3.1.12 \(\int (c+d x)^2 \tanh ^3(e+f x) \, dx\) [12]

Optimal. Leaf size=157 \[ \frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^2}-\frac {d^2 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^3}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f} \]

[Out]

c*d*x/f+1/2*d^2*x^2/f-1/3*(d*x+c)^3/d+(d*x+c)^2*ln(1+exp(2*f*x+2*e))/f+d^2*ln(cosh(f*x+e))/f^3+d*(d*x+c)*polyl
og(2,-exp(2*f*x+2*e))/f^2-1/2*d^2*polylog(3,-exp(2*f*x+2*e))/f^3-d*(d*x+c)*tanh(f*x+e)/f^2-1/2*(d*x+c)^2*tanh(
f*x+e)^2/f

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Rubi [A]
time = 0.17, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3801, 3556, 3799, 2221, 2611, 2320, 6724} \begin {gather*} \frac {d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {d (c+d x) \tanh (e+f x)}{f^2}+\frac {(c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}+\frac {c d x}{f}-\frac {(c+d x)^3}{3 d}-\frac {d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d^2 x^2}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Tanh[e + f*x]^3,x]

[Out]

(c*d*x)/f + (d^2*x^2)/(2*f) - (c + d*x)^3/(3*d) + ((c + d*x)^2*Log[1 + E^(2*(e + f*x))])/f + (d^2*Log[Cosh[e +
f*x]])/f^3 + (d*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))])/f^2 - (d^2*PolyLog[3, -E^(2*(e + f*x))])/(2*f^3) - (d
*(c + d*x)*Tanh[e + f*x])/f^2 - ((c + d*x)^2*Tanh[e + f*x]^2)/(2*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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && 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*} \int (c+d x)^2 \tanh ^3(e+f x) \, dx &=-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}+\frac {d \int (c+d x) \tanh ^2(e+f x) \, dx}{f}+\int (c+d x)^2 \tanh (e+f x) \, dx\\ &=-\frac {(c+d x)^3}{3 d}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}+2 \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx+\frac {d^2 \int \tanh (e+f x) \, dx}{f^2}+\frac {d \int (c+d x) \, dx}{f}\\ &=\frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {(2 d) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {d^2 \int \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}-\frac {d^2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}\\ &=\frac {c d x}{f}+\frac {d^2 x^2}{2 f}-\frac {(c+d x)^3}{3 d}+\frac {(c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f}+\frac {d^2 \log (\cosh (e+f x))}{f^3}+\frac {d (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^2}-\frac {d^2 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^3}-\frac {d (c+d x) \tanh (e+f x)}{f^2}-\frac {(c+d x)^2 \tanh ^2(e+f x)}{2 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.31, size = 463, normalized size = 2.95 \begin {gather*} \frac {d^2 e^{-e} \left (-2 f^2 x^2 \left (2 e^{2 e} f x-3 \left (1+e^{2 e}\right ) \log \left (1+e^{2 (e+f x)}\right )\right )+6 \left (1+e^{2 e}\right ) f x \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )-3 \left (1+e^{2 e}\right ) \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )\right ) \text {sech}(e)}{12 f^3}+\frac {(c+d x)^2 \text {sech}^2(e+f x)}{2 f}+\frac {d^2 \text {sech}(e) (\cosh (e) \log (\cosh (e) \cosh (f x)+\sinh (e) \sinh (f x))-f x \sinh (e))}{f^3 \left (\cosh ^2(e)-\sinh ^2(e)\right )}+\frac {c^2 \text {sech}(e) (\cosh (e) \log (\cosh (e) \cosh (f x)+\sinh (e) \sinh (f x))-f x \sinh (e))}{f \left (\cosh ^2(e)-\sinh ^2(e)\right )}+\frac {c d \text {csch}(e) \left (e^{-\tanh ^{-1}(\coth (e))} f^2 x^2-\frac {i \coth (e) \left (-f x \left (-\pi +2 i \tanh ^{-1}(\coth (e))\right )-\pi \log \left (1+e^{2 f x}\right )-2 \left (i f x+i \tanh ^{-1}(\coth (e))\right ) \log \left (1-e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )+\pi \log (\cosh (f x))+2 i \tanh ^{-1}(\coth (e)) \log \left (i \sinh \left (f x+\tanh ^{-1}(\coth (e))\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (i f x+i \tanh ^{-1}(\coth (e))\right )}\right )\right )}{\sqrt {1-\coth ^2(e)}}\right ) \text {sech}(e)}{f^2 \sqrt {\text {csch}^2(e) \left (-\cosh ^2(e)+\sinh ^2(e)\right )}}+\frac {\text {sech}(e) \text {sech}(e+f x) \left (-c d \sinh (f x)-d^2 x \sinh (f x)\right )}{f^2}+\frac {1}{3} x \left (3 c^2+3 c d x+d^2 x^2\right ) \tanh (e) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2*Tanh[e + f*x]^3,x]

