3.1.71 \(\int \text {sech}^3(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [71]

Optimal. Leaf size=196 \[ \frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {ArcTan}(\sinh (c+d x))}{128 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {sech}(c+d x) \tanh (c+d x)}{128 d}+\frac {b \left (72 a^2+92 a b+35 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {b (12 a+7 b) \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}+\frac {b \text {sech}^7(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d} \]

[Out]

1/128*(64*a^3+144*a^2*b+120*a*b^2+35*b^3)*arctan(sinh(d*x+c))/d+1/128*(64*a^3+144*a^2*b+120*a*b^2+35*b^3)*sech
(d*x+c)*tanh(d*x+c)/d+1/192*b*(72*a^2+92*a*b+35*b^2)*sech(d*x+c)^3*tanh(d*x+c)/d+1/48*b*(12*a+7*b)*sech(d*x+c)
^5*(a+b+a*sinh(d*x+c)^2)*tanh(d*x+c)/d+1/8*b*sech(d*x+c)^7*(a+b+a*sinh(d*x+c)^2)^2*tanh(d*x+c)/d

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Rubi [A]
time = 0.15, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4232, 424, 540, 393, 205, 209} \begin {gather*} \frac {b \left (72 a^2+92 a b+35 b^2\right ) \tanh (c+d x) \text {sech}^3(c+d x)}{192 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {ArcTan}(\sinh (c+d x))}{128 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \tanh (c+d x) \text {sech}(c+d x)}{128 d}+\frac {b \tanh (c+d x) \text {sech}^7(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{8 d}+\frac {b (12 a+7 b) \tanh (c+d x) \text {sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{48 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*ArcTan[Sinh[c + d*x]])/(128*d) + ((64*a^3 + 144*a^2*b + 120*a*b^2 +
35*b^3)*Sech[c + d*x]*Tanh[c + d*x])/(128*d) + (b*(72*a^2 + 92*a*b + 35*b^2)*Sech[c + d*x]^3*Tanh[c + d*x])/(
192*d) + (b*(12*a + 7*b)*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]^2)*Tanh[c + d*x])/(48*d) + (b*Sech[c + d*x]^
7*(a + b + a*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(8*d)

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 209

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

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 424

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 540

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*n*(p + 1))), x] + Dist[1/(a*b*n*(p + 1)), Int[(a + b*x
^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*(p + 1) + (b*e - a*f)*(n*q + 1))
*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && GtQ[q, 0]

