3.4.14 \(\int \text {sech}^7(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [314]

Optimal. Leaf size=154 \[ \frac {(a+b) \left (5 a^2-2 a b+5 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{16 d}+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 \left (a^2-b^2\right ) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {(a-b) \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d} \]

[Out]

1/16*(a+b)*(5*a^2-2*a*b+5*b^2)*arctan(sinh(d*x+c))/d+1/48*(a-b)*(15*a^2+14*a*b+15*b^2)*sech(d*x+c)*tanh(d*x+c)
/d+5/24*(a^2-b^2)*sech(d*x+c)^3*(a+b*sinh(d*x+c)^2)*tanh(d*x+c)/d+1/6*(a-b)*sech(d*x+c)^5*(a+b*sinh(d*x+c)^2)^
2*tanh(d*x+c)/d

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Rubi [A]
time = 0.10, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3269, 424, 540, 393, 209} \begin {gather*} \frac {(a+b) \left (5 a^2-2 a b+5 b^2\right ) \text {ArcTan}(\sinh (c+d x))}{16 d}+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \tanh (c+d x) \text {sech}(c+d x)}{48 d}+\frac {5 \left (a^2-b^2\right ) \tanh (c+d x) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )}{24 d}+\frac {(a-b) \tanh (c+d x) \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((a + b)*(5*a^2 - 2*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((a - b)*(15*a^2 + 14*a*b + 15*b^2)*Sech[c +
d*x]*Tanh[c + d*x])/(48*d) + (5*(a^2 - b^2)*Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2)*Tanh[c + d*x])/(24*d) + ((
a - b)*Sech[c + d*x]^5*(a + b*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(6*d)

