3.2.69 \(\int (a+b \tanh ^2(c+d x))^5 \, dx\) [169]

Optimal. Leaf size=160 \[ (a+b)^5 x-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^5 \tanh ^9(c+d x)}{9 d} \]

[Out]

(a+b)^5*x-b*(5*a^4+10*a^3*b+10*a^2*b^2+5*a*b^3+b^4)*tanh(d*x+c)/d-1/3*b^2*(10*a^3+10*a^2*b+5*a*b^2+b^3)*tanh(d
*x+c)^3/d-1/5*b^3*(10*a^2+5*a*b+b^2)*tanh(d*x+c)^5/d-1/7*b^4*(5*a+b)*tanh(d*x+c)^7/d-1/9*b^5*tanh(d*x+c)^9/d

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 398, 212} \begin {gather*} -\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}+x (a+b)^5-\frac {b^5 \tanh ^9(c+d x)}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tanh[c + d*x]^2)^5,x]

[Out]

(a + b)^5*x - (b*(5*a^4 + 10*a^3*b + 10*a^2*b^2 + 5*a*b^3 + b^4)*Tanh[c + d*x])/d - (b^2*(10*a^3 + 10*a^2*b +
5*a*b^2 + b^3)*Tanh[c + d*x]^3)/(3*d) - (b^3*(10*a^2 + 5*a*b + b^2)*Tanh[c + d*x]^5)/(5*d) - (b^4*(5*a + b)*Ta
nh[c + d*x]^7)/(7*d) - (b^5*Tanh[c + d*x]^9)/(9*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 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3742

Int[((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[(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, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps

\begin {align*} \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^5}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right )-b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) x^2-b^3 \left (10 a^2+5 a b+b^2\right ) x^4-b^4 (5 a+b) x^6-b^5 x^8+\frac {(a+b)^5}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^5 \tanh ^9(c+d x)}{9 d}+\frac {(a+b)^5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^5 x-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^5 \tanh ^9(c+d x)}{9 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.58, size = 170, normalized size = 1.06 \begin {gather*} \frac {\tanh (c+d x) \left (\frac {315 (a+b)^5 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}-b \left (315 \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right )+105 b \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^2(c+d x)+63 b^2 \left (10 a^2+5 a b+b^2\right ) \tanh ^4(c+d x)+45 b^3 (5 a+b) \tanh ^6(c+d x)+35 b^4 \tanh ^8(c+d x)\right )\right )}{315 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tanh[c + d*x]^2)^5,x]

[Out]

(Tanh[c + d*x]*((315*(a + b)^5*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2] - b*(315*(5*a^4 + 10*a^3*
b + 10*a^2*b^2 + 5*a*b^3 + b^4) + 105*b*(10*a^3 + 10*a^2*b + 5*a*b^2 + b^3)*Tanh[c + d*x]^2 + 63*b^2*(10*a^2 +
5*a*b + b^2)*Tanh[c + d*x]^4 + 45*b^3*(5*a + b)*Tanh[c + d*x]^6 + 35*b^4*Tanh[c + d*x]^8)))/(315*d)

