3.3.35 \(\int \frac {(e+f x) \sinh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [235]

Optimal. Leaf size=335 \[ \frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {a^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a^3 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2} \]

[Out]

a^2*e*x/b^3-1/2*e*x/b+1/2*a^2*f*x^2/b^3-1/4*f*x^2/b-a*(f*x+e)*cosh(d*x+c)/b^2/d+a*f*sinh(d*x+c)/b^2/d^2+1/2*(f
*x+e)*cosh(d*x+c)*sinh(d*x+c)/b/d-1/4*f*sinh(d*x+c)^2/b/d^2-a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))
/b^3/d/(a^2+b^2)^(1/2)+a^3*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d/(a^2+b^2)^(1/2)-a^3*f*polylog(
2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^2/(a^2+b^2)^(1/2)+a^3*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2))
)/b^3/d^2/(a^2+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.42, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5676, 3391, 3377, 2717, 3403, 2296, 2221, 2317, 2438} \begin {gather*} \frac {a^2 e x}{b^3}+\frac {a^2 f x^2}{2 b^3}-\frac {a^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}+\frac {a^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2 \sqrt {a^2+b^2}}-\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d \sqrt {a^2+b^2}}+\frac {a f \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d}-\frac {e x}{2 b}-\frac {f x^2}{4 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*e*x)/b^3 - (e*x)/(2*b) + (a^2*f*x^2)/(2*b^3) - (f*x^2)/(4*b) - (a*(e + f*x)*Cosh[c + d*x])/(b^2*d) - (a^3
*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) + (a^3*(e + f*x)*Log[1 + (b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*Sqrt[a^2 + b^2]*d) - (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[
a^2 + b^2]))])/(b^3*Sqrt[a^2 + b^2]*d^2) + (a^3*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*S
qrt[a^2 + b^2]*d^2) + (a*f*Sinh[c + d*x])/(b^2*d^2) + ((e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b*d) - (f*Sin
h[c + d*x]^2)/(4*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 3391

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

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 ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \sinh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}-\frac {a \int (e+f x) \sinh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {\int (e+f x) \, dx}{2 b}\\ &=-\frac {e x}{2 b}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {a^2 \int (e+f x) \, dx}{b^3}-\frac {a^3 \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {(a f) \int \cosh (c+d x) \, dx}{b^2 d}\\ &=\frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\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^3}\\ &=\frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}-\frac {\left (2 a^3\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^2 \sqrt {a^2+b^2}}+\frac {\left (2 a^3\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^2 \sqrt {a^2+b^2}}\\ &=\frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (a^3 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d}-\frac {\left (a^3 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 \sqrt {a^2+b^2} d}\\ &=\frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}+\frac {\left (a^3 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^3 \sqrt {a^2+b^2} d^2}-\frac {\left (a^3 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^3 \sqrt {a^2+b^2} d^2}\\ &=\frac {a^2 e x}{b^3}-\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}-\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}+\frac {a^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d}-\frac {a^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a^3 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 \sqrt {a^2+b^2} d^2}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {f \sinh ^2(c+d x)}{4 b d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.64, size = 307, normalized size = 0.92 \begin {gather*} \frac {-2 \left (2 a^2-b^2\right ) (c+d x) (c f-d (2 e+f x))-8 a b d (e+f x) \cosh (c+d x)-b^2 f \cosh (2 (c+d x))+\frac {8 a^3 \left (2 d e \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 c f \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+8 a b f \sinh (c+d x)+2 b^2 d (e+f x) \sinh (2 (c+d x))}{8 b^3 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*a^2 - b^2)*(c + d*x)*(c*f - d*(2*e + f*x)) - 8*a*b*d*(e + f*x)*Cosh[c + d*x] - b^2*f*Cosh[2*(c + d*x)]
+ (8*a^3*(2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*c*f*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^
2]] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sq
rt[a^2 + b^2])] - f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*PolyLog[2, -((b*E^(c + d*x))/(a + S
qrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*f*Sinh[c + d*x] + 2*b^2*d*(e + f*x)*Sinh[2*(c + d*x)])/(8*b^3*d^2)