Rule 4232

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b+a x^2\right )^3}{\left (1+x^2\right )^5} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \text {sech}^7(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}+\frac {\text {Subst}\left (\int \frac {\left (a+b+a x^2\right ) \left ((a+b) (8 a+7 b)+a (8 a+3 b) x^2\right )}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac {b (12 a+7 b) \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}+\frac {b \text {sech}^7(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}-\frac {\text {Subst}\left (\int \frac {-(a+b) \left (48 a^2+78 a b+35 b^2\right )-3 a \left (16 a^2+18 a b+7 b^2\right ) x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{48 d}\\ &=\frac {b \left (72 a^2+92 a b+35 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {b (12 a+7 b) \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}+\frac {b \text {sech}^7(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{64 d}\\ &=\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {sech}(c+d x) \tanh (c+d x)}{128 d}+\frac {b \left (72 a^2+92 a b+35 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {b (12 a+7 b) \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}+\frac {b \text {sech}^7(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{128 d}\\ &=\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \tan ^{-1}(\sinh (c+d x))}{128 d}+\frac {\left (64 a^3+144 a^2 b+120 a b^2+35 b^3\right ) \text {sech}(c+d x) \tanh (c+d x)}{128 d}+\frac {b \left (72 a^2+92 a b+35 b^2\right ) \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {b (12 a+7 b) \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{48 d}+\frac {b \text {sech}^7(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{8 d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.22, size = 1618, normalized size = 8.26 \begin {gather*} \frac {\coth ^6(c+d x) \text {csch}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left (344123325 (a+b)^3 \sinh ^2(c+d x)[phone] a (a+b)^2 \sinh ^4(c+d x)[phone] (a+b)^3 \sinh ^4(c+d x)[phone] a^2 (a+b) \sinh ^6(c+d x)[phone] a (a+b)^2 \sinh ^6(c+d x)[phone] (a+b)^3 \sinh ^6(c+d x)[phone] a^3 \sinh ^8(c+d x)[phone] a^2 (a+b) \sinh ^8(c+d x)[phone] a (a+b)^2 \sinh ^8(c+d x)[phone] a^3 \sinh ^{10}(c+d x)[phone] a^2 (a+b) \sinh ^{10}(c+d x)-257600 (a+b)^3 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{10}(c+d x)-65408 (a+b)^3 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{10}(c+d x)-8960 (a+b)^3 \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{10}(c+d x)-512 (a+b)^3 \, _8F_7\left (\frac {3}{2},2,2,2,2,2,2,2;1,1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{10}(c+d x)+6134499 a^3 \sinh ^{12}(c+d x)-613440 a (a+b)^2 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{12}(c+d x)-171648 a (a+b)^2 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{12}(c+d x)-25344 a (a+b)^2 \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{12}(c+d x)-1536 a (a+b)^2 \, _8F_7\left (\frac {3}{2},2,2,2,2,2,2,2;1,1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{12}(c+d x)-495552 a^2 (a+b) \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{14}(c+d x)-150144 a^2 (a+b) \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{14}(c+d x)-23808 a^2 (a+b) \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{14}(c+d x)-1536 a^2 (a+b) \, _8F_7\left (\frac {3}{2},2,2,2,2,2,2,2;1,1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{14}(c+d x)-136640 a^3 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{16}(c+d x)-43904 a^3 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{16}(c+d x)-7424 a^3 \, _7F_6\left (\frac {3}{2},2,2,2,2,2,2;1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{16}(c+d x)-512 a^3 \, _8F_7\left (\frac {3}{2},2,2,2,2,2,2,2;1,1,1,1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^{16}(c+d x)[phone] (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sqrt {-\sinh ^2(c+d x)}[phone] a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}+656775 a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{10}(c+d x) \sqrt {-\sinh ^2(c+d x)}+524475 a^2 (a+b) \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{10}(c+d x) \sqrt {-\sinh ^2(c+d x)}+143325 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{12}(c+d x) \sqrt {-\sinh ^2(c+d x)}-760096575 a (a+b)^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (-\sinh ^2(c+d x)\right )^{3/2}-327796875 (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (-\sinh ^2(c+d x)\right )^{3/2}+117495 (a+b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}\right )}{181440 d (a+2 b+a \cosh (2 c+2 d x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Coth[c + d*x]^6*Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^3*(344123325*(a + b)^3*Sinh[c + d*x]^2 + 760096575*a*(a
+ b)^2*Sinh[c + d*x]^4 + 213089100*(a + b)^3*Sinh[c + d*x]^4 + 578580975*a^2*(a + b)*Sinh[c + d*x]^6 + 481962
600*a*(a + b)^2*Sinh[c + d*x]^6 + 12757815*(a + b)^3*Sinh[c + d*x]^6 + 153475245*a^3*Sinh[c + d*x]^8 + 3722657
40*a^2*(a + b)*Sinh[c + d*x]^8 + 28676025*a*(a + b)^2*Sinh[c + d*x]^8 + 99450960*a^3*Sinh[c + d*x]^10 + 226393
65*a^2*(a + b)*Sinh[c + d*x]^10 - 257600*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh
[c + d*x]^2]*Sinh[c + d*x]^10 - 65408*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -S
inh[c + d*x]^2]*Sinh[c + d*x]^10 - 8960*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 1
1/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10 - 512*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2, 2}, {1, 1,
1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^10 + 6134499*a^3*Sinh[c + d*x]^12 - 613440*a*(a + b)^2*Hype
rgeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 171648*a*(a + b)^2*Hype
rgeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 25344*a*(a + b)^2
*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 1536*a
*(a + b)^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c +
d*x]^12 - 495552*a^2*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c +
d*x]^14 - 150144*a^2*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sin
h[c + d*x]^14 - 23808*a^2*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c +
d*x]^2]*Sinh[c + d*x]^14 - 1536*a^2*(a + b)*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 1, 1
1/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^14 - 136640*a^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -S
inh[c + d*x]^2]*Sinh[c + d*x]^16 - 43904*a^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh
[c + d*x]^2]*Sinh[c + d*x]^16 - 7424*a^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Si
nh[c + d*x]^2]*Sinh[c + d*x]^16 - 512*a^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 1, 11/
2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^16 + 344123325*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sqrt[-Sinh[c + d*
x]^2] + 578580975*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 7353281
25*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 53198775*(a + b)^3*Arc
Tanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 153475245*a^3*ArcTanh[Sqrt[-Sinh[c + d*x
]^2]]*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 565126065*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c +
d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 121766085*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6*Sqrt[-Si
nh[c + d*x]^2] + 150609375*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 952985
25*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 656775*a*(a + b)^2*Arc
Tanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 25642575*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]
^2]]*Sinh[c + d*x]^10*Sqrt[-Sinh[c + d*x]^2] + 524475*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x
]^10*Sqrt[-Sinh[c + d*x]^2] + 143325*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^12*Sqrt[-Sinh[c + d*x]^
2] - 760096575*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(-Sinh[c + d*x]^2)^(3/2) - 327796875*(a + b)^3*ArcT
anh[Sqrt[-Sinh[c + d*x]^2]]*(-Sinh[c + d*x]^2)^(3/2) + 117495*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c
+ d*x]^4*(-Sinh[c + d*x]^2)^(3/2)))/(181440*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)