________________________________________________________________________________________

Maple [A]
time = 0.34, size = 303, normalized size = 1.89

method	result	size
derivativedivides	\(\frac {-5 a \,b^{4} \tanh \left (d x +c \right )-10 a^{2} b^{3} \tanh \left (d x +c \right )-10 a^{3} b^{2} \tanh \left (d x +c \right )-5 a^{4} b \tanh \left (d x +c \right )-\frac {5 a \,b^{4} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {10 a^{2} b^{3} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {10 a^{3} b^{2} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-2 a^{2} b^{3} \left (\tanh ^{5}\left (d x +c \right )\right )-a \,b^{4} \left (\tanh ^{5}\left (d x +c \right )\right )-\frac {5 a \,b^{4} \left (\tanh ^{7}\left (d x +c \right )\right )}{7}-\frac {b^{5} \left (\tanh ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {b^{5} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {b^{5} \left (\tanh ^{7}\left (d x +c \right )\right )}{7}-\frac {b^{5} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}-b^{5} \tanh \left (d x +c \right )+\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}}{d}\)	\(303\)
default	\(\frac {-5 a \,b^{4} \tanh \left (d x +c \right )-10 a^{2} b^{3} \tanh \left (d x +c \right )-10 a^{3} b^{2} \tanh \left (d x +c \right )-5 a^{4} b \tanh \left (d x +c \right )-\frac {5 a \,b^{4} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {10 a^{2} b^{3} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {10 a^{3} b^{2} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-2 a^{2} b^{3} \left (\tanh ^{5}\left (d x +c \right )\right )-a \,b^{4} \left (\tanh ^{5}\left (d x +c \right )\right )-\frac {5 a \,b^{4} \left (\tanh ^{7}\left (d x +c \right )\right )}{7}-\frac {b^{5} \left (\tanh ^{9}\left (d x +c \right )\right )}{9}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {b^{5} \left (\tanh ^{3}\left (d x +c \right )\right )}{3}-\frac {b^{5} \left (\tanh ^{7}\left (d x +c \right )\right )}{7}-\frac {b^{5} \left (\tanh ^{5}\left (d x +c \right )\right )}{5}-b^{5} \tanh \left (d x +c \right )+\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (1+\tanh \left (d x +c \right )\right )}{2}}{d}\)	\(303\)
risch	\(a^{5} x +5 a^{4} x b +10 a^{3} b^{2} x +10 a^{2} b^{3} x +5 a \,b^{4} x +b^{5} x +\frac {2 b \left (88200 a^{4} {\mathrm e}^{10 d x +10 c}+39438 b^{4} {\mathrm e}^{8 d x +8 c}+44100 a^{4} {\mathrm e}^{12 d x +12 c}+1575 a^{4}+88200 a^{4} {\mathrm e}^{6 d x +6 c}+90300 a \,b^{3} {\mathrm e}^{12 d x +12 c}+245700 a^{3} b \,{\mathrm e}^{10 d x +10 c}+2640 a \,b^{3}+563 b^{4}+4830 a^{2} b^{2}+283500 a^{2} b^{2} {\mathrm e}^{10 d x +10 c}+157500 a \,b^{3} {\mathrm e}^{10 d x +10 c}+161700 a^{2} b^{2} {\mathrm e}^{12 d x +12 c}+6300 b^{4} {\mathrm e}^{14 d x +14 c}+4200 a^{3} b +44100 a^{4} {\mathrm e}^{4 d x +4 c}+325080 a^{2} b^{2} {\mathrm e}^{8 d x +8 c}+175140 a \,b^{3} {\mathrm e}^{8 d x +8 c}+216300 a^{3} b \,{\mathrm e}^{6 d x +6 c}+244020 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}+131460 a \,b^{3} {\mathrm e}^{6 d x +6 c}+107100 a^{3} b \,{\mathrm e}^{4 d x +4 c}+9450 a^{2} b^{2} {\mathrm e}^{16 d x +16 c}+6300 a \,b^{3} {\mathrm e}^{16 d x +16 c}+56700 a^{2} b^{2} {\mathrm e}^{14 d x +14 c}+31500 a \,b^{3} {\mathrm e}^{14 d x +14 c}+117180 a^{2} b^{2} {\mathrm e}^{4 d x +4 c}+63540 a \,b^{3} {\mathrm e}^{4 d x +4 c}+31500 a^{3} b \,{\mathrm e}^{2 d x +2 c}+34020 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}+17460 a \,b^{3} {\mathrm e}^{2 d x +2 c}+283500 a^{3} b \,{\mathrm e}^{8 d x +8 c}+136500 a^{3} b \,{\mathrm e}^{12 d x +12 c}+6300 a^{3} b \,{\mathrm e}^{16 d x +16 c}+44100 a^{3} b \,{\mathrm e}^{14 d x +14 c}+13968 b^{4} {\mathrm e}^{4 d x +4 c}+3492 b^{4} {\mathrm e}^{2 d x +2 c}+12600 a^{4} {\mathrm e}^{2 d x +2 c}+1575 b^{4} {\mathrm e}^{16 d x +16 c}+21000 b^{4} {\mathrm e}^{12 d x +12 c}+110250 a^{4} {\mathrm e}^{8 d x +8 c}+1575 a^{4} {\mathrm e}^{16 d x +16 c}+12600 a^{4} {\mathrm e}^{14 d x +14 c}+26292 b^{4} {\mathrm e}^{6 d x +6 c}+31500 b^{4} {\mathrm e}^{10 d x +10 c}\right )}{315 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{9}}\)	\(694\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tanh(d*x+c)^2)^5,x,method=_RETURNVERBOSE)