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 3269

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

Rubi steps

\begin {align*} \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a-b) \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}+\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right ) \left (a (5 a+b)+b (a+5 b) x^2\right )}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{6 d}\\ &=\frac {5 \left (a^2-b^2\right ) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {(a-b) \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-a \left (15 a^2+4 a b+5 b^2\right )-b \left (5 a^2+4 a b+15 b^2\right ) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{24 d}\\ &=\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 \left (a^2-b^2\right ) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {(a-b) \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}+\frac {\left ((a+b) \left (5 a^2-2 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=\frac {(a+b) \left (5 a^2-2 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 \left (a^2-b^2\right ) \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {(a-b) \text {sech}^5(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 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 = 13.52, size = 1192, normalized size = 7.74 \begin {gather*} \frac {\text {csch}^5(c+d x) \left (-117228825 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right )-109265625 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^2(c+d x)-274542345 a^2 b \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^2(c+d x)-17069535 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x)-260465625 a^2 b \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x)-215549775 a b^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x)+142065 a^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x)-41427855 a^2 b \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x)-207173295 a b^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x)-58009455 b^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x)-210735 a^2 b \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x)-33756345 a b^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x)-56109375 b^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x)-174825 a b^2 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{10}(c+d x)-9261945 b^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{10}(c+d x)-48825 b^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^{12}(c+d x)[phone] a^3 \sqrt {-\sinh ^2(c+d x)}+4093425 a^3 \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a^2 b \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a b^2 \sinh ^4(c+d x) \sqrt {-\sinh ^2(c+d x)}+9514449 a^2 b \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] a b^2 \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] b^3 \sinh ^6(c+d x) \sqrt {-\sinh ^2(c+d x)}+7808535 a b^2 \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}[phone] b^3 \sinh ^8(c+d x) \sqrt {-\sinh ^2(c+d x)}+2160711 b^3 \sinh ^{10}(c+d x) \sqrt {-\sinh ^2(c+d x)}-70189350 a^3 \left (-\sinh ^2(c+d x)\right )^{3/2}-274542345 a^2 b \left (-\sinh ^2(c+d x)\right )^{3/2}+1024 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 ^6(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}+3072 a^2 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 ^8(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}+3072 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 ^{10}(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}+1024 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 ^{12}(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}+1536 \, _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 ^6(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2} \left (a+b \sinh ^2(c+d x)\right )^2 \left (9 a+7 b \sinh ^2(c+d x)\right )+256 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^6(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2} \left (295 a^3+741 a^2 b \sinh ^2(c+d x)+621 a b^2 \sinh ^4(c+d x)+175 b^3 \sinh ^6(c+d x)\right )\right )}{725760 d \sqrt {-\sinh ^2(c+d x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(Csch[c + d*x]^5*(-117228825*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] - 109265625*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2
]]*Sinh[c + d*x]^2 - 274542345*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 17069535*a^3*ArcTanh[Sq
rt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 260465625*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 2155
49775*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 142065*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[
c + d*x]^6 - 41427855*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 207173295*a*b^2*ArcTanh[Sqrt[-Si
nh[c + d*x]^2]]*Sinh[c + d*x]^6 - 58009455*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 210735*a^2*b*
ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 33756345*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]
^8 - 56109375*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 174825*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2
]]*Sinh[c + d*x]^10 - 9261945*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10 - 48825*b^3*ArcTanh[Sqrt[-S
inh[c + d*x]^2]]*Sinh[c + d*x]^12 + 117228825*a^3*Sqrt[-Sinh[c + d*x]^2] + 4093425*a^3*Sinh[c + d*x]^4*Sqrt[-S
inh[c + d*x]^2] + 168951510*a^2*b*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 215549775*a*b^2*Sinh[c + d*x]^4*Sqr
t[-Sinh[c + d*x]^2] + 9514449*a^2*b*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 135323370*a*b^2*Sinh[c + d*x]^6*S
qrt[-Sinh[c + d*x]^2] + 58009455*b^3*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 7808535*a*b^2*Sinh[c + d*x]^8*Sq
rt[-Sinh[c + d*x]^2] + 36772890*b^3*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 2160711*b^3*Sinh[c + d*x]^10*Sqrt
[-Sinh[c + d*x]^2] - 70189350*a^3*(-Sinh[c + d*x]^2)^(3/2) - 274542345*a^2*b*(-Sinh[c + d*x]^2)^(3/2) + 1024*a
^3*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]^6*(-Sinh[
c + d*x]^2)^(3/2) + 3072*a^2*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]^8*(-Sinh[c + d*x]^2)^(3/2) + 3072*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]^10*(-Sinh[c + d*x]^2)^(3/2) + 1024*b^3*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*(-Sinh[c + d*x]^2)^(3/2) + 1536*H
ypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2
)^(3/2)*(a + b*Sinh[c + d*x]^2)^2*(9*a + 7*b*Sinh[c + d*x]^2) + 256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1
, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2)^(3/2)*(295*a^3 + 741*a^2*b*Sinh[c + d*x]^2 +
621*a*b^2*Sinh[c + d*x]^4 + 175*b^3*Sinh[c + d*x]^6)))/(725760*d*Sqrt[-Sinh[c + d*x]^2])

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Maple [C] Result contains complex when optimal does not.
time = 1.78, size = 495, normalized size = 3.21