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Maple [C] Result contains complex when optimal does not.
time = 1.82, size = 611, normalized size = 3.12

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left (-360 a \,b^{2}+360 a \,b^{2} {\mathrm e}^{14 d x +14 c}+3312 a^{2} b \,{\mathrm e}^{12 d x +12 c}+2760 a \,b^{2} {\mathrm e}^{12 d x +12 c}+7344 a^{2} b \,{\mathrm e}^{10 d x +10 c}+9192 a \,b^{2} {\mathrm e}^{10 d x +10 c}+432 a^{2} b \,{\mathrm e}^{14 d x +14 c}-7344 a^{2} b \,{\mathrm e}^{4 d x +4 c}-3312 a^{2} b \,{\mathrm e}^{2 d x +2 c}-432 a^{2} b -192 a^{3}-105 b^{3}+6792 a \,b^{2} {\mathrm e}^{8 d x +8 c}-6792 a \,b^{2} {\mathrm e}^{6 d x +6 c}-9192 a \,b^{2} {\mathrm e}^{4 d x +4 c}+4464 a^{2} b \,{\mathrm e}^{8 d x +8 c}-2760 a \,b^{2} {\mathrm e}^{2 d x +2 c}-4464 a^{2} b \,{\mathrm e}^{6 d x +6 c}+192 a^{3} {\mathrm e}^{14 d x +14 c}+105 b^{3} {\mathrm e}^{14 d x +14 c}-960 a^{3} {\mathrm e}^{2 d x +2 c}+960 a^{3} {\mathrm e}^{12 d x +12 c}+805 b^{3} {\mathrm e}^{12 d x +12 c}+1728 a^{3} {\mathrm e}^{10 d x +10 c}-805 b^{3} {\mathrm e}^{2 d x +2 c}+5053 b^{3} {\mathrm e}^{8 d x +8 c}-1728 a^{3} {\mathrm e}^{4 d x +4 c}-2681 b^{3} {\mathrm e}^{4 d x +4 c}+2681 b^{3} {\mathrm e}^{10 d x +10 c}+960 a^{3} {\mathrm e}^{8 d x +8 c}-960 a^{3} {\mathrm e}^{6 d x +6 c}-5053 b^{3} {\mathrm e}^{6 d x +6 c}\right )}{192 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{8}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{2 d}+\frac {9 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{8 d}+\frac {15 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{16 d}+\frac {35 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{2 d}-\frac {9 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{8 d}-\frac {15 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{16 d}-\frac {35 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}\)	\(611\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*exp(d*x+c)*(-360*a*b^2+360*a*b^2*exp(14*d*x+14*c)+3312*a^2*b*exp(12*d*x+12*c)+2760*a*b^2*exp(12*d*x+12*c
)+7344*a^2*b*exp(10*d*x+10*c)+9192*a*b^2*exp(10*d*x+10*c)+432*a^2*b*exp(14*d*x+14*c)-7344*a^2*b*exp(4*d*x+4*c)
-3312*a^2*b*exp(2*d*x+2*c)-432*a^2*b-192*a^3-105*b^3+6792*a*b^2*exp(8*d*x+8*c)-6792*a*b^2*exp(6*d*x+6*c)-9192*
a*b^2*exp(4*d*x+4*c)+4464*a^2*b*exp(8*d*x+8*c)-2760*a*b^2*exp(2*d*x+2*c)-4464*a^2*b*exp(6*d*x+6*c)+192*a^3*exp
(14*d*x+14*c)+105*b^3*exp(14*d*x+14*c)-960*a^3*exp(2*d*x+2*c)+960*a^3*exp(12*d*x+12*c)+805*b^3*exp(12*d*x+12*c
)+1728*a^3*exp(10*d*x+10*c)-805*b^3*exp(2*d*x+2*c)+5053*b^3*exp(8*d*x+8*c)-1728*a^3*exp(4*d*x+4*c)-2681*b^3*ex
p(4*d*x+4*c)+2681*b^3*exp(10*d*x+10*c)+960*a^3*exp(8*d*x+8*c)-960*a^3*exp(6*d*x+6*c)-5053*b^3*exp(6*d*x+6*c))/
d/(1+exp(2*d*x+2*c))^8+1/2*I/d*ln(exp(d*x+c)+I)*a^3+9/8*I/d*ln(exp(d*x+c)+I)*a^2*b+15/16*I/d*ln(exp(d*x+c)+I)*
a*b^2+35/128*I/d*ln(exp(d*x+c)+I)*b^3-1/2*I/d*ln(exp(d*x+c)-I)*a^3-9/8*I/d*ln(exp(d*x+c)-I)*a^2*b-15/16*I/d*ln
(exp(d*x+c)-I)*a*b^2-35/128*I/d*ln(exp(d*x+c)-I)*b^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (186) = 372\).
time = 0.54, size = 556, normalized size = 2.84 \begin {gather*} -\frac {1}{192} \, b^{3} {\left (\frac {105 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {105 \, e^{\left (-d x - c\right )} + 805 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2681 \, e^{\left (-5 \, d x - 5 \, c\right )} + 5053 \, e^{\left (-7 \, d x - 7 \, c\right )} - 5053 \, e^{\left (-9 \, d x - 9 \, c\right )} - 2681 \, e^{\left (-11 \, d x - 11 \, c\right )} - 805 \, e^{\left (-13 \, d x - 13 \, c\right )} - 105 \, e^{\left (-15 \, d x - 15 \, c\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} + 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} + 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} + 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} + e^{\left (-16 \, d x - 16 \, c\right )} + 1\right )}}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {3}{4} \, a^{2} b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*b^3*(105*arctan(e^(-d*x - c))/d - (105*e^(-d*x - c) + 805*e^(-3*d*x - 3*c) + 2681*e^(-5*d*x - 5*c) + 50
