\(\int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx\) [111]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=\frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-e^{e+f x}\right )}{a f^4}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \]

[Out]

(d*x+c)^3/a/f-6*d*(d*x+c)^2*ln(1+exp(f*x+e))/a/f^2-12*d^2*(d*x+c)*polylog(2,-exp(f*x+e))/a/f^3+12*d^3*polylog(
3,-exp(f*x+e))/a/f^4+(d*x+c)^3*tanh(1/2*f*x+1/2*e)/a/f

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3399, 4269, 3799, 2221, 2611, 2320, 6724} \[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}-\frac {6 d (c+d x)^2 \log \left (e^{e+f x}+1\right )}{a f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {(c+d x)^3}{a f}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-e^{e+f x}\right )}{a f^4} \]

[In]

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

[Out]

(c + d*x)^3/(a*f) - (6*d*(c + d*x)^2*Log[1 + E^(e + f*x)])/(a*f^2) - (12*d^2*(c + d*x)*PolyLog[2, -E^(e + f*x)
])/(a*f^3) + (12*d^3*PolyLog[3, -E^(e + f*x)])/(a*f^4) + ((c + d*x)^3*Tanh[e/2 + (f*x)/2])/(a*f)

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 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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[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*} \text {integral}& = \frac {\int (c+d x)^3 \csc ^2\left (\frac {1}{2} (i e+\pi )+\frac {i f x}{2}\right ) \, dx}{2 a} \\ & = \frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(3 d) \int (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a f} \\ & = \frac {(c+d x)^3}{a f}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}-\frac {(6 d) \int \frac {e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^2\right ) \int (c+d x) \log \left (1+e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^2} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^3\right ) \int \operatorname {PolyLog}\left (2,-e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^3} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\left (12 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^4} \\ & = \frac {(c+d x)^3}{a f}-\frac {6 d (c+d x)^2 \log \left (1+e^{e+f x}\right )}{a f^2}-\frac {12 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{e+f x}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,-e^{e+f x}\right )}{a f^4}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.52 \[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=\frac {2 \cosh \left (\frac {1}{2} (e+f x)\right ) \left (-\frac {6 d e^e \cosh \left (\frac {1}{2} (e+f x)\right ) \left (\frac {e^{-e} (c+d x)^3}{3 d}+\frac {\left (1+e^{-e}\right ) (c+d x)^2 \log \left (1+e^{-e-f x}\right )}{f}-\frac {2 d e^{-e} \left (1+e^e\right ) \left (f (c+d x) \operatorname {PolyLog}\left (2,-e^{-e-f x}\right )+d \operatorname {PolyLog}\left (3,-e^{-e-f x}\right )\right )}{f^3}\right )}{1+e^e}+(c+d x)^3 \text {sech}\left (\frac {e}{2}\right ) \sinh \left (\frac {f x}{2}\right )\right )}{a f (1+\cosh (e+f x))} \]

[In]

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

[Out]

(2*Cosh[(e + f*x)/2]*((-6*d*E^e*Cosh[(e + f*x)/2]*((c + d*x)^3/(3*d*E^e) + ((1 + E^(-e))*(c + d*x)^2*Log[1 + E
^(-e - f*x)])/f - (2*d*(1 + E^e)*(f*(c + d*x)*PolyLog[2, -E^(-e - f*x)] + d*PolyLog[3, -E^(-e - f*x)]))/(E^e*f
^3)))/(1 + E^e) + (c + d*x)^3*Sech[e/2]*Sinh[(f*x)/2]))/(a*f*(1 + Cosh[e + f*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(324\) vs. \(2(110)=220\).

Time = 0.22 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.78

method	result	size
risch	\(-\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f a \left (1+{\mathrm e}^{f x +e}\right )}+\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{2}}-\frac {6 d \,c^{2} \ln \left (1+{\mathrm e}^{f x +e}\right )}{a \,f^{2}}+\frac {2 d^{3} x^{3}}{a f}-\frac {4 d^{3} e^{3}}{a \,f^{4}}-\frac {6 d^{3} \ln \left (1+{\mathrm e}^{f x +e}\right ) x^{2}}{a \,f^{2}}+\frac {12 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{f x +e}\right )}{a \,f^{4}}+\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{4}}-\frac {6 d^{3} e^{2} x}{a \,f^{3}}-\frac {12 d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right ) x}{a \,f^{3}}+\frac {6 d^{2} c \,x^{2}}{a f}+\frac {6 d^{2} c \,e^{2}}{a \,f^{3}}-\frac {12 d^{2} c \ln \left (1+{\mathrm e}^{f x +e}\right ) x}{a \,f^{2}}-\frac {12 d^{2} c \operatorname {polylog}\left (2, -{\mathrm e}^{f x +e}\right )}{a \,f^{3}}+\frac {12 d^{2} c e x}{a \,f^{2}}-\frac {12 d^{2} c e \ln \left ({\mathrm e}^{f x +e}\right )}{a \,f^{3}}\)	\(325\)

[In]

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

[Out]

-2/f/a*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/(1+exp(f*x+e))+6/a/f^2*d*c^2*ln(exp(f*x+e))-6/a/f^2*d*c^2*ln(1+exp(
f*x+e))+2/a/f*d^3*x^3-4/a/f^4*d^3*e^3-6/a/f^2*d^3*ln(1+exp(f*x+e))*x^2+12*d^3*polylog(3,-exp(f*x+e))/a/f^4+6/a
/f^4*d^3*e^2*ln(exp(f*x+e))-6/a/f^3*d^3*e^2*x-12/a/f^3*d^3*polylog(2,-exp(f*x+e))*x+6/a/f*d^2*c*x^2+6/a/f^3*d^
2*c*e^2-12/a/f^2*d^2*c*ln(1+exp(f*x+e))*x-12/a/f^3*d^2*c*polylog(2,-exp(f*x+e))+12/a/f^2*d^2*c*e*x-12/a/f^3*d^
2*c*e*ln(exp(f*x+e))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (109) = 218\).

