3.1.6 \(\int (c+d x)^3 \tanh ^2(e+f x) \, dx\) [6]

Optimal. Leaf size=119 \[ -\frac {(c+d x)^3}{f}+\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 d^2 (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}-\frac {3 d^3 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )}{2 f^4}-\frac {(c+d x)^3 \tanh (e+f x)}{f} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3801, 3799, 2221, 2611, 2320, 6724, 32} \begin {gather*} \frac {3 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}+\frac {3 d (c+d x)^2 \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {(c+d x)^3 \tanh (e+f x)}{f}-\frac {(c+d x)^3}{f}+\frac {(c+d x)^4}{4 d}-\frac {3 d^3 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Tanh[e + f*x]^2,x]

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 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)^3 \tanh ^2(e+f x) \, dx &=-\frac {(c+d x)^3 \tanh (e+f x)}{f}+\frac {(3 d) \int (c+d x)^2 \tanh (e+f x) \, dx}{f}+\int (c+d x)^3 \, dx\\ &=-\frac {(c+d x)^3}{f}+\frac {(c+d x)^4}{4 d}-\frac {(c+d x)^3 \tanh (e+f x)}{f}+\frac {(6 d) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1+e^{2 (e+f x)}} \, dx}{f}\\ &=-\frac {(c+d x)^3}{f}+\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}-\frac {(c+d x)^3 \tanh (e+f x)}{f}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1+e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {(c+d x)^3}{f}+\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}-\frac {(c+d x)^3 \tanh (e+f x)}{f}-\frac {\left (3 d^3\right ) \int \text {Li}_2\left (-e^{2 (e+f x)}\right ) \, dx}{f^3}\\ &=-\frac {(c+d x)^3}{f}+\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}-\frac {(c+d x)^3 \tanh (e+f x)}{f}-\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^4}\\ &=-\frac {(c+d x)^3}{f}+\frac {(c+d x)^4}{4 d}+\frac {3 d (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3}-\frac {3 d^3 \text {Li}_3\left (-e^{2 (e+f x)}\right )}{2 f^4}-\frac {(c+d x)^3 \tanh (e+f x)}{f}\\ \end {align*}

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Mathematica [A]
time = 1.61, size = 169, normalized size = 1.42 \begin {gather*} \frac {1}{4} \left (x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+\frac {2 d \left (-\frac {4 e^{2 e} f^3 x \left (3 c^2+3 c d x+d^2 x^2\right )}{1+e^{2 e}}+6 f^2 (c+d x)^2 \log \left (1+e^{2 (e+f x)}\right )+6 d f (c+d x) \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )-3 d^2 \text {PolyLog}\left (3,-e^{2 (e+f x)}\right )\right )}{f^4}-\frac {4 (c+d x)^3 \text {sech}(e) \text {sech}(e+f x) \sinh (f x)}{f}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + (2*d*((-4*E^(2*e)*f^3*x*(3*c^2 + 3*c*d*x + d^2*x^2))/(1 + E^(
2*e)) + 6*f^2*(c + d*x)^2*Log[1 + E^(2*(e + f*x))] + 6*d*f*(c + d*x)*PolyLog[2, -E^(2*(e + f*x))] - 3*d^2*Poly
Log[3, -E^(2*(e + f*x))]))/f^4 - (4*(c + d*x)^3*Sech[e]*Sech[e + f*x]*Sinh[f*x])/f)/4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(335\) vs. \(2(115)=230\).
time = 1.52, size = 336, normalized size = 2.82