________________________________________________________________________________________

Maple [A]
time = 1.59, size = 589, normalized size = 1.76

method	result	size
risch	\(\frac {a^{2} f \,x^{2}}{2 b^{3}}-\frac {f \,x^{2}}{4 b}+\frac {a^{2} e x}{b^{3}}-\frac {e x}{2 b}+\frac {\left (2 d x f +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 d^{2} b}-\frac {a \left (d x f +d e -f \right ) {\mathrm e}^{d x +c}}{2 b^{2} d^{2}}-\frac {a \left (d x f +d e +f \right ) {\mathrm e}^{-d x -c}}{2 b^{2} d^{2}}-\frac {\left (2 d x f +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 d^{2} b}+\frac {2 a^{3} e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d \,b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}-\frac {a^{3} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}+\frac {a^{3} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}-\frac {2 a^{3} f c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{3} \sqrt {a^{2}+b^{2}}}\)	\(589\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*f*x^2/b^3-1/4*f*x^2/b+a^2*e*x/b^3-1/2*e*x/b+1/16*(2*d*f*x+2*d*e-f)/d^2/b*exp(2*d*x+2*c)-1/2*a*(d*f*x+d
*e-f)/b^2/d^2*exp(d*x+c)-1/2*a*(d*f*x+d*e+f)/b^2/d^2*exp(-d*x-c)-1/16*(2*d*f*x+2*d*e+f)/d^2/b*exp(-2*d*x-2*c)+
2/d*a^3/b^3*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^3/b^3*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^3/b^3*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^3/b^3*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^3/b^3*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^3/b^3*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^3/b^
3*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^3/b^3*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))

________________________________________________________________________________________

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

[Out]

-1/16*(32*a^3*integrate(x*e^(d*x + c)/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x) - (4*(2*a^2*d^2*e^
(2*c) - b^2*d^2*e^(2*c))*x^2 + (2*b^2*d*x*e^(4*c) - b^2*e^(4*c))*e^(2*d*x) - 8*(a*b*d*x*e^(3*c) - a*b*e^(3*c))
*e^(d*x) - 8*(a*b*d*x*e^c + a*b*e^c)*e^(-d*x) - (2*b^2*d*x + b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^2))*f - 1/8*(8*a
^3*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^3*d)
+ (4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) - 4*(2*a^2 - b^2)*(d*x + c)/(b^3*d) + (4*a*e^(-d*x - c) + b*e
^(-2*d*x - 2*c))/(b^2*d))*e