method	result	size
risch	\(\frac {{\mathrm e}^{d x +c} \left (15 a^{3} {\mathrm e}^{10 d x +10 c}+9 a^{2} b \,{\mathrm e}^{10 d x +10 c}+9 a \,b^{2} {\mathrm e}^{10 d x +10 c}-33 b^{3} {\mathrm e}^{10 d x +10 c}+85 a^{3} {\mathrm e}^{8 d x +8 c}+51 a^{2} b \,{\mathrm e}^{8 d x +8 c}-141 a \,b^{2} {\mathrm e}^{8 d x +8 c}+5 b^{3} {\mathrm e}^{8 d x +8 c}+198 a^{3} {\mathrm e}^{6 d x +6 c}-342 a^{2} b \,{\mathrm e}^{6 d x +6 c}+234 a \,b^{2} {\mathrm e}^{6 d x +6 c}-90 b^{3} {\mathrm e}^{6 d x +6 c}-198 a^{3} {\mathrm e}^{4 d x +4 c}+342 a^{2} b \,{\mathrm e}^{4 d x +4 c}-234 a \,b^{2} {\mathrm e}^{4 d x +4 c}+90 b^{3} {\mathrm e}^{4 d x +4 c}-85 a^{3} {\mathrm e}^{2 d x +2 c}-51 a^{2} b \,{\mathrm e}^{2 d x +2 c}+141 a \,b^{2} {\mathrm e}^{2 d x +2 c}-5 b^{3} {\mathrm e}^{2 d x +2 c}-15 a^{3}-9 a^{2} b -9 a \,b^{2}+33 b^{3}\right )}{24 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{6}}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{16 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{16 d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{16 d}+\frac {5 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{16 d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{16 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{16 d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{16 d}-\frac {5 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{16 d}\)	\(495\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x,method=_RETURNVERBOSE)

[Out]

