^2*b*c*f)*cosh(d*x + c) + (a^2*b*d*f*x + a^2*b*c*f)*sinh(d
*x + c))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*s
qrt((a^2 + b^2)/b^2) - b)/b) + (a^2*b + b^3)*f - ((a^3 + a*b^2)*d^2*f*x^2 + 2*(a^3 + a*b^2)*d^2*x*cosh(1) + 2*
(a^3 + a*b^2)*d^2*x*sinh(1))*cosh(d*x + c) - ((a^3 + a*b^2)*d^2*f*x^2 + 2*(a^3 + a*b^2)*d^2*x*cosh(1) + 2*(a^3
+ a*b^2)*d^2*x*sinh(1) - 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*cosh(1) + (a^2*b + b^3)*d*sinh(1) - (a^2*b
+ b^3)*f)*cosh(d*x + c))*sinh(d*x + c))/((a^2*b^2 + b^4)*d^2*cosh(d*x + c) + (a^2*b^2 + b^4)*d^2*sinh(d*x + c)
)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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