53*e^(-7*d*x - 7*c) - 5053*e^(-9*d*x - 9*c) - 2681*e^(-11*d*x - 11*c) - 805*e^(-13*d*x - 13*c) - 105*e^(-15*d*
x - 15*c))/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 70*e^(-8*d*x - 8*c) + 56*e^(-1
0*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^(-14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1))) - 1/8*a*b^2*(15*arcta
n(e^(-d*x - c))/d - (15*e^(-d*x - c) + 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) - 198*e^(-7*d*x - 7*c) - 85*
e^(-9*d*x - 9*c) - 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20*e^(-6*d*x - 6*c) +
15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) - 3/4*a^2*b*(3*arctan(e^(-d*x - c))/d
- (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c) + 6
*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - a^3*(arctan(e^(-d*x - c))/d - (e^(-d*x - c)
- e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6114 vs. \(2 (186) = 372\).
time = 0.40, size = 6114, normalized size = 31.19 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^15 + 45*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*
b^3)*cosh(d*x + c)*sinh(d*x + c)^14 + 3*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*sinh(d*x + c)^15 + (960*a^3
+ 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^13 + (960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3 + 315*(64
*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^13 + 13*(105*(64*a^3 + 144*a^2*b + 120*a
*b^2 + 35*b^3)*cosh(d*x + c)^3 + (960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c))*sinh(d*x + c)^12
+ (1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^11 + (4095*(64*a^3 + 144*a^2*b + 120*a*b^2 +
35*b^3)*cosh(d*x + c)^4 + 1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3 + 78*(960*a^3 + 3312*a^2*b + 2760*a*b^
2 + 805*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 11*(819*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x +
c)^5 + 26*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^3 + (1728*a^3 + 7344*a^2*b + 9192*a*b^2
+ 2681*b^3)*cosh(d*x + c))*sinh(d*x + c)^10 + (960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c)^9 +
(15015*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^6 + 715*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 8
05*b^3)*cosh(d*x + c)^4 + 960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3 + 55*(1728*a^3 + 7344*a^2*b + 9192*a*b^
2 + 2681*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 3*(6435*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x +
c)^7 + 429*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^5 + 55*(1728*a^3 + 7344*a^2*b + 9192*a*
b^2 + 2681*b^3)*cosh(d*x + c)^3 + 3*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c))*sinh(d*x + c
)^8 - (960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c)^7 + (19305*(64*a^3 + 144*a^2*b + 120*a*b^2
+ 35*b^3)*cosh(d*x + c)^8 + 1716*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^6 + 330*(1728*a^3
+ 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^4 - 960*a^3 - 4464*a^2*b - 6792*a*b^2 - 5053*b^3 + 36*(96
0*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + (15015*(64*a^3 + 144*a^2*b + 12
0*a*b^2 + 35*b^3)*cosh(d*x + c)^9 + 1716*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^7 + 462*(
1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^5 + 84*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*
b^3)*cosh(d*x + c)^3 - 7*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - (1728
*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^5 + (9009*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*c
osh(d*x + c)^10 + 1287*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^8 + 462*(1728*a^3 + 7344*a^
2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^6 + 126*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c
)^4 - 1728*a^3 - 7344*a^2*b - 9192*a*b^2 - 2681*b^3 - 21*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d
*x + c)^2)*sinh(d*x + c)^5 + (4095*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^11 + 715*(960*a^3 +
3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^9 + 330*(1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh
(d*x + c)^7 + 126*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c)^5 - 35*(960*a^3 + 4464*a^2*b +
6792*a*b^2 + 5053*b^3)*cosh(d*x + c)^3 - 5*(1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c))*sinh
(d*x + c)^4 - (960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^3 + (1365*(64*a^3 + 144*a^2*b + 120*
a*b^2 + 35*b^3)*cosh(d*x + c)^12 + 286*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 805*b^3)*cosh(d*x + c)^10 + 165*(1
728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^8 + 84*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b
^3)*cosh(d*x + c)^6 - 35*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c)^4 - 960*a^3 - 3312*a^2*b
- 2760*a*b^2 - 805*b^3 - 10*(1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3
+ (315*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^13 + 78*(960*a^3 + 3312*a^2*b + 2760*a*b^2 + 80
5*b^3)*cosh(d*x + c)^11 + 55*(1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^9 + 36*(960*a^3 + 4
464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*x + c)^7 - 21*(960*a^3 + 4464*a^2*b + 6792*a*b^2 + 5053*b^3)*cosh(d*
x + c)^5 - 10*(1728*a^3 + 7344*a^2*b + 9192*a*b^2 + 2681*b^3)*cosh(d*x + c)^3 - 3*(960*a^3 + 3312*a^2*b + 2760
*a*b^2 + 805*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^
16 + 16*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)*sinh(d*x + c)^15 + (64*a^3 + 144*a^2*b + 120*a
*b^2 + 35*b^3)*sinh(d*x + c)^16 + 8*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^14 + 8*(64*a^3 + 1
44*a^2*b + 120*a*b^2 + 35*b^3 + 15*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^14
+ 112*(5*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^3 + (64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3
)*cosh(d*x + c))*sinh(d*x + c)^13 + 28*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x + c)^12 + 28*(65*(64
*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3)*cosh(d*x...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sech(c + d*x)**2)**3*sech(c + d*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 485 vs. \(2 (186) = 372\).
time = 0.42, size = 485, normalized size = 2.47 \begin {gather*} \frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (64 \, a^{3} + 144 \, a^{2} b + 120 \, a b^{2} + 35 \, b^{3}\right )} + \frac {4 \, {\left (192 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 432 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 360 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 105 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{7} + 2304 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 6336 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 5280 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 1540 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9216 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 29952 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 28032 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 8176 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12288 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 46080 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 50688 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 17856 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{4}}}{768 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(64*a^3 + 144*a^2*b + 120*a*b^2 + 35*b^3) + 4
*(192*a^3*(e^(d*x + c) - e^(-d*x - c))^7 + 432*a^2*b*(e^(d*x + c) - e^(-d*x - c))^7 + 360*a*b^2*(e^(d*x + c) -
e^(-d*x - c))^7 + 105*b^3*(e^(d*x + c) - e^(-d*x - c))^7 + 2304*a^3*(e^(d*x + c) - e^(-d*x - c))^5 + 6336*a^2
*b*(e^(d*x + c) - e^(-d*x - c))^5 + 5280*a*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 1540*b^3*(e^(d*x + c) - e^(-d*
x - c))^5 + 9216*a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 29952*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3 + 28032*a*b^2
*(e^(d*x + c) - e^(-d*x - c))^3 + 8176*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 12288*a^3*(e^(d*x + c) - e^(-d*x -
c)) + 46080*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 50688*a*b^2*(e^(d*x + c) - e^(-d*x - c)) + 17856*b^3*(e^(d*x
+ c) - e^(-d*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4)^4)/d