[Out]

1/d*(-5*a*b^4*tanh(d*x+c)-10*a^2*b^3*tanh(d*x+c)-10*a^3*b^2*tanh(d*x+c)-5*a^4*b*tanh(d*x+c)-5/3*a*b^4*tanh(d*x
+c)^3-10/3*a^2*b^3*tanh(d*x+c)^3-10/3*a^3*b^2*tanh(d*x+c)^3-2*a^2*b^3*tanh(d*x+c)^5-a*b^4*tanh(d*x+c)^5-5/7*a*
b^4*tanh(d*x+c)^7-1/9*b^5*tanh(d*x+c)^9-1/2*(a^5+5*a^4*b+10*a^3*b^2+10*a^2*b^3+5*a*b^4+b^5)*ln(tanh(d*x+c)-1)-
1/3*b^5*tanh(d*x+c)^3-1/7*b^5*tanh(d*x+c)^7-1/5*b^5*tanh(d*x+c)^5-b^5*tanh(d*x+c)+1/2*(a^5+5*a^4*b+10*a^3*b^2+
10*a^2*b^3+5*a*b^4+b^5)*ln(1+tanh(d*x+c)))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (152) = 304\).
time = 0.31, size = 624, normalized size = 3.90 \begin {gather*} \frac {1}{315} \, b^{5} {\left (315 \, x + \frac {315 \, c}{d} - \frac {2 \, {\left (3492 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13968 \, e^{\left (-4 \, d x - 4 \, c\right )} + 26292 \, e^{\left (-6 \, d x - 6 \, c\right )} + 39438 \, e^{\left (-8 \, d x - 8 \, c\right )} + 31500 \, e^{\left (-10 \, d x - 10 \, c\right )} + 21000 \, e^{\left (-12 \, d x - 12 \, c\right )} + 6300 \, e^{\left (-14 \, d x - 14 \, c\right )} + 1575 \, e^{\left (-16 \, d x - 16 \, c\right )} + 563\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} + 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} + 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} + 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} + 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} + 1\right )}}\right )} + \frac {1}{21} \, a b^{4} {\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 {2}{3} \, a^{2} b^{3} {\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 )} + \frac {10}{3} \, a^{3} b^{2} {\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 )} + 5 \, a^{4} b {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="maxima")

[Out]

1/315*b^5*(315*x + 315*c/d - 2*(3492*e^(-2*d*x - 2*c) + 13968*e^(-4*d*x - 4*c) + 26292*e^(-6*d*x - 6*c) + 3943
8*e^(-8*d*x - 8*c) + 31500*e^(-10*d*x - 10*c) + 21000*e^(-12*d*x - 12*c) + 6300*e^(-14*d*x - 14*c) + 1575*e^(-
16*d*x - 16*c) + 563)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c
) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x
- 18*c) + 1))) + 1/21*a*b^4*(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 - 14*c) + 1))) + 2/3*a^2*b^3*(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))) + 10/3*a^3*b^2*(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))) + 5*a^4*b*(x + c/d
- 2/(d*(e^(-2*d*x - 2*c) + 1))) + a^5*x

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2133 vs. \(2 (152) = 304\).
time = 0.40, size = 2133, normalized size = 13.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="fricas")

[Out]