[Out]

(d^2*(-2*f^2*x^2*(2*E^(2*e)*f*x - 3*(1 + E^(2*e))*Log[1 + E^(2*(e + f*x))]) + 6*(1 + E^(2*e))*f*x*PolyLog[2, -
E^(2*(e + f*x))] - 3*(1 + E^(2*e))*PolyLog[3, -E^(2*(e + f*x))])*Sech[e])/(12*E^e*f^3) + ((c + d*x)^2*Sech[e +
f*x]^2)/(2*f) + (d^2*Sech[e]*(Cosh[e]*Log[Cosh[e]*Cosh[f*x] + Sinh[e]*Sinh[f*x]] - f*x*Sinh[e]))/(f^3*(Cosh[e
]^2 - Sinh[e]^2)) + (c^2*Sech[e]*(Cosh[e]*Log[Cosh[e]*Cosh[f*x] + Sinh[e]*Sinh[f*x]] - f*x*Sinh[e]))/(f*(Cosh[
e]^2 - Sinh[e]^2)) + (c*d*Csch[e]*((f^2*x^2)/E^ArcTanh[Coth[e]] - (I*Coth[e]*(-(f*x*(-Pi + (2*I)*ArcTanh[Coth[
e]])) - Pi*Log[1 + E^(2*f*x)] - 2*(I*f*x + I*ArcTanh[Coth[e]])*Log[1 - E^((2*I)*(I*f*x + I*ArcTanh[Coth[e]]))]
+ Pi*Log[Cosh[f*x]] + (2*I)*ArcTanh[Coth[e]]*Log[I*Sinh[f*x + ArcTanh[Coth[e]]]] + I*PolyLog[2, E^((2*I)*(I*f
*x + I*ArcTanh[Coth[e]]))]))/Sqrt[1 - Coth[e]^2])*Sech[e])/(f^2*Sqrt[Csch[e]^2*(-Cosh[e]^2 + Sinh[e]^2)]) + (S
ech[e]*Sech[e + f*x]*(-(c*d*Sinh[f*x]) - d^2*x*Sinh[f*x]))/f^2 + (x*(3*c^2 + 3*c*d*x + d^2*x^2)*Tanh[e])/3

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(149)=298\).
time = 1.83, size = 375, normalized size = 2.39