method	result	size
risch	\(\frac {d^{3} x^{4}}{4}+d^{2} c \,x^{3}+\frac {3 d \,c^{2} x^{2}}{2}+c^{3} x +\frac {c^{4}}{4 d}+\frac {2 d^{3} x^{3}+6 c \,d^{2} x^{2}+6 c^{2} d x +2 c^{3}}{f \left (1+{\mathrm e}^{2 f x +2 e}\right )}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {3 d \,c^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}-\frac {6 d^{3} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{4}}-\frac {2 d^{3} x^{3}}{f}+\frac {6 d^{3} e^{2} x}{f^{3}}+\frac {4 d^{3} e^{3}}{f^{4}}+\frac {3 d^{3} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x^{2}}{f^{2}}+\frac {3 d^{3} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right ) x}{f^{3}}-\frac {3 d^{3} \polylog \left (3, -{\mathrm e}^{2 f x +2 e}\right )}{2 f^{4}}+\frac {12 d^{2} e c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {6 d^{2} c \,x^{2}}{f}-\frac {12 d^{2} c e x}{f^{2}}-\frac {6 d^{2} c \,e^{2}}{f^{3}}+\frac {6 d^{2} c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}+\frac {3 d^{2} c \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}\)	\(336\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (118) = 236\).
time = 0.42, size = 360, normalized size = 3.03 \begin {gather*} c^{3} {\left (x + \frac {e}{f} - \frac {2}{f {\left (e^{\left (-2 \, f x - 2 \, e\right )} + 1\right )}}\right )} - \frac {3}{2} \, c^{2} d {\left (\frac {2 \, x e^{\left (2 \, f x + 2 \, e\right )}}{f e^{\left (2 \, f x + 2 \, e\right )} + f} - \frac {f x^{2} + {\left (f x^{2} e^{\left (2 \, e\right )} - 2 \, x e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f e^{\left (2 \, f x + 2 \, e\right )} + f} - \frac {2 \, \log \left ({\left (e^{\left (2 \, f x + 2 \, e\right )} + 1\right )} e^{\left (-2 \, e\right )}\right )}{f^{2}}\right )} + \frac {3 \, {\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^{2}}{f^{3}} + \frac {d^{3} f x^{4} + 24 \, c d^{2} x^{2} + 4 \, {\left (c d^{2} f + 2 \, d^{3}\right )} x^{3} + {\left (d^{3} f x^{4} e^{\left (2 \, e\right )} + 4 \, c d^{2} f x^{3} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{4 \, {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )}} + \frac {3 \, {\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^{3}}{2 \, f^{4}} - \frac {2 \, {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2}\right )}}{f^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(x + e/f - 2/(f*(e^(-2*f*x - 2*e) + 1))) - 3/2*c^2*d*(2*x*e^(2*f*x + 2*e)/(f*e^(2*f*x + 2*e) + f) - (f*x^2
+ (f*x^2*e^(2*e) - 2*x*e^(2*e))*e^(2*f*x))/(f*e^(2*f*x + 2*e) + f) - 2*log((e^(2*f*x + 2*e) + 1)*e^(-2*e))/f^
2) + 3*(2*f*x*log(e^(2*f*x + 2*e) + 1) + dilog(-e^(2*f*x + 2*e)))*c*d^2/f^3 + 1/4*(d^3*f*x^4 + 24*c*d^2*x^2 +
4*(c*d^2*f + 2*d^3)*x^3 + (d^3*f*x^4*e^(2*e) + 4*c*d^2*f*x^3*e^(2*e))*e^(2*f*x))/(f*e^(2*f*x + 2*e) + f) + 3/2
*(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^3/f^4 -
2*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2)/f^4