1/315*((1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10
*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^9 + 9*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 56
3*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)*sinh(d*x + c)^8 - (15
75*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*sinh(d*x + c)^9 + 9*(1575*a^4*b + 4200*a^3*b^2
+ 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cos
h(d*x + c)^7 - 9*(1225*a^4*b + 2800*a^3*b^2 + 2730*a^2*b^3 + 1240*a*b^4 + 213*b^5 + 4*(1575*a^4*b + 4200*a^3*b
^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(4*(1575*a^4*b + 4200*a^3*b^2
+ 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cos
h(d*x + c)^3 + 3*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^
3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^6 + 36*(1575*a^4*b + 4200*a^3*b^2 + 4830
*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x
+ c)^5 - 9*(3500*a^4*b + 7000*a^3*b^2 + 6720*a^2*b^3 + 3560*a*b^4 + 852*b^5 + 14*(1575*a^4*b + 4200*a^3*b^2 +
4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^4 + 21*(1225*a^4*b + 2800*a^3*b^2 + 2730*a^2*b^3 + 1240*a*b
^4 + 213*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4
+ 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^5 + 35*(1575*a^4*
b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^
4 + b^5)*d*x)*cosh(d*x + c)^3 + 20*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5
+ 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 84*(1575*a^4*b + 4
200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b
^5)*d*x)*cosh(d*x + c)^3 - 3*(28*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x +
c)^6 + 14700*a^4*b + 26600*a^3*b^2 + 27440*a^2*b^3 + 13720*a*b^4 + 1764*b^5 + 105*(1225*a^4*b + 2800*a^3*b^2 +
2730*a^2*b^3 + 1240*a*b^4 + 213*b^5)*cosh(d*x + c)^4 + 120*(875*a^4*b + 1750*a^3*b^2 + 1680*a^2*b^3 + 890*a*b
^4 + 213*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(4*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 +
563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^7 + 21*(1575*a^4*b
+ 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4
+ b^5)*d*x)*cosh(d*x + c)^5 + 40*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5
+ 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c)^3 + 28*(1575*a^4*b + 4200*a^3*b^2 + 48
30*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*
x + c))*sinh(d*x + c)^2 + 126*(1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5 + 315*(a^5 + 5*
a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*d*x)*cosh(d*x + c) - 9*((1575*a^4*b + 4200*a^3*b^2 + 4830*a^2
*b^3 + 2640*a*b^4 + 563*b^5)*cosh(d*x + c)^8 + 7*(1225*a^4*b + 2800*a^3*b^2 + 2730*a^2*b^3 + 1240*a*b^4 + 213*
b^5)*cosh(d*x + c)^6 + 2450*a^4*b + 4200*a^3*b^2 + 4620*a^2*b^3 + 1960*a*b^4 + 882*b^5 + 20*(875*a^4*b + 1750*
a^3*b^2 + 1680*a^2*b^3 + 890*a*b^4 + 213*b^5)*cosh(d*x + c)^4 + 28*(525*a^4*b + 950*a^3*b^2 + 980*a^2*b^3 + 49
0*a*b^4 + 63*b^5)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + 9*d
*cosh(d*x + c)^7 + 21*(4*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^6 + 36*d*cosh(d*x + c)^5 + 9*(14
*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 + 20*d*cosh(d*x + c))*sinh(d*x + c)^4 + 84*d*cosh(d*x + c)^3 + 9*(4*
d*cosh(d*x + c)^7 + 21*d*cosh(d*x + c)^5 + 40*d*cosh(d*x + c)^3 + 28*d*cosh(d*x + c))*sinh(d*x + c)^2 + 126*d*
cosh(d*x + c))

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (148) = 296\).
time = 0.38, size = 308, normalized size = 1.92 \begin {gather*} \begin {cases} a^{5} x + 5 a^{4} b x - \frac {5 a^{4} b \tanh {\left (c + d x \right )}}{d} + 10 a^{3} b^{2} x - \frac {10 a^{3} b^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {10 a^{3} b^{2} \tanh {\left (c + d x \right )}}{d} + 10 a^{2} b^{3} x - \frac {2 a^{2} b^{3} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac {10 a^{2} b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {10 a^{2} b^{3} \tanh {\left (c + d x \right )}}{d} + 5 a b^{4} x - \frac {5 a b^{4} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {a b^{4} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac {5 a b^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {5 a b^{4} \tanh {\left (c + d x \right )}}{d} + b^{5} x - \frac {b^{5} \tanh ^{9}{\left (c + d x \right )}}{9 d} - \frac {b^{5} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{5} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{5} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{5} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)**2)**5,x)

