3.2.58 \(\int \tanh ^2(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [158]

Optimal. Leaf size=94 \[ (a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d} \]

[Out]

(a+b)^3*x-(a+b)^3*tanh(d*x+c)/d-1/3*b*(3*a^2+3*a*b+b^2)*tanh(d*x+c)^3/d-1/5*b^2*(3*a+b)*tanh(d*x+c)^5/d-1/7*b^
3*tanh(d*x+c)^7/d

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Rubi [A]
time = 0.07, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 472, 212} \begin {gather*} -\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac {b^3 \tanh ^7(c+d x)}{7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^3)/(3*d) - (b^2*(3*a + b)*T
anh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^7)/(7*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 472

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[(e*x)^m*((a + b*x^n)^p/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a+b x^2\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-(a+b)^3-b \left (3 a^2+3 a b+b^2\right ) x^2-b^2 (3 a+b) x^4-b^3 x^6+\frac {a^3+3 a^2 b+3 a b^2+b^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d}+\frac {(a+b)^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac {b^3 \tanh ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]
time = 1.19, size = 108, normalized size = 1.15 \begin {gather*} \frac {\tanh (c+d x) \left (-105 (a+b)^3-35 b \left (3 a^2+3 a b+b^2\right ) \tanh ^2(c+d x)-21 b^2 (3 a+b) \tanh ^4(c+d x)-15 b^3 \tanh ^6(c+d x)+\frac {105 (a+b)^3 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}\right )}{105 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]^2*(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(Tanh[c + d*x]*(-105*(a + b)^3 - 35*b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^2 - 21*b^2*(3*a + b)*Tanh[c + d*x]^4
- 15*b^3*Tanh[c + d*x]^6 + (105*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2]))/(105*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(192\) vs. \(2(88)=176\).
time = 0.40, size = 193, normalized size = 2.05

