3.4.40 \(\int \frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [340]

Optimal. Leaf size=327 \[ \frac {a^2 e x}{b^3}+\frac {e x}{2 b}+\frac {a^2 f x^2}{2 b^3}+\frac {f x^2}{4 b}-\frac {a (e+f x) \cosh (c+d x)}{b^2 d}-\frac {f \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a \sqrt {a^2+b^2} f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a \sqrt {a^2+b^2} f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a f \sinh (c+d x)}{b^2 d^2}+\frac {(e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d} \]

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

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Rubi [A]
time = 0.39, antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5698, 3391, 5684, 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 f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a f \sqrt {a^2+b^2} \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {a \sqrt {a^2+b^2} (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}+\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 \cosh ^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)*Cosh[c + d*x]^2*Sinh[c + d*x])/(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) - (f*C
osh[c + d*x]^2)/(4*b*d^2) - (a*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^3*
d) + (a*Sqrt[a^2 + b^2]*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) - (a*Sqrt[a^2 + b^2]
*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (a*Sqrt[a^2 + b^2]*f*PolyLog[2, -((b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*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)

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 5684

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[-a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*
x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 5698

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*(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] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 3.14, size = 1551, normalized size = 4.74 \begin {gather*} \frac {2 b^2 e \left (\frac {c}{d}+x-\frac {2 a \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2} d}\right )+b^2 f \left (x^2+\frac {2 i a \pi \tanh ^{-1}\left (\frac {-b+a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} d^2}+\frac {2 a \left (2 \left (-i c+\text {ArcCos}\left (-\frac {i a}{b}\right )\right ) \tanh ^{-1}\left (\frac {(a+i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )+(-2 i c+\pi -2 i d x) \tanh ^{-1}\left (\frac {(a-i b) \tan \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )-\left (\text {ArcCos}\left (-\frac {i a}{b}\right )+2 i \tanh ^{-1}\left (\frac {(a+i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {(i a+b) \left (a+i \left (b+\sqrt {-a^2-b^2}\right )\right ) \left (-i+\cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{b \left (i a+b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {i a}{b}\right )-2 i \tanh ^{-1}\left (\frac {(a+i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (\frac {(i a+b) \left (i a-b+\sqrt {-a^2-b^2}\right ) \left (i+\cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{b \left (a-i b+\sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )+\left (\text {ArcCos}\left (-\frac {i a}{b}\right )-2 i \tanh ^{-1}\left (\frac {(a+i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )-2 i \tanh ^{-1}\left (\frac {(a-i b) \tan \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} \sqrt {-a^2-b^2} e^{-\frac {c}{2}-\frac {d x}{2}}}{\sqrt {2} \sqrt {-i b} \sqrt {a+b \sinh (c+d x)}}\right )+\left (\text {ArcCos}\left (-\frac {i a}{b}\right )+2 i \left (\tanh ^{-1}\left (\frac {(a+i b) \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )+\tanh ^{-1}\left (\frac {(a-i b) \tan \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )}{\sqrt {-a^2-b^2}}\right )\right )\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {-a^2-b^2} e^{\frac {1}{2} (c+d x)}}{\sqrt {2} \sqrt {-i b} \sqrt {a+b \sinh (c+d x)}}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (i a+\sqrt {-a^2-b^2}\right ) \left (i a+b-i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{b \left (i a+b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (a+i \sqrt {-a^2-b^2}\right ) \left (-a+i b+\sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}{b \left (i a+b+i \sqrt {-a^2-b^2} \cot \left (\frac {1}{4} (2 i c+\pi +2 i d x)\right )\right )}\right )\right )\right )}{\sqrt {-a^2-b^2} d^2}\right )+\frac {2 e \left (\left (4 a^2+b^2\right ) (c+d x)-\frac {2 a \left (4 a^2+3 b^2\right ) \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}-4 a b \cosh (c+d x)+b^2 \sinh (2 (c+d x))\right )}{d}+\frac {f \left (\left (4 a^2+b^2\right ) (-c+d x) (c+d x)-8 a b d x \cosh (c+d x)-b^2 \cosh (2 (c+d x))-\frac {2 a \left (4 a^2+3 b^2\right ) \left (2 c \tanh ^{-1}\left (\frac {a+b \cosh (c+d x)+b \sinh (c+d x)}{\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (1+\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a-\sqrt {a^2+b^2}}\right )-(c+d x) \log \left (1+\frac {b (\cosh (c+d x)+\sinh (c+d x))}{a+\sqrt {a^2+b^2}}\right )+\text {PolyLog}\left (2,\frac {b (\cosh (c+d x)+\sinh (c+d x))}{-a+\sqrt {a^2+b^2}}\right )-\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}}+8 a b \sinh (c+d x)+2 b^2 d x \sinh (2 (c+d x))\right )}{d^2}}{8 b^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*b^2*e*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d)) + b^2*f*(x^
2 + ((2*I)*a*Pi*ArcTanh[(-b + a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d^2) + (2*a*(2*((-I)*c +
ArcCos[((-I)*a)/b])*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] + ((-2*I)*c + Pi
- (2*I)*d*x)*ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (ArcCos[((-I)*a)/b] + (
2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((I*a + b)*(a + I*(b + Sqrt[
-a^2 - b^2]))*(-I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (
2*I)*d*x)/4]))] - (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2
- b^2]])*Log[((I*a + b)*(I*a - b + Sqrt[-a^2 - b^2])*(I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(a - I*b + Sq
rt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] + (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot[((2*I
)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (2*I)*ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt
[-a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[-a^2 - b^2]*E^(-1/2*c - (d*x)/2))/(Sqrt[2]*Sqrt[(-I)*b]*Sqrt[a + b*Sinh[
c + d*x]]))] + (ArcCos[((-I)*a)/b] + (2*I)*(ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 -
b^2]] + ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]]))*Log[((-1)^(1/4)*Sqrt[-a^2 -
b^2]*E^((c + d*x)/2))/(Sqrt[2]*Sqrt[(-I)*b]*Sqrt[a + b*Sinh[c + d*x]])] + I*(PolyLog[2, ((I*a + Sqrt[-a^2 - b^
2])*(I*a + b - I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2
*I)*c + Pi + (2*I)*d*x)/4]))] - PolyLog[2, ((a + I*Sqrt[-a^2 - b^2])*(-a + I*b + Sqrt[-a^2 - b^2]*Cot[((2*I)*c
+ Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))])))/(Sqrt[-a^2 -
b^2]*d^2)) + (2*e*((4*a^2 + b^2)*(c + d*x) - (2*a*(4*a^2 + 3*b^2)*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 -
b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + b^2*Sinh[2*(c + d*x)]))/d + (f*((4*a^2 + b^2)*(-c + d*x)*(c +
d*x) - 8*a*b*d*x*Cosh[c + d*x] - b^2*Cosh[2*(c + d*x)] - (2*a*(4*a^2 + 3*b^2)*(2*c*ArcTanh[(a + b*Cosh[c + d*
x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 +
b^2])] - (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + PolyLog[2, (b*(Cosh[c
+ d*x] + Sinh[c + d*x]))/(-a + Sqrt[a^2 + b^2])] - PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt
[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + 8*a*b*Sinh[c + d*x] + 2*b^2*d*x*Sinh[2*(c + d*x)]))/d^2)/(8*b^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1011\) vs. \(2(297)=594\).
time = 1.58, size = 1012, normalized size = 3.09