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2068 vs. \(2 (309) = 618\).
time = 0.39, size = 2068, normalized size = 6.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((2*(a^2*b^2 + b^4)*d*f*x + 2*(a^2*b^2 + b^4)*d*cosh(1) + 2*(a^2*b^2 + b^4)*d*sinh(1) - (a^2*b^2 + b^4)*f
)*cosh(d*x + c)^4 + (2*(a^2*b^2 + b^4)*d*f*x + 2*(a^2*b^2 + b^4)*d*cosh(1) + 2*(a^2*b^2 + b^4)*d*sinh(1) - (a^
2*b^2 + b^4)*f)*sinh(d*x + c)^4 - 2*(a^2*b^2 + b^4)*d*f*x - 8*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(
1) + (a^3*b + a*b^3)*d*sinh(1) - (a^3*b + a*b^3)*f)*cosh(d*x + c)^3 - 4*(2*(a^3*b + a*b^3)*d*f*x + 2*(a^3*b +
a*b^3)*d*cosh(1) + 2*(a^3*b + a*b^3)*d*sinh(1) - 2*(a^3*b + a*b^3)*f - (2*(a^2*b^2 + b^4)*d*f*x + 2*(a^2*b^2 +
b^4)*d*cosh(1) + 2*(a^2*b^2 + b^4)*d*sinh(1) - (a^2*b^2 + b^4)*f)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(a^2*b^2
+ b^4)*d*cosh(1) + 4*((2*a^4 + a^2*b^2 - b^4)*d^2*f*x^2 + 2*(2*a^4 + a^2*b^2 - b^4)*d^2*x*cosh(1) + 2*(2*a^4
+ a^2*b^2 - b^4)*d^2*x*sinh(1))*cosh(d*x + c)^2 - 2*(a^2*b^2 + b^4)*d*sinh(1) + 2*(2*(2*a^4 + a^2*b^2 - b^4)*d
^2*f*x^2 + 4*(2*a^4 + a^2*b^2 - b^4)*d^2*x*cosh(1) + 4*(2*a^4 + a^2*b^2 - b^4)*d^2*x*sinh(1) + 3*(2*(a^2*b^2 +
b^4)*d*f*x + 2*(a^2*b^2 + b^4)*d*cosh(1) + 2*(a^2*b^2 + b^4)*d*sinh(1) - (a^2*b^2 + b^4)*f)*cosh(d*x + c)^2 -
12*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1) - (a^3*b + a*b^3)*f)*cosh(d
*x + c))*sinh(d*x + c)^2 - 16*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a^3*b*f*sinh(
d*x + c)^2)*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) + 16*(a^3*b*f*cosh(d*x + c)^2 + 2*a^3*b*f*cosh(d*x + c)*sinh(d*x + c) + a
^3*b*f*sinh(d*x + c)^2)*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) - 16*((a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*
x + c)^2 + 2*(a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*c*f - a^3*b*
d*cosh(1) - a^3*b*d*sinh(1))*sinh(d*x + c)^2)*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) + 16*((a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x + c)^2 + 2*(
a^3*b*c*f - a^3*b*d*cosh(1) - a^3*b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*c*f - a^3*b*d*cosh(1) - a^
3*b*d*sinh(1))*sinh(d*x + c)^2)*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) - 16*((a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x +
c)*sinh(d*x + c) + (a^3*b*d*f*x + a^3*b*c*f)*sinh(d*x + c)^2)*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) + 16*((a^3*b*d*f*x + a^3*b*c
*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f*x + a^3*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f*x + a^3*b*c*f)*sinh
(d*x + c)^2)*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) - (a^2*b^2 + b^4)*f - 8*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) +
(a^3*b + a*b^3)*d*sinh(1) + (a^3*b + a*b^3)*f)*cosh(d*x + c) - 4*(2*(a^3*b + a*b^3)*d*f*x - (2*(a^2*b^2 + b^4)
*d*f*x + 2*(a^2*b^2 + b^4)*d*cosh(1) + 2*(a^2*b^2 + b^4)*d*sinh(1) - (a^2*b^2 + b^4)*f)*cosh(d*x + c)^3 + 2*(a
^3*b + a*b^3)*d*cosh(1) + 6*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1) - (
a^3*b + a*b^3)*f)*cosh(d*x + c)^2 + 2*(a^3*b + a*b^3)*d*sinh(1) + 2*(a^3*b + a*b^3)*f - 2*((2*a^4 + a^2*b^2 -
b^4)*d^2*f*x^2 + 2*(2*a^4 + a^2*b^2 - b^4)*d^2*x*cosh(1) + 2*(2*a^4 + a^2*b^2 - b^4)*d^2*x*sinh(1))*cosh(d*x +
c))*sinh(d*x + c))/((a^2*b^3 + b^5)*d^2*cosh(d*x + c)^2 + 2*(a^2*b^3 + b^5)*d^2*cosh(d*x + c)*sinh(d*x + c) +
(a^2*b^3 + b^5)*d^2*sinh(d*x + c)^2)

________________________________________________________________________________________

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)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^3\,\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)^3*(e + f*x))/(a + b*sinh(c + d*x)),x)

[Out]

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

________________________________________________________________________________________