3.1.7 \(\int (c+d x)^2 \tanh ^2(e+f x) \, dx\) [7]

Optimal. Leaf size=88 \[ -\frac {(c+d x)^2}{f}+\frac {(c+d x)^3}{3 d}+\frac {2 d (c+d x) \log \left (1+e^{2 (e+f x)}\right )}{f^2}+\frac {d^2 \text {PolyLog}\left (2,-e^{2 (e+f x)}\right )}{f^3}-\frac {(c+d x)^2 \tanh (e+f x)}{f} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3801, 3799, 2221, 2317, 2438, 32} \begin {gather*} \frac {2 d (c+d x) \log \left (e^{2 (e+f x)}+1\right )}{f^2}-\frac {(c+d x)^2 \tanh (e+f x)}{f}-\frac {(c+d x)^2}{f}+\frac {(c+d x)^3}{3 d}+\frac {d^2 \text {Li}_2\left (-e^{2 (e+f x)}\right )}{f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Tanh[e + f*x]^2,x]

[Out]

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 6.15, size = 205, normalized size = 2.33 \begin {gather*} c^2 x+c d x^2+\frac {d^2 x^3}{3}-\frac {(c+d x)^2 \text {sech}(e) \text {sech}(e+f x) \sinh (f x)}{f}+\frac {2 c d (\log (\cosh (e+f x))-f x \tanh (e))}{f^2}+\frac {d^2 \left (-i \pi \log \left (1+e^{2 f x}\right )+2 f x \log \left (1-e^{-2 \left (f x+\tanh ^{-1}(\coth (e))\right )}\right )+i \pi (f x+\log (\cosh (f x)))+2 \tanh ^{-1}(\coth (e)) \left (f x+\log \left (1-e^{-2 \left (f x+\tanh ^{-1}(\coth (e))\right )}\right )-\log \left (i \sinh \left (f x+\tanh ^{-1}(\coth (e))\right )\right )\right )-\text {PolyLog}\left (2,e^{-2 \left (f x+\tanh ^{-1}(\coth (e))\right )}\right )-e^{-\tanh ^{-1}(\coth (e))} f^2 x^2 \sqrt {-\text {csch}^2(e)} \tanh (e)\right )}{f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(86)=172\).
time = 1.29, size = 185, normalized size = 2.10

method	result	size
risch	\(\frac {d^{2} x^{3}}{3}+d c \,x^{2}+c^{2} x +\frac {c^{3}}{3 d}+\frac {2 d^{2} x^{2}+4 c d x +2 c^{2}}{f \left (1+{\mathrm e}^{2 f x +2 e}\right )}+\frac {2 d c \ln \left (1+{\mathrm e}^{2 f x +2 e}\right )}{f^{2}}-\frac {4 d c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {2 d^{2} x^{2}}{f}-\frac {4 d^{2} e x}{f^{2}}-\frac {2 d^{2} e^{2}}{f^{3}}+\frac {2 d^{2} \ln \left (1+{\mathrm e}^{2 f x +2 e}\right ) x}{f^{2}}+\frac {d^{2} \polylog \left (2, -{\mathrm e}^{2 f x +2 e}\right )}{f^{3}}+\frac {4 d^{2} e \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

[Out]

c^2*(x + e/f - 2/(f*(e^(-2*f*x - 2*e) + 1))) - c*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) + 1
/3*d^2*((f*x^3*e^(2*f*x + 2*e) + f*x^3 + 6*x^2)/(f*e^(2*f*x + 2*e) + f) - 12*integrate(x/(f*e^(2*f*x + 2*e) +
f), x))