1/24*exp(d*x+c)*(15*a^3*exp(10*d*x+10*c)+9*a^2*b*exp(10*d*x+10*c)+9*a*b^2*exp(10*d*x+10*c)-33*b^3*exp(10*d*x+1
0*c)+85*a^3*exp(8*d*x+8*c)+51*a^2*b*exp(8*d*x+8*c)-141*a*b^2*exp(8*d*x+8*c)+5*b^3*exp(8*d*x+8*c)+198*a^3*exp(6
*d*x+6*c)-342*a^2*b*exp(6*d*x+6*c)+234*a*b^2*exp(6*d*x+6*c)-90*b^3*exp(6*d*x+6*c)-198*a^3*exp(4*d*x+4*c)+342*a
^2*b*exp(4*d*x+4*c)-234*a*b^2*exp(4*d*x+4*c)+90*b^3*exp(4*d*x+4*c)-85*a^3*exp(2*d*x+2*c)-51*a^2*b*exp(2*d*x+2*
c)+141*a*b^2*exp(2*d*x+2*c)-5*b^3*exp(2*d*x+2*c)-15*a^3-9*a^2*b-9*a*b^2+33*b^3)/d/(1+exp(2*d*x+2*c))^6+5/16*I/
d*ln(exp(d*x+c)+I)*a^3+3/16*I/d*ln(exp(d*x+c)+I)*a^2*b+3/16*I/d*ln(exp(d*x+c)+I)*a*b^2+5/16*I/d*ln(exp(d*x+c)+
I)*b^3-5/16*I/d*ln(exp(d*x+c)-I)*a^3-3/16*I/d*ln(exp(d*x+c)-I)*a^2*b-3/16*I/d*ln(exp(d*x+c)-I)*a*b^2-5/16*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. 646 vs. \(2 (146) = 292\).
time = 0.50, size = 646, normalized size = 4.19 \begin {gather*} -\frac {1}{24} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {33 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 90 \, e^{\left (-5 \, d x - 5 \, c\right )} - 90 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} - 33 \, 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 {1}{24} \, a^{3} {\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 {1}{8} \, a^{2} b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 17 \, e^{\left (-3 \, d x - 3 \, c\right )} - 114 \, e^{\left (-5 \, d x - 5 \, c\right )} + 114 \, e^{\left (-7 \, d x - 7 \, c\right )} - 17 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, 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 {1}{8} \, a b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} - 47 \, e^{\left (-3 \, d x - 3 \, c\right )} + 78 \, e^{\left (-5 \, d x - 5 \, c\right )} - 78 \, e^{\left (-7 \, d x - 7 \, c\right )} + 47 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/24*b^3*(15*arctan(e^(-d*x - c))/d + (33*e^(-d*x - c) - 5*e^(-3*d*x - 3*c) + 90*e^(-5*d*x - 5*c) - 90*e^(-7*
d*x - 7*c) + 5*e^(-9*d*x - 9*c) - 33*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))) - 1/24*a^3*(15*arctan(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))) - 1/8*a^2*b*(3*arctan(e^(-d*x - c))/d - (
3*e^(-d*x - c) + 17*e^(-3*d*x - 3*c) - 114*e^(-5*d*x - 5*c) + 114*e^(-7*d*x - 7*c) - 17*e^(-9*d*x - 9*c) - 3*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))) - 1/8*a*b^2*(3*arctan(e^(-d*x - c))/d - (3*e^(-d*x - c) - 47*
e^(-3*d*x - 3*c) + 78*e^(-5*d*x - 5*c) - 78*e^(-7*d*x - 7*c) + 47*e^(-9*d*x - 9*c) - 3*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)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3675 vs. \(2 (146) = 292\).
time = 0.44, size = 3675, normalized size = 23.86 \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)^7*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/24*(3*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^11 + 33*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d
*x + c)*sinh(d*x + c)^10 + 3*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*sinh(d*x + c)^11 + (85*a^3 + 51*a^2*b - 141*
a*b^2 + 5*b^3)*cosh(d*x + c)^9 + (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3 + 165*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 9*(55*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^3 + (85*a^3 +
51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 18*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh
(d*x + c)^7 + 18*(55*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^4 + 11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b
^3 + 2*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 42*(33*(5*a^3 + 3*a^2*b + 3*
a*b^2 - 11*b^3)*cosh(d*x + c)^5 + 2*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(11*a^3 - 19*a
^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 18*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x +
c)^5 + 18*(77*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^6 + 7*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3
)*cosh(d*x + c)^4 - 11*a^3 + 19*a^2*b - 13*a*b^2 + 5*b^3 + 21*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x
+ c)^2)*sinh(d*x + c)^5 + 18*(55*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^7 + 7*(85*a^3 + 51*a^2*b -
141*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 35*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^3 - 5*(11*a^3 -
19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x
+ c)^3 + (495*(5*a^3 + 3*a^2*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^8 + 84*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^
3)*cosh(d*x + c)^6 + 630*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^4 - 85*a^3 - 51*a^2*b + 141*a*b^
2 - 5*b^3 - 180*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 3*(55*(5*a^3 + 3*a^2
*b + 3*a*b^2 - 11*b^3)*cosh(d*x + c)^9 + 12*(85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 126*(11*
a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^5 - 60*(11*a^3 - 19*a^2*b + 13*a*b^2 - 5*b^3)*cosh(d*x + c)^3
- (85*a^3 + 51*a^2*b - 141*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((5*a^3 + 3*a^2*b + 3*a*b^2 + 5*
b^3)*cosh(d*x + c)^12 + 12*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)*sinh(d*x + c)^11 + (5*a^3 + 3*a^2
*b + 3*a*b^2 + 5*b^3)*sinh(d*x + c)^12 + 6*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 6*(5*a^3 + 3
*a^2*b + 3*a*b^2 + 5*b^3 + 11*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(
5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh
(d*x + c)^9 + 15*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 15*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b
^3)*cosh(d*x + c)^4 + 5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3 + 18*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^8 + 24*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 30*(5*a^3 + 3*a^2*b + 3*a*b
^2 + 5*b^3)*cosh(d*x + c)^3 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 20*(5*a^3
+ 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 4*(231*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 3
15*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 25*a^3 + 15*a^2*b + 15*a*b^2 + 25*b^3 + 105*(5*a^3 +
3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(
d*x + c)^7 + 63*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 35*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*c
osh(d*x + c)^3 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(5*a^3 + 3*a^2*b +
3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 15*(33*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 84*(5*a^3 + 3*
a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 70*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 5*a^3 + 3*
a^2*b + 3*a*b^2 + 5*b^3 + 20*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 20*(11*(5*
a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 36*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 42
*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 20*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^3
+ 3*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3 + 6
*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2 + 6*(11*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)
^10 + 45*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 70*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x
+ c)^6 + 50*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3 + 15*(5*a
^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*((5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cos
h(d*x + c)^11 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 10*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)
*cosh(d*x + c)^7 + 10*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 5*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b
^3)*cosh(d*x + c)^3 + (5*a^3 + 3*a^2*b + 3*a*b^...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**7*(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (146) = 292\).
time = 0.46, size = 383, normalized size = 2.49 \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 (5 \, a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (15 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} - 33 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 160 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 96 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 96 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 160 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 240 \, 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 )}^{3}}}{96 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/96*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(5*a^3 + 3*a^2*b + 3*a*b^2 + 5*b^3) + 4*(15*a^
3*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a^2*b*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a*b^2*(e^(d*x + c) - e^(-d*x - c
))^5 - 33*b^3*(e^(d*x + c) - e^(-d*x - c))^5 + 160*a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 96*a^2*b*(e^(d*x + c)
- e^(-d*x - c))^3 - 96*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 - 160*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 528*a^3
*(e^(d*x + c) - e^(-d*x - c)) - 144*a^2*b*(e^(d*x + c) - e^(-d*x - c)) - 144*a*b^2*(e^(d*x + c) - e^(-d*x - c)
) - 240*b^3*(e^(d*x + c) - e^(-d*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3)/d