Time = 0.25 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.74 \[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=\frac {2 \, {\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3} + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} \cosh \left (f x + e\right ) - 6 \, {\left (d^{3} f x + c d^{2} f + {\left (d^{3} f x + c d^{2} f\right )} \cosh \left (f x + e\right ) + {\left (d^{3} f x + c d^{2} f\right )} \sinh \left (f x + e\right )\right )} {\rm Li}_2\left (-\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} \cosh \left (f x + e\right ) + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2}\right )} \sinh \left (f x + e\right )\right )} \log \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1\right ) + 6 \, {\left (d^{3} \cosh \left (f x + e\right ) + d^{3} \sinh \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cosh \left (f x + e\right ) - \sinh \left (f x + e\right )\right ) + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2}\right )} \sinh \left (f x + e\right )\right )}}{a f^{4} \cosh \left (f x + e\right ) + a f^{4} \sinh \left (f x + e\right ) + a f^{4}} \]

[In]

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

[Out]

2*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3 + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + d^3*e^
3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2)*cosh(f*x + e) - 6*(d^3*f*x + c*d^2*f + (d^3*f*x + c*d^2*f)*cosh(f*x + e) +
(d^3*f*x + c*d^2*f)*sinh(f*x + e))*dilog(-cosh(f*x + e) - sinh(f*x + e)) - 3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^
2*d*f^2 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2)*cosh(f*x + e) + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2)*
sinh(f*x + e))*log(cosh(f*x + e) + sinh(f*x + e) + 1) + 6*(d^3*cosh(f*x + e) + d^3*sinh(f*x + e) + d^3)*polylo
g(3, -cosh(f*x + e) - sinh(f*x + e)) + (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + 3*c^2*d*f^3*x + d^3*e^3 - 3*c*d^2*e^2*
f + 3*c^2*d*e*f^2)*sinh(f*x + e))/(a*f^4*cosh(f*x + e) + a*f^4*sinh(f*x + e) + a*f^4)

Sympy [F]

\[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=\frac {\int \frac {c^{3}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\cosh {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\cosh {\left (e + f x \right )} + 1}\, dx}{a} \]

[In]

integrate((d*x+c)**3/(a+a*cosh(f*x+e)),x)

[Out]

(Integral(c**3/(cosh(e + f*x) + 1), x) + Integral(d**3*x**3/(cosh(e + f*x) + 1), x) + Integral(3*c*d**2*x**2/(
cosh(e + f*x) + 1), x) + Integral(3*c**2*d*x/(cosh(e + f*x) + 1), x))/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (109) = 218\).

Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.95 \[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=6 \, c^{2} d {\left (\frac {x e^{\left (f x + e\right )}}{a f e^{\left (f x + e\right )} + a f} - \frac {\log \left ({\left (e^{\left (f x + e\right )} + 1\right )} e^{\left (-e\right )}\right )}{a f^{2}}\right )} + \frac {2 \, c^{3}}{{\left (a e^{\left (-f x - e\right )} + a\right )} f} - \frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2}\right )}}{a f e^{\left (f x + e\right )} + a f} - \frac {12 \, {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} c d^{2}}{a f^{3}} - \frac {6 \, {\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} d^{3}}{a f^{4}} + \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{a f^{4}} \]

[In]

integrate((d*x+c)^3/(a+a*cosh(f*x+e)),x, algorithm="maxima")

[Out]

6*c^2*d*(x*e^(f*x + e)/(a*f*e^(f*x + e) + a*f) - log((e^(f*x + e) + 1)*e^(-e))/(a*f^2)) + 2*c^3/((a*e^(-f*x -
e) + a)*f) - 2*(d^3*x^3 + 3*c*d^2*x^2)/(a*f*e^(f*x + e) + a*f) - 12*(f*x*log(e^(f*x + e) + 1) + dilog(-e^(f*x
+ e)))*c*d^2/(a*f^3) - 6*(f^2*x^2*log(e^(f*x + e) + 1) + 2*f*x*dilog(-e^(f*x + e)) - 2*polylog(3, -e^(f*x + e)
))*d^3/(a*f^4) + 2*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2)/(a*f^4)

Giac [F]

\[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{a \cosh \left (f x + e\right ) + a} \,d x } \]

[In]

integrate((d*x+c)^3/(a+a*cosh(f*x+e)),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{a+a \cosh (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+a\,\mathrm {cosh}\left (e+f\,x\right )} \,d x \]

[In]

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

[Out]

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