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 {2 a e \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d b \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 {a 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 \sqrt {a^{2}+b^{2}}}-\frac {a 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 \sqrt {a^{2}+b^{2}}}+\frac {a 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 \sqrt {a^{2}+b^{2}}}+\frac {a 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 \sqrt {a^{2}+b^{2}}}-\frac {a 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 \sqrt {a^{2}+b^{2}}}+\frac {a 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 \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}}}-\frac {2 a f c \arctanh \left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{d^{2} b \sqrt {a^{2}+b^{2}}}\)	\(1012\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/(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))+2/d*a/b*e/(a^2+b^2)^(1/2)*arct
anh(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)))-1/d*a/b*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/b*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/b*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/b*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/b*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/b*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))-2/d^
2*a/b*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)*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1484 vs. \(2 (301) = 602\).
time = 0.42, size = 1484, normalized size = 4.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*b^2*d*f*x - (2*b^2*d*f*x + 2*b^2*d*cosh(1) + 2*b^2*d*sinh(1) - b^2*f)*cosh(d*x + c)^4 - (2*b^2*d*f*x
+ 2*b^2*d*cosh(1) + 2*b^2*d*sinh(1) - b^2*f)*sinh(d*x + c)^4 + 2*b^2*d*cosh(1) + 8*(a*b*d*f*x + a*b*d*cosh(1)
+ a*b*d*sinh(1) - a*b*f)*cosh(d*x + c)^3 + 2*b^2*d*sinh(1) + 4*(2*a*b*d*f*x + 2*a*b*d*cosh(1) + 2*a*b*d*sinh(1
) - 2*a*b*f - (2*b^2*d*f*x + 2*b^2*d*cosh(1) + 2*b^2*d*sinh(1) - b^2*f)*cosh(d*x + c))*sinh(d*x + c)^3 + b^2*f
- 4*((2*a^2 + b^2)*d^2*f*x^2 + 2*(2*a^2 + b^2)*d^2*x*cosh(1) + 2*(2*a^2 + b^2)*d^2*x*sinh(1))*cosh(d*x + c)^2
- 2*(2*(2*a^2 + b^2)*d^2*f*x^2 + 4*(2*a^2 + b^2)*d^2*x*cosh(1) + 4*(2*a^2 + b^2)*d^2*x*sinh(1) + 3*(2*b^2*d*f
*x + 2*b^2*d*cosh(1) + 2*b^2*d*sinh(1) - b^2*f)*cosh(d*x + c)^2 - 12*(a*b*d*f*x + a*b*d*cosh(1) + a*b*d*sinh(1
) - a*b*f)*cosh(d*x + c))*sinh(d*x + c)^2 + 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d*x + c)*sinh(d*x + c) +
a*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*s
inh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 16*(a*b*f*cosh(d*x + c)^2 + 2*a*b*f*cosh(d*x + c)*sinh(d*x +
c) + a*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*b*c*f - a*b*d*cosh(1) - a*b*d*sinh(1))*cosh(d*x
+ c)^2 + 2*(a*b*c*f - a*b*d*cosh(1) - a*b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a*b*c*f - a*b*d*cosh(1) -
a*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*b*c*f - a*b*d*cosh(1) - a*b*d*sinh(1))*cosh(d*x + c)^2 + 2*(a*b*c*f - a*b*d*cos
h(1) - a*b*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a*b*c*f - a*b*d*cosh(1) - a*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*b
*d*f*x + a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*f*x + a*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d*f*x + a*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*b*d*f*x + a*b*c*f)*cosh(d*x + c)^2 + 2*(a*b*d*f*x + a*b*c*f)*c
osh(d*x + c)*sinh(d*x + c) + (a*b*d*f*x + a*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) + 8*(a*b*d*f*x + a*b*d
*cosh(1) + a*b*d*sinh(1) + a*b*f)*cosh(d*x + c) + 4*(2*a*b*d*f*x + 2*a*b*d*cosh(1) - (2*b^2*d*f*x + 2*b^2*d*co
sh(1) + 2*b^2*d*sinh(1) - b^2*f)*cosh(d*x + c)^3 + 2*a*b*d*sinh(1) + 2*a*b*f + 6*(a*b*d*f*x + a*b*d*cosh(1) +
a*b*d*sinh(1) - a*b*f)*cosh(d*x + c)^2 - 2*((2*a^2 + b^2)*d^2*f*x^2 + 2*(2*a^2 + b^2)*d^2*x*cosh(1) + 2*(2*a^2
+ b^2)*d^2*x*sinh(1))*cosh(d*x + c))*sinh(d*x + c))/(b^3*d^2*cosh(d*x + c)^2 + 2*b^3*d^2*cosh(d*x + c)*sinh(d
*x + c) + b^3*d^2*sinh(d*x + c)^2)

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

[Out]

integrate((f*x + e)*cosh(d*x + c)^2*sinh(d*x + c)/(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 {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\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((cosh(c + d*x)^2*sinh(c + d*x)*(e + f*x))/(a + b*sinh(c + d*x)),x)

[Out]

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

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