method	result	size
derivativedivides	\(\frac {-3 a \,b^{2} \tanh \left (d x +c \right )-3 a^{2} b \tanh \left (d x +c \right )-a^{2} b \left (\tanh ^{3}\left (d x +c \right )\right )-a \,b^{2} \left (\tanh ^{3}\left (d x +c \right )\right )-\frac {3 a \,b^{2} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-b^{3} \tanh \left (d x +c \right )-a^{3} \tanh \left (d x +c \right )-\frac {b^{3} \left (\tanh ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {b^{3} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {b^{3} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(193\)
default	\(\frac {-3 a \,b^{2} \tanh \left (d x +c \right )-3 a^{2} b \tanh \left (d x +c \right )-a^{2} b \left (\tanh ^{3}\left (d x +c \right )\right )-a \,b^{2} \left (\tanh ^{3}\left (d x +c \right )\right )-\frac {3 a \,b^{2} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}-b^{3} \tanh \left (d x +c \right )-a^{3} \tanh \left (d x +c \right )-\frac {b^{3} \left (\tanh ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {b^{3} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {b^{3} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(193\)
risch	\(a^{3} x +3 a^{2} b x +3 a \,b^{2} x +b^{3} x +\frac {\frac {46 a \,b^{2}}{5}+12 a^{2} b \,{\mathrm e}^{12 d x +12 c}+18 a \,b^{2} {\mathrm e}^{12 d x +12 c}+60 a^{2} b \,{\mathrm e}^{10 d x +10 c}+72 a \,b^{2} {\mathrm e}^{10 d x +10 c}+108 a^{2} b \,{\mathrm e}^{4 d x +4 c}+44 a^{2} b \,{\mathrm e}^{2 d x +2 c}+8 a^{2} b +2 a^{3}+\frac {352 b^{3}}{105}+146 a \,b^{2} {\mathrm e}^{8 d x +8 c}+176 a \,b^{2} {\mathrm e}^{6 d x +6 c}+\frac {606 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{5}+128 a^{2} b \,{\mathrm e}^{8 d x +8 c}+\frac {232 a \,b^{2} {\mathrm e}^{2 d x +2 c}}{5}+152 a^{2} b \,{\mathrm e}^{6 d x +6 c}+12 a^{3} {\mathrm e}^{2 d x +2 c}+2 a^{3} {\mathrm e}^{12 d x +12 c}+8 b^{3} {\mathrm e}^{12 d x +12 c}+12 a^{3} {\mathrm e}^{10 d x +10 c}+\frac {232 b^{3} {\mathrm e}^{2 d x +2 c}}{15}+\frac {176 b^{3} {\mathrm e}^{8 d x +8 c}}{3}+30 a^{3} {\mathrm e}^{4 d x +4 c}+\frac {232 b^{3} {\mathrm e}^{4 d x +4 c}}{5}+24 b^{3} {\mathrm e}^{10 d x +10 c}+30 a^{3} {\mathrm e}^{8 d x +8 c}+40 a^{3} {\mathrm e}^{6 d x +6 c}+\frac {176 b^{3} {\mathrm e}^{6 d x +6 c}}{3}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{7}}\)	\(415\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-3*a*b^2*tanh(d*x+c)-3*a^2*b*tanh(d*x+c)-a^2*b*tanh(d*x+c)^3-a*b^2*tanh(d*x+c)^3-3/5*a*b^2*tanh(d*x+c)^5+
1/2*(a^3+3*a^2*b+3*a*b^2+b^3)*ln(1+tanh(d*x+c))-b^3*tanh(d*x+c)-a^3*tanh(d*x+c)-1/7*b^3*tanh(d*x+c)^7-1/2*(a^3
+3*a^2*b+3*a*b^2+b^3)*ln(tanh(d*x+c)-1)-1/3*b^3*tanh(d*x+c)^3-1/5*b^3*tanh(d*x+c)^5)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (88) = 176\).
time = 0.29, size = 400, normalized size = 4.26 \begin {gather*} \frac {1}{105} \, b^{3} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {1}{5} \, a b^{2} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + a^{2} b {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + a^{3} {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*b^3*(105*x + 105*c/d - 8*(203*e^(-2*d*x - 2*c) + 609*e^(-4*d*x - 4*c) + 770*e^(-6*d*x - 6*c) + 770*e^(-8
*d*x - 8*c) + 315*e^(-10*d*x - 10*c) + 105*e^(-12*d*x - 12*c) + 44)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*
c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 1
4*c) + 1))) + 1/5*a*b^2*(15*x + 15*c/d - 2*(70*e^(-2*d*x - 2*c) + 140*e^(-4*d*x - 4*c) + 90*e^(-6*d*x - 6*c) +
45*e^(-8*d*x - 8*c) + 23)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x -
8*c) + e^(-10*d*x - 10*c) + 1))) + a^2*b*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 2)/(d*(3*
e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) + a^3*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1036 vs. \(2 (88) = 176\).
time = 0.40, size = 1036, normalized size = 11.02 \begin {gather*} \frac {{\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{5} - 7 \, {\left (75 \, a^{3} + 240 \, a^{2} b + 213 \, a b^{2} + 56 \, b^{3} + 3 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left ({\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} + {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} - 7 \, {\left (5 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 135 \, a^{3} + 360 \, a^{2} b + 369 \, a b^{2} + 168 \, b^{3} + 10 \, {\left (75 \, a^{3} + 240 \, a^{2} b + 213 \, a b^{2} + 56 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (3 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3} + 105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) - 7 \, {\left ({\left (105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (75 \, a^{3} + 240 \, a^{2} b + 213 \, a b^{2} + 56 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 75 \, a^{3} + 180 \, a^{2} b + 225 \, a b^{2} + 9 \, {\left (45 \, a^{3} + 120 \, a^{2} b + 123 \, a b^{2} + 56 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{105 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + 35 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, d \cosh \left (d x + c\right )^{3} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*((105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 +
7*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)*sinh(d*