method	result	size
risch	\(-\frac {d^{2} x^{3}}{3}-d c \,x^{2}+c^{2} x +\frac {c^{3}}{3 d}+\frac {2 d^{2} f \,x^{2} {\mathrm e}^{2 f x +2 e}+4 c d f x \,{\mathrm e}^{2 f x +2 e}+2 c^{2} f \,{\mathrm e}^{2 f x +2 e}+2 d^{2} x \,{\mathrm e}^{2 f x +2 e}+2 c d \,{\mathrm e}^{2 f x +2 e}+2 d^{2} x +2 c d}{f^{2} \left (1+{\mathrm e}^{2 f x +2 e}\right )^{2}}+\frac {c d \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}+\frac {\polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) d^{2} x}{f^{2}}-\frac {4 c d e x}{f}-\frac {2 d c \,e^{2}}{f^{2}}+\frac {\ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) d^{2} x^{2}}{f}+\frac {2 d^{2} e^{2} x}{f^{2}}+\frac {4 d e c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 c d \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f}+\frac {c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f}-\frac {2 c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}-\frac {2 d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}-\frac {2 d^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {4 d^{2} e^{3}}{3 f^{3}}-\frac {d^{2} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{3}}\)	\(375\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2*tanh(f*x+e)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (154) = 308\).
time = 0.55, size = 402, normalized size = 2.56 \begin {gather*} c^{2} {\left (x + \frac {e}{f} + \frac {\log \left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}{f} + \frac {2 \, e^{\left (-2 \, f x - 2 \, e\right )}}{f {\left (2 \, e^{\left (-2 \, f x - 2 \, e\right )} + e^{\left (-4 \, f x - 4 \, e\right )} + 1\right )}}\right )} + \frac {{\left (2 \, f x \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right )\right )} c d}{f^{2}} - \frac {2 \, d^{2} x}{f^{2}} + \frac {d^{2} f^{2} x^{3} + 3 \, c d f^{2} x^{2} + 6 \, d^{2} x + 6 \, c d + {\left (d^{2} f^{2} x^{3} e^{\left (4 \, e\right )} + 3 \, c d f^{2} x^{2} e^{\left (4 \, e\right )}\right )} e^{\left (4 \, f x\right )} + 2 \, {\left (d^{2} f^{2} x^{3} e^{\left (2 \, e\right )} + 3 \, {\left (c d f^{2} + d^{2} f\right )} x^{2} e^{\left (2 \, e\right )} + 3 \, c d e^{\left (2 \, e\right )} + 3 \, {\left (2 \, c d f + d^{2}\right )} x e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{3 \, {\left (f^{2} e^{\left (4 \, f x + 4 \, e\right )} + 2 \, f^{2} e^{\left (2 \, f x + 2 \, e\right )} + f^{2}\right )}} + \frac {{\left (2 \, f^{2} x^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (2 \, f x + 2 \, e\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, f x + 2 \, e\right )})\right )} d^{2}}{2 \, f^{3}} + \frac {d^{2} \log \left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )}{f^{3}} - \frac {2 \, {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2}\right )}}{3 \, f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*tanh(f*x+e)^3,x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 0.46, size = 4901, normalized size = 31.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2*tanh(f*x+e)^3,x, algorithm="fricas")

[Out]