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Mupad [B]
time = 1.60, size = 931, normalized size = 4.75 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (64\,a^3\,\sqrt {d^2}+35\,b^3\,\sqrt {d^2}+120\,a\,b^2\,\sqrt {d^2}+144\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {4096\,a^6+18432\,a^5\,b+36096\,a^4\,b^2+39040\,a^3\,b^3+24480\,a^2\,b^4+8400\,a\,b^5+1225\,b^6}}\right )\,\sqrt {4096\,a^6+18432\,a^5\,b+36096\,a^4\,b^2+39040\,a^3\,b^3+24480\,a^2\,b^4+8400\,a\,b^5+1225\,b^6}}{64\,\sqrt {d^2}}-\frac {\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}+\frac {2\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (5\,a^3+18\,a^2\,b+24\,a\,b^2+16\,b^3\right )}{d}+\frac {a^3\,{\mathrm {e}}^{13\,c+13\,d\,x}}{2\,d}+\frac {3\,a\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{2\,d}+\frac {3\,a\,{\mathrm {e}}^{9\,c+9\,d\,x}\,\left (5\,a^2+16\,a\,b+16\,b^2\right )}{2\,d}+\frac {3\,a^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+2\,b\right )}{d}+\frac {3\,a^2\,{\mathrm {e}}^{11\,c+11\,d\,x}\,\left (a+2\,b\right )}{d}}{8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (48\,a\,b^2-37\,b^3\right )}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b-120\,a\,b^2+b^3\right )}{4\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {16\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (64\,a^3+144\,a^2\,b+120\,a\,b^2+35\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (6\,a\,b^2-29\,b^3\right )}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (-144\,a^3+144\,a^2\,b+120\,a\,b^2+35\,b^3\right )}{96\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (-288\,a^2\,b+24\,a\,b^2+7\,b^3\right )}{24\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((exp(d*x)*exp(c)*(64*a^3*(d^2)^(1/2) + 35*b^3*(d^2)^(1/2) + 120*a*b^2*(d^2)^(1/2) + 144*a^2*b*(d^2)^(1/2
)))/(d*(8400*a*b^5 + 18432*a^5*b + 4096*a^6 + 1225*b^6 + 24480*a^2*b^4 + 39040*a^3*b^3 + 36096*a^4*b^2)^(1/2))
)*(8400*a*b^5 + 18432*a^5*b + 4096*a^6 + 1225*b^6 + 24480*a^2*b^4 + 39040*a^3*b^3 + 36096*a^4*b^2)^(1/2))/(64*
(d^2)^(1/2)) - ((a^3*exp(c + d*x))/(2*d) + (2*exp(7*c + 7*d*x)*(24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/d + (a^
3*exp(13*c + 13*d*x))/(2*d) + (3*a*exp(5*c + 5*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(2*d) + (3*a*exp(9*c + 9*d*x)*(
16*a*b + 5*a^2 + 16*b^2))/(2*d) + (3*a^2*exp(3*c + 3*d*x)*(a + 2*b))/d + (3*a^2*exp(11*c + 11*d*x)*(a + 2*b))/
d)/(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) + 56*exp(10*c + 10*d*
x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1) + (2*exp(c + d*x)*(48*a*b^2 - 37*b
^3))/(3*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10
*d*x) + 1)) + (exp(c + d*x)*(24*a^2*b - 120*a*b^2 + b^3))/(4*d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*ex
p(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (16*b^3*exp(c + d*x))/(d*(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) +
35*exp(6*c + 6*d*x) + 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x)
+ 1)) + (exp(c + d*x)*(120*a*b^2 + 144*a^2*b + 64*a^3 + 35*b^3))/(64*d*(exp(2*c + 2*d*x) + 1)) - (4*exp(c + d
*x)*(6*a*b^2 - 29*b^3))/(3*d*(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*
d*x) + 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) + (exp(c + d*x)*(120*a*b^2 + 144*a^2*b - 144*a^3 + 35*b
^3))/(96*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (exp(c + d*x)*(24*a*b^2 - 288*a^2*b + 7*b^3))/(24*d*
(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1))

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