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Fricas [C] Result contains complex when optimal does not.
time = 0.58, size = 1200, normalized size = 13.64 \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)^2,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^2*f^2 - 12*c*d*f*cosh(1) + 6*d^2*cosh(1)^2 + 6*d^2*sinh(1
)^2 + (d^2*f^3*x^3 - 12*c*d*f*cosh(1) + 6*d^2*cosh(1)^2 + 6*d^2*sinh(1)^2 + 3*(c*d*f^3 - 2*d^2*f^2)*x^2 + 3*(c
^2*f^3 - 4*c*d*f^2)*x - 12*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + 2*(d^2*f^3*x^3 - 1
2*c*d*f*cosh(1) + 6*d^2*cosh(1)^2 + 6*d^2*sinh(1)^2 + 3*(c*d*f^3 - 2*d^2*f^2)*x^2 + 3*(c^2*f^3 - 4*c*d*f^2)*x
- 12*(c*d*f - d^2*cosh(1))*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (d^2*f^3*x^3
- 12*c*d*f*cosh(1) + 6*d^2*cosh(1)^2 + 6*d^2*sinh(1)^2 + 3*(c*d*f^3 - 2*d^2*f^2)*x^2 + 3*(c^2*f^3 - 4*c*d*f^2
)*x - 12*(c*d*f - d^2*cosh(1))*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2 + 6*(d^2*cosh(f*x + cosh(1) + sinh(1))
^2 + 2*d^2*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + d^2*sinh(f*x + cosh(1) + sinh(1))^2 +
d^2)*dilog(I*cosh(f*x + cosh(1) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1))) + 6*(d^2*cosh(f*x + cosh(1) + s
inh(1))^2 + 2*d^2*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + d^2*sinh(f*x + cosh(1) + sinh(
1))^2 + d^2)*dilog(-I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x + cosh(1) + sinh(1))) + 6*(c*d*f - d^2*cosh(1
) + (c*d*f - d^2*cosh(1) - d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - d^2*sinh(1) + 2*(c*d*f - d^2*cosh(1)
- d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (c*d*f - d^2*cosh(1) - d^2*sinh(
1))*sinh(f*x + cosh(1) + sinh(1))^2)*log(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)) + I) +
6*(c*d*f - d^2*cosh(1) + (c*d*f - d^2*cosh(1) - d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 - d^2*sinh(1) + 2
*(c*d*f - d^2*cosh(1) - d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (c*d*f - d^
2*cosh(1) - d^2*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2)*log(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(
1) + sinh(1)) - I) + 6*(d^2*f*x + d^2*cosh(1) + (d^2*f*x + d^2*cosh(1) + d^2*sinh(1))*cosh(f*x + cosh(1) + sin
h(1))^2 + d^2*sinh(1) + 2*(d^2*f*x + d^2*cosh(1) + d^2*sinh(1))*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(
1) + sinh(1)) + (d^2*f*x + d^2*cosh(1) + d^2*sinh(1))*sinh(f*x + cosh(1) + sinh(1))^2)*log(I*cosh(f*x + cosh(1
) + sinh(1)) + I*sinh(f*x + cosh(1) + sinh(1)) + 1) + 6*(d^2*f*x + d^2*cosh(1) + (d^2*f*x + d^2*cosh(1) + d^2*
sinh(1))*cosh(f*x + cosh(1) + sinh(1))^2 + d^2*sinh(1) + 2*(d^2*f*x + d^2*cosh(1) + d^2*sinh(1))*cosh(f*x + co
sh(1) + sinh(1))*sinh(f*x + cosh(1) + sinh(1)) + (d^2*f*x + d^2*cosh(1) + d^2*sinh(1))*sinh(f*x + cosh(1) + si
nh(1))^2)*log(-I*cosh(f*x + cosh(1) + sinh(1)) - I*sinh(f*x + cosh(1) + sinh(1)) + 1) - 12*(c*d*f - d^2*cosh(1
))*sinh(1))/(f^3*cosh(f*x + cosh(1) + sinh(1))^2 + 2*f^3*cosh(f*x + cosh(1) + sinh(1))*sinh(f*x + cosh(1) + si
nh(1)) + f^3*sinh(f*x + cosh(1) + sinh(1))^2 + f^3)

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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 ^{2}{\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)**2,x)

[Out]

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

[Out]

integrate((d*x + c)^2*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 )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(e + f*x)^2*(c + d*x)^2,x)

[Out]

int(tanh(e + f*x)^2*(c + d*x)^2, x)

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