-1/3*(d^2*f^3*x^3 + 3*c*d*f^3*x^2 + 3*c^2*f^3*x - 6*c*d*f*cosh(1)^2 + 2*d^2*cosh(1)^3 + (d^2*f^3*x^3 + 3*c*d*f
^3*x^2 - 6*c*d*f*cosh(1)^2 + 2*d^2*cosh(1)^3 + 2*d^2*sinh(1)^3 - 6*(c*d*f - d^2*cosh(1))*sinh(1)^2 + 3*(c^2*f^
3 + 2*d^2*f)*x + 6*(c^2*f^2 + d^2)*cosh(1) + 6*(c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2)*sinh(1))*cosh
(f*x + cosh(1) + sinh(1))^4 + 2*d^2*sinh(1)^3 + 4*(d^2*f^3*x^3 + 3*c*d*f^3*x^2 - 6*c*d*f*cosh(1)^2 + 2*d^2*cos
h(1)^3 + 2*d^2*sinh(1)^3 - 6*(c*d*f - d^2*cosh(1))*sinh(1)^2 + 3*(c^2*f^3 + 2*d^2*f)*x + 6*(c^2*f^2 + d^2)*cos
h(1) + 6*(c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + c
osh(1) + sinh(1))^3 + (d^2*f^3*x^3 + 3*c*d*f^3*x^2 - 6*c*d*f*cosh(1)^2 + 2*d^2*cosh(1)^3 + 2*d^2*sinh(1)^3 - 6
*(c*d*f - d^2*cosh(1))*sinh(1)^2 + 3*(c^2*f^3 + 2*d^2*f)*x + 6*(c^2*f^2 + d^2)*cosh(1) + 6*(c^2*f^2 - 2*c*d*f*
cosh(1) + d^2*cosh(1)^2 + d^2)*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^4 - 6*c*d*f + 2*(d^2*f^3*x^3 - 6*c*d*f*c
osh(1)^2 + 2*d^2*cosh(1)^3 + 2*d^2*sinh(1)^3 - 3*c^2*f^2 - 3*c*d*f + 3*(c*d*f^3 - d^2*f^2)*x^2 - 6*(c*d*f - d^
2*cosh(1))*sinh(1)^2 + 3*(c^2*f^3 - 2*c*d*f^2 + d^2*f)*x + 6*(c^2*f^2 + d^2)*cosh(1) + 6*(c^2*f^2 - 2*c*d*f*co
sh(1) + d^2*cosh(1)^2 + d^2)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 6*(c*d*f - d^2*cosh(1))*sinh(1)^2 + 2*
(d^2*f^3*x^3 - 6*c*d*f*cosh(1)^2 + 2*d^2*cosh(1)^3 + 2*d^2*sinh(1)^3 - 3*c^2*f^2 - 3*c*d*f + 3*(c*d*f^3 - d^2*
f^2)*x^2 + 3*(d^2*f^3*x^3 + 3*c*d*f^3*x^2 - 6*c*d*f*cosh(1)^2 + 2*d^2*cosh(1)^3 + 2*d^2*sinh(1)^3 - 6*(c*d*f -
d^2*cosh(1))*sinh(1)^2 + 3*(c^2*f^3 + 2*d^2*f)*x + 6*(c^2*f^2 + d^2)*cosh(1) + 6*(c^2*f^2 - 2*c*d*f*cosh(1) +
d^2*cosh(1)^2 + d^2)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - 6*(c*d*f - d^2*cosh(1))*sinh(1)^2 + 3*(c^2*f^
3 - 2*c*d*f^2 + d^2*f)*x + 6*(c^2*f^2 + d^2)*cosh(1) + 6*(c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2)*sin
h(1))*sinh(f*x + cosh(1) + sinh(1))^2 + 6*(c^2*f^2 + d^2)*cosh(1) - 6*((d^2*f*x + c*d*f)*cosh(f*x + cosh(1) +
sinh(1))^4 + 4*(d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (d^2*f*x + c*
d*f)*sinh(f*x + cosh(1) + sinh(1))^4 + d^2*f*x + c*d*f + 2*(d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1))^2 +
2*(d^2*f*x + c*d*f + 3*(d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1))^2)*sinh(f*x + cosh(1) + sinh(1))^2 + 4
*((d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1))^3 + (d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1)))*sinh(f*
x + cosh(1) + sinh(1)))*dilog(I*cosh(f*x + cosh(1) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1))) - 6*((d^2*f*x
+ c*d*f)*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(
1) + sinh(1))^3 + (d^2*f*x + c*d*f)*sinh(f*x + cosh(1) + sinh(1))^4 + d^2*f*x + c*d*f + 2*(d^2*f*x + c*d*f)*co
sh(f*x + cosh(1) + sinh(1))^2 + 2*(d^2*f*x + c*d*f + 3*(d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1))^2)*sinh
(f*x + cosh(1) + sinh(1))^2 + 4*((d^2*f*x + c*d*f)*cosh(f*x + cosh(1) + sinh(1))^3 + (d^2*f*x + c*d*f)*cosh(f*
x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*dilog(-I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x + c
osh(1) + sinh(1))) - 3*((c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh
(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 +
d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (c^2*f
^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*sinh(f*x + cosh(
1) + sinh(1))^4 + c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + 2*(c^2*f^2 - 2*c*d*f*cosh(1) + d
^2*cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(c^2
*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + 3*(c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*si
nh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + d^2 - 2*(c*d*f - d^2*cosh(1
))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1) + 4*((c^2*f^2 - 2*c*d*f*co
sh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^3
+ (c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*
x + cosh(1) + sinh(1)))*sinh(f*x + cosh(1) + sinh(1)))*log(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1)
+ sinh(1)) + I) - 3*((c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh(1)
)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^4 + 4*(c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + d^
2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1))^3 + (c^2*f^2
- 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^2*cosh(1))*sinh(1))*sinh(f*x + cosh(1)
+ sinh(1))^4 + c^2*f^2 - 2*c*d*f*cosh(1) + d^2*cosh(1)^2 + d^2*sinh(1)^2 + 2*(c^2*f^2 - 2*c*d*f*cosh(1) + d^2*
cosh(1)^2 + d^2*sinh(1)^2 + d^2 - 2*(c*d*f - d^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c + d x\right )^{2} \tanh ^{3}{\left (e + f x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2*tanh(f*x+e)**3,x)

[Out]

Integral((c + d*x)**2*tanh(e + f*x)**3, 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*tanh(f*x+e)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^2*tanh(f*x + e)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tanh}\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^3*(c + d*x)^2,x)

[Out]

int(tanh(e + f*x)^3*(c + d*x)^2, x)

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