[Out]

Piecewise((a**5*x + 5*a**4*b*x - 5*a**4*b*tanh(c + d*x)/d + 10*a**3*b**2*x - 10*a**3*b**2*tanh(c + d*x)**3/(3*
d) - 10*a**3*b**2*tanh(c + d*x)/d + 10*a**2*b**3*x - 2*a**2*b**3*tanh(c + d*x)**5/d - 10*a**2*b**3*tanh(c + d*
x)**3/(3*d) - 10*a**2*b**3*tanh(c + d*x)/d + 5*a*b**4*x - 5*a*b**4*tanh(c + d*x)**7/(7*d) - a*b**4*tanh(c + d*
x)**5/d - 5*a*b**4*tanh(c + d*x)**3/(3*d) - 5*a*b**4*tanh(c + d*x)/d + b**5*x - b**5*tanh(c + d*x)**9/(9*d) -
b**5*tanh(c + d*x)**7/(7*d) - b**5*tanh(c + d*x)**5/(5*d) - b**5*tanh(c + d*x)**3/(3*d) - b**5*tanh(c + d*x)/d
, Ne(d, 0)), (x*(a + b*tanh(c)**2)**5, True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (152) = 304\).
time = 0.47, size = 721, normalized size = 4.51 \begin {gather*} \frac {315 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (1575 \, a^{4} b e^{\left (16 \, d x + 16 \, c\right )} + 6300 \, a^{3} b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 9450 \, a^{2} b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 6300 \, a b^{4} e^{\left (16 \, d x + 16 \, c\right )} + 1575 \, b^{5} e^{\left (16 \, d x + 16 \, c\right )} + 12600 \, a^{4} b e^{\left (14 \, d x + 14 \, c\right )} + 44100 \, a^{3} b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 56700 \, a^{2} b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 31500 \, a b^{4} e^{\left (14 \, d x + 14 \, c\right )} + 6300 \, b^{5} e^{\left (14 \, d x + 14 \, c\right )} + 44100 \, a^{4} b e^{\left (12 \, d x + 12 \, c\right )} + 136500 \, a^{3} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 161700 \, a^{2} b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 90300 \, a b^{4} e^{\left (12 \, d x + 12 \, c\right )} + 21000 \, b^{5} e^{\left (12 \, d x + 12 \, c\right )} + 88200 \, a^{4} b e^{\left (10 \, d x + 10 \, c\right )} + 245700 \, a^{3} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 283500 \, a^{2} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 157500 \, a b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 31500 \, b^{5} e^{\left (10 \, d x + 10 \, c\right )} + 110250 \, a^{4} b e^{\left (8 \, d x + 8 \, c\right )} + 283500 \, a^{3} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 325080 \, a^{2} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 175140 \, a b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 39438 \, b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 88200 \, a^{4} b e^{\left (6 \, d x + 6 \, c\right )} + 216300 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 244020 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 131460 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 26292 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 44100 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 107100 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 117180 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 63540 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 13968 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 12600 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 31500 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 34020 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 17460 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 3492 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 1575 \, a^{4} b + 4200 \, a^{3} b^{2} + 4830 \, a^{2} b^{3} + 2640 \, a b^{4} + 563 \, b^{5}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}}}{315 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tanh(d*x+c)^2)^5,x, algorithm="giac")

[Out]