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Fricas [C] Result contains complex when optimal does not.
time = 0.42, size = 2377, normalized size = 19.97 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 6*c^2*d*f^4*x^2 + 4*c^3*f^4*x + 8*c^3*f^3 - 24*c^2*d*f^2*cosh(1) + 24*c*d
^2*f*cosh(1)^2 - 8*d^3*cosh(1)^3 - 8*d^3*sinh(1)^3 + (d^3*f^4*x^4 - 24*c^2*d*f^2*cosh(1) + 24*c*d^2*f*cosh(1)^
2 - 8*d^3*cosh(1)^3 - 8*d^3*sinh(1)^3 + 4*(c*d^2*f^4 - 2*d^3*f^3)*x^3 + 6*(c^2*d*f^4 - 4*c*d^2*f^3)*x^2 + 24*(
c*d^2*f - d^3*cosh(1))*sinh(1)^2 + 4*(c^3*f^4 - 6*c^2*d*f^3)*x - 24*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(
1)^2)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 24*(c*d^2*f - d^3*cosh(1))*sinh(1)^2 + 2*(d^3*f^4*x^4 - 24*c^
2*d*f^2*cosh(1) + 24*c*d^2*f*cosh(1)^2 - 8*d^3*cosh(1)^3 - 8*d^3*sinh(1)^3 + 4*(c*d^2*f^4 - 2*d^3*f^3)*x^3 + 6
*(c^2*d*f^4 - 4*c*d^2*f^3)*x^2 + 24*(c*d^2*f - d^3*cosh(1))*sinh(1)^2 + 4*(c^3*f^4 - 6*c^2*d*f^3)*x - 24*(c^2*
d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2)*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)
) + (d^3*f^4*x^4 - 24*c^2*d*f^2*cosh(1) + 24*c*d^2*f*cosh(1)^2 - 8*d^3*cosh(1)^3 - 8*d^3*sinh(1)^3 + 4*(c*d^2*
f^4 - 2*d^3*f^3)*x^3 + 6*(c^2*d*f^4 - 4*c*d^2*f^3)*x^2 + 24*(c*d^2*f - d^3*cosh(1))*sinh(1)^2 + 4*(c^3*f^4 - 6
*c^2*d*f^3)*x - 24*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2)*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 +
24*(d^3*f*x + c*d^2*f + (d^3*f*x + c*d^2*f)*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(d^3*f*x + c*d^2*f)*cosh(f*x +
cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (d^3*f*x + c*d^2*f)*sinh(f*x + cosh(1) + sinh(1))^2)*dilog
(I*cosh(f*x + cosh(1) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1))) + 24*(d^3*f*x + c*d^2*f + (d^3*f*x + c*d^2
*f)*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(d^3*f*x + c*d^2*f)*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) +
sinh(1)) + (d^3*f*x + c*d^2*f)*sinh(f*x + cosh(1) + sinh(1))^2)*dilog(-I*cosh(f*x + cosh(1) + sinh(1)) - I*si
nh(f*x + cosh(1) + sinh(1))) + 12*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3*sinh(1)^2 + (c^2*d*f^2
- 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3*sinh(1)^2 - 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*cosh(f*x + cosh(1) +
sinh(1))^2 + 2*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3*sinh(1)^2 - 2*(c*d^2*f - d^3*cosh(1))*sinh
(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1
)^2 + d^3*sinh(1)^2 - 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 - 2*(c*d^2*f - d^3*co
sh(1))*sinh(1))*log(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)) + I) + 12*(c^2*d*f^2 - 2*c*d
^2*f*cosh(1) + d^3*cosh(1)^2 + d^3*sinh(1)^2 + (c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3*sinh(1)^2
- 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*
cosh(1)^2 + d^3*sinh(1)^2 - 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1
) + sinh(1)) + (c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2 + d^3*sinh(1)^2 - 2*(c*d^2*f - d^3*cosh(1))*sinh
(1))*sinh(f*x + cosh(1) + sinh(1))^2 - 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*log(cosh(f*x + cosh(1) + sinh(1)) +
sinh(f*x + cosh(1) + sinh(1)) - I) + 12*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*cosh(1) - d^3*cosh(1)^2 - d^3
*sinh(1)^2 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*cosh(1) - d^3*cosh(1)^2 - d^3*sinh(1)^2 + 2*(c*d^2*f - d
^3*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*cosh(1) - d^
3*cosh(1)^2 - d^3*sinh(1)^2 + 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh
(1) + sinh(1)) + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*cosh(1) - d^3*cosh(1)^2 - d^3*sinh(1)^2 + 2*(c*d^2*f
- d^3*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*log(I*cosh(f*x +
cosh(1) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1)) + 1) + 12*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*cosh(
1) - d^3*cosh(1)^2 - d^3*sinh(1)^2 + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*cosh(1) - d^3*cosh(1)^2 - d^3*si
nh(1)^2 + 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(d^3*f^2*x^2 + 2*c*d^2*f^2*x
+ 2*c*d^2*f*cosh(1) - d^3*cosh(1)^2 - d^3*sinh(1)^2 + 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*cosh(f*x + cosh(1) +
sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (d^3*f^2*x^2 + 2*c*d^2*f^2*x + 2*c*d^2*f*cosh(1) - d^3*cosh(1)^2 - d^
3*sinh(1)^2 + 2*(c*d^2*f - d^3*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + 2*(c*d^2*f - d^3*cosh(1))*s
inh(1))*log(-I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x + cosh(1) + sinh(1)) + 1) - 24*(d^3*cosh(f*x + cosh(
1) + sinh(1))^2 + 2*d^3*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + d^3*sinh(f*x + cosh(1) +
sinh(1))^2 + d^3)*polylog(3, I*cosh(f*x + cosh(1) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1))) - 24*(d^3*cos
h(f*x + cosh(1) + sinh(1))^2 + 2*d^3*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + d^3*sinh(f*
x + cosh(1) + sinh(1))^2 + d^3)*polylog(3, -I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x + cosh(1) + sinh(1)))
- 24*(c^2*d*f^2 - 2*c*d^2*f*cosh(1) + d^3*cosh(1)^2)*sinh(1))/(f^4*cosh(f*x + cosh(1) + sinh(1))^2 + 2*f^4*co
sh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) ...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((d*x + c)^3*tanh(f*x + e)^2, 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 )}^2\,{\left (c+d\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^2*(c + d*x)^3,x)

[Out]

int(tanh(e + f*x)^2*(c + d*x)^3, x)

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