x + c)^6 - (105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*sinh(d*x + c)^7 + 7*(105*a^3 + 420*a^2*b + 483*a*b^2 +
176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 - 7*(75*a^3 + 240*a^2*b + 213*a*b^2 + 56*b^
3 + 3*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 35*((105*a^3 + 420*a^2*b
+ 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + (105*a^3 + 420*a^2*b + 483*
a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 21*(105*a^3 + 420*
a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - 7*(5*(105*a^3 + 420*a
^2*b + 483*a*b^2 + 176*b^3)*cosh(d*x + c)^4 + 135*a^3 + 360*a^2*b + 369*a*b^2 + 168*b^3 + 10*(75*a^3 + 240*a^2
*b + 213*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 7*(3*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 +
105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 + 10*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 9*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^
3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b
^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c) - 7*((105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*c
osh(d*x + c)^6 + 5*(75*a^3 + 240*a^2*b + 213*a*b^2 + 56*b^3)*cosh(d*x + c)^4 + 75*a^3 + 180*a^2*b + 225*a*b^2
+ 9*(45*a^3 + 120*a^2*b + 123*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*
x + c)*sinh(d*x + c)^6 + 7*d*cosh(d*x + c)^5 + 35*(d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^4 + 21*d
*cosh(d*x + c)^3 + 7*(3*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^2 + 35*d*c
osh(d*x + c))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (82) = 164\).
time = 0.25, size = 192, normalized size = 2.04 \begin {gather*} \begin {cases} a^{3} x - \frac {a^{3} \tanh {\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac {a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{2} b \tanh {\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac {3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac {3 a b^{2} \tanh {\left (c + d x \right )}}{d} + b^{3} x - \frac {b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{3} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)**2*(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Piecewise((a**3*x - a**3*tanh(c + d*x)/d + 3*a**2*b*x - a**2*b*tanh(c + d*x)**3/d - 3*a**2*b*tanh(c + d*x)/d +
3*a*b**2*x - 3*a*b**2*tanh(c + d*x)**5/(5*d) - a*b**2*tanh(c + d*x)**3/d - 3*a*b**2*tanh(c + d*x)/d + b**3*x
- b**3*tanh(c + d*x)**7/(7*d) - b**3*tanh(c + d*x)**5/(5*d) - b**3*tanh(c + d*x)**3/(3*d) - b**3*tanh(c + d*x)
/d, Ne(d, 0)), (x*(a + b*tanh(c)**2)**3*tanh(c)**2, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (88) = 176\).
time = 0.55, size = 418, normalized size = 4.45 \begin {gather*} \frac {105 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (105 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 945 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 420 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3150 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 3780 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 7665 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3080 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7980 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 9240 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 5670 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6363 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2436 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2310 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2436 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*x+c)^2*(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/105*(105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(d*x + c) + 2*(105*a^3*e^(12*d*x + 12*c) + 630*a^2*b*e^(12*d*x + 12
*c) + 945*a*b^2*e^(12*d*x + 12*c) + 420*b^3*e^(12*d*x + 12*c) + 630*a^3*e^(10*d*x + 10*c) + 3150*a^2*b*e^(10*d
*x + 10*c) + 3780*a*b^2*e^(10*d*x + 10*c) + 1260*b^3*e^(10*d*x + 10*c) + 1575*a^3*e^(8*d*x + 8*c) + 6720*a^2*b
*e^(8*d*x + 8*c) + 7665*a*b^2*e^(8*d*x + 8*c) + 3080*b^3*e^(8*d*x + 8*c) + 2100*a^3*e^(6*d*x + 6*c) + 7980*a^2
*b*e^(6*d*x + 6*c) + 9240*a*b^2*e^(6*d*x + 6*c) + 3080*b^3*e^(6*d*x + 6*c) + 1575*a^3*e^(4*d*x + 4*c) + 5670*a
^2*b*e^(4*d*x + 4*c) + 6363*a*b^2*e^(4*d*x + 4*c) + 2436*b^3*e^(4*d*x + 4*c) + 630*a^3*e^(2*d*x + 2*c) + 2310*
a^2*b*e^(2*d*x + 2*c) + 2436*a*b^2*e^(2*d*x + 2*c) + 812*b^3*e^(2*d*x + 2*c) + 105*a^3 + 420*a^2*b + 483*a*b^2
+ 176*b^3)/(e^(2*d*x + 2*c) + 1)^7)/d

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Mupad [B]
time = 1.22, size = 106, normalized size = 1.13 \begin {gather*} x\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )-\frac {\mathrm {tanh}\left (c+d\,x\right )\,{\left (a+b\right )}^3}{d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (3\,a^2\,b+3\,a\,b^2+b^3\right )}{3\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (b^3+3\,a\,b^2\right )}{5\,d}-\frac {b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^7}{7\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(c + d*x)^2*(a + b*tanh(c + d*x)^2)^3,x)

[Out]

x*(3*a*b^2 + 3*a^2*b + a^3 + b^3) - (tanh(c + d*x)*(a + b)^3)/d - (tanh(c + d*x)^3*(3*a*b^2 + 3*a^2*b + b^3))/
(3*d) - (tanh(c + d*x)^5*(3*a*b^2 + b^3))/(5*d) - (b^3*tanh(c + d*x)^7)/(7*d)

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