1/315*(315*(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5)*(d*x + c) + 2*(1575*a^4*b*e^(16*d*x + 16*
c) + 6300*a^3*b^2*e^(16*d*x + 16*c) + 9450*a^2*b^3*e^(16*d*x + 16*c) + 6300*a*b^4*e^(16*d*x + 16*c) + 1575*b^5
*e^(16*d*x + 16*c) + 12600*a^4*b*e^(14*d*x + 14*c) + 44100*a^3*b^2*e^(14*d*x + 14*c) + 56700*a^2*b^3*e^(14*d*x
+ 14*c) + 31500*a*b^4*e^(14*d*x + 14*c) + 6300*b^5*e^(14*d*x + 14*c) + 44100*a^4*b*e^(12*d*x + 12*c) + 136500
*a^3*b^2*e^(12*d*x + 12*c) + 161700*a^2*b^3*e^(12*d*x + 12*c) + 90300*a*b^4*e^(12*d*x + 12*c) + 21000*b^5*e^(1
2*d*x + 12*c) + 88200*a^4*b*e^(10*d*x + 10*c) + 245700*a^3*b^2*e^(10*d*x + 10*c) + 283500*a^2*b^3*e^(10*d*x +
10*c) + 157500*a*b^4*e^(10*d*x + 10*c) + 31500*b^5*e^(10*d*x + 10*c) + 110250*a^4*b*e^(8*d*x + 8*c) + 283500*a
^3*b^2*e^(8*d*x + 8*c) + 325080*a^2*b^3*e^(8*d*x + 8*c) + 175140*a*b^4*e^(8*d*x + 8*c) + 39438*b^5*e^(8*d*x +
8*c) + 88200*a^4*b*e^(6*d*x + 6*c) + 216300*a^3*b^2*e^(6*d*x + 6*c) + 244020*a^2*b^3*e^(6*d*x + 6*c) + 131460*
a*b^4*e^(6*d*x + 6*c) + 26292*b^5*e^(6*d*x + 6*c) + 44100*a^4*b*e^(4*d*x + 4*c) + 107100*a^3*b^2*e^(4*d*x + 4*
c) + 117180*a^2*b^3*e^(4*d*x + 4*c) + 63540*a*b^4*e^(4*d*x + 4*c) + 13968*b^5*e^(4*d*x + 4*c) + 12600*a^4*b*e^
(2*d*x + 2*c) + 31500*a^3*b^2*e^(2*d*x + 2*c) + 34020*a^2*b^3*e^(2*d*x + 2*c) + 17460*a*b^4*e^(2*d*x + 2*c) +
3492*b^5*e^(2*d*x + 2*c) + 1575*a^4*b + 4200*a^3*b^2 + 4830*a^2*b^3 + 2640*a*b^4 + 563*b^5)/(e^(2*d*x + 2*c) +
1)^9)/d

________________________________________________________________________________________

Mupad [B]
time = 1.32, size = 188, normalized size = 1.18 \begin {gather*} x\,\left (a^5+5\,a^4\,b+10\,a^3\,b^2+10\,a^2\,b^3+5\,a\,b^4+b^5\right )-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (10\,a^3\,b^2+10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{3\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{5\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^7\,\left (b^5+5\,a\,b^4\right )}{7\,d}-\frac {b^5\,{\mathrm {tanh}\left (c+d\,x\right )}^9}{9\,d}-\frac {b\,\mathrm {tanh}\left (c+d\,x\right )\,\left (5\,a^4+10\,a^3\,b+10\,a^2\,b^2+5\,a\,b^3+b^4\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tanh(c + d*x)^2)^5,x)

[Out]

x*(5*a*b^4 + 5*a^4*b + a^5 + b^5 + 10*a^2*b^3 + 10*a^3*b^2) - (tanh(c + d*x)^3*(5*a*b^4 + b^5 + 10*a^2*b^3 + 1
0*a^3*b^2))/(3*d) - (tanh(c + d*x)^5*(5*a*b^4 + b^5 + 10*a^2*b^3))/(5*d) - (tanh(c + d*x)^7*(5*a*b^4 + b^5))/(
7*d) - (b^5*tanh(c + d*x)^9)/(9*d) - (b*tanh(c + d*x)*(5*a*b^3 + 10*a^3*b + 5*a^4 + b^4 + 10*a^2*b^2))/d

________________________________________________________________________________________