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Mupad [B]
time = 0.20, size = 601, normalized size = 3.90 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^3\,\sqrt {d^2}+5\,b^3\,\sqrt {d^2}+3\,a\,b^2\,\sqrt {d^2}+3\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {25\,a^6+30\,a^5\,b+39\,a^4\,b^2+68\,a^3\,b^3+39\,a^2\,b^4+30\,a\,b^5+25\,b^6}}\right )\,\sqrt {25\,a^6+30\,a^5\,b+39\,a^4\,b^2+68\,a^3\,b^3+39\,a^2\,b^4+30\,a\,b^5+25\,b^6}}{8\,\sqrt {d^2}}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (3\,a^3-11\,a^2\,b+13\,a\,b^2-5\,b^3\right )}{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 {{\mathrm {e}}^{c+d\,x}\,\left (a^3-57\,a^2\,b+111\,a\,b^2-55\,b^3\right )}{3\,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 )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+3\,a^2\,b+3\,a\,b^2-11\,b^3\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+3\,a^2\,b-93\,a\,b^2+85\,b^3\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {80\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-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 {32\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^2)^3/cosh(c + d*x)^7,x)

[Out]

(atan((exp(d*x)*exp(c)*(5*a^3*(d^2)^(1/2) + 5*b^3*(d^2)^(1/2) + 3*a*b^2*(d^2)^(1/2) + 3*a^2*b*(d^2)^(1/2)))/(d
*(30*a*b^5 + 30*a^5*b + 25*a^6 + 25*b^6 + 39*a^2*b^4 + 68*a^3*b^3 + 39*a^4*b^2)^(1/2)))*(30*a*b^5 + 30*a^5*b +
25*a^6 + 25*b^6 + 39*a^2*b^4 + 68*a^3*b^3 + 39*a^4*b^2)^(1/2))/(8*(d^2)^(1/2)) - (6*exp(c + d*x)*(13*a*b^2 -
11*a^2*b + 3*a^3 - 5*b^3))/(d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x)
+ 1)) + (exp(c + d*x)*(111*a*b^2 - 57*a^2*b + a^3 - 55*b^3))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) +
exp(6*c + 6*d*x) + 1)) + (exp(c + d*x)*(3*a*b^2 + 3*a^2*b + 5*a^3 - 11*b^3))/(8*d*(exp(2*c + 2*d*x) + 1)) + (e
xp(c + d*x)*(3*a^2*b - 93*a*b^2 + 5*a^3 + 85*b^3))/(12*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (80*ex
p(c + d*x)*(3*a*b^2 - 3*a^2*b + a^3 - 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)) - (32*exp(c + d*x)*(3*a*b^2 - 3*a^2*b + a^3 - 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) + e
xp(12*c + 12*d*x) + 1))

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