3.3.17 \(\int \sinh ^2(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [217]

Optimal. Leaf size=255 \[ -\frac {\left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) x}{2048}+\frac {\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{2048 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d} \]

[Out]

-1/2048*(1024*a^3+1920*a^2*b+1512*a*b^2+429*b^3)*x+1/2048*(1024*a^3+4224*a^2*b+4632*a*b^2+1619*b^3)*cosh(d*x+c
)*sinh(d*x+c)/d-1/3072*b*(4992*a^2+10728*a*b+5549*b^2)*cosh(d*x+c)^3*sinh(d*x+c)/d+1/3840*b*(1920*a^2+12312*a*
b+10579*b^2)*cosh(d*x+c)^5*sinh(d*x+c)/d-1/4480*b^2*(6888*a+11821*b)*cosh(d*x+c)^7*sinh(d*x+c)/d+1/1680*b^2*(5
04*a+2593*b)*cosh(d*x+c)^9*sinh(d*x+c)/d-85/168*b^3*cosh(d*x+c)^11*sinh(d*x+c)/d+1/14*b^3*cosh(d*x+c)^13*sinh(
d*x+c)/d

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Rubi [A]
time = 0.40, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3296, 1271, 1828, 1171, 393, 212} \begin {gather*} \frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \sinh (c+d x) \cosh ^5(c+d x)}{3840 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \sinh (c+d x) \cosh ^3(c+d x)}{3072 d}+\frac {\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{2048 d}-\frac {x \left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right )}{2048}+\frac {b^2 (504 a+2593 b) \sinh (c+d x) \cosh ^9(c+d x)}{1680 d}-\frac {b^2 (6888 a+11821 b) \sinh (c+d x) \cosh ^7(c+d x)}{4480 d}+\frac {b^3 \sinh (c+d x) \cosh ^{13}(c+d x)}{14 d}-\frac {85 b^3 \sinh (c+d x) \cosh ^{11}(c+d x)}{168 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/2048*((1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*x) + ((1024*a^3 + 4224*a^2*b + 4632*a*b^2 + 1619*b^3)*
Cosh[c + d*x]*Sinh[c + d*x])/(2048*d) - (b*(4992*a^2 + 10728*a*b + 5549*b^2)*Cosh[c + d*x]^3*Sinh[c + d*x])/(3
072*d) + (b*(1920*a^2 + 12312*a*b + 10579*b^2)*Cosh[c + d*x]^5*Sinh[c + d*x])/(3840*d) - (b^2*(6888*a + 11821*
b)*Cosh[c + d*x]^7*Sinh[c + d*x])/(4480*d) + (b^2*(504*a + 2593*b)*Cosh[c + d*x]^9*Sinh[c + d*x])/(1680*d) - (
85*b^3*Cosh[c + d*x]^11*Sinh[c + d*x])/(168*d) + (b^3*Cosh[c + d*x]^13*Sinh[c + d*x])/(14*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 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 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1271

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Dist[1/(2*e^(2*p +
m/2)*(q + 1)), Int[(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*e^(2*p + m/2)*(q + 1)*x^m*(a +
b*x^2 + c*x^4)^p - (-d)^(m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*(d + e*(2*q + 3)*x^2))], x], x], x] /; FreeQ[{a, b
, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && IGtQ[m/2, 0]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 3296

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*
x^2)^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a-2 a x^2+(a+b) x^4\right )^3}{\left (1-x^2\right )^8} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac {\text {Subst}\left (\int \frac {-b^3+14 \left (a^3-b^3\right ) x^2-14 \left (5 a^3+b^3\right ) x^4+14 \left (10 a^3+3 a^2 b-b^3\right ) x^6-14 \left (10 a^3+9 a^2 b+b^3\right ) x^8+14 (5 a-b) (a+b)^2 x^{10}-14 (a+b)^3 x^{12}}{\left (1-x^2\right )^7} \, dx,x,\tanh (c+d x)\right )}{14 d}\\ &=-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\text {Subst}\left (\int \frac {-73 b^3-168 \left (a^3+5 b^3\right ) x^2+672 \left (a^3-b^3\right ) x^4-504 \left (2 a^3+a^2 b+b^3\right ) x^6+336 (2 a-b) (a+b)^2 x^8-168 (a+b)^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\tanh (c+d x)\right )}{168 d}\\ &=\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac {\text {Subst}\left (\int \frac {-9 b^2 (56 a+207 b)+1680 \left (a^3-3 a b^2-10 b^3\right ) x^2-5040 \left (a^3+a b^2+2 b^3\right ) x^4+5040 (a-b) (a+b)^2 x^6-1680 (a+b)^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\tanh (c+d x)\right )}{1680 d}\\ &=-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\text {Subst}\left (\int \frac {-231 b^2 (72 a+89 b)-13440 \left (a^3+9 a b^2+10 b^3\right ) x^2+26880 (a-2 b) (a+b)^2 x^4-13440 (a+b)^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{13440 d}\\ &=\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}+\frac {\text {Subst}\left (\int \frac {-105 b \left (384 a^2+1512 a b+941 b^2\right )+80640 (a-5 b) (a+b)^2 x^2-80640 (a+b)^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{80640 d}\\ &=-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\text {Subst}\left (\int \frac {-315 b \left (1152 a^2+1560 a b+595 b^2\right )-322560 (a+b)^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{322560 d}\\ &=\frac {\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{2048 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}-\frac {\left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2048 d}\\ &=-\frac {\left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) x}{2048}+\frac {\left (1024 a^3+4224 a^2 b+4632 a b^2+1619 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{2048 d}-\frac {b \left (4992 a^2+10728 a b+5549 b^2\right ) \cosh ^3(c+d x) \sinh (c+d x)}{3072 d}+\frac {b \left (1920 a^2+12312 a b+10579 b^2\right ) \cosh ^5(c+d x) \sinh (c+d x)}{3840 d}-\frac {b^2 (6888 a+11821 b) \cosh ^7(c+d x) \sinh (c+d x)}{4480 d}+\frac {b^2 (504 a+2593 b) \cosh ^9(c+d x) \sinh (c+d x)}{1680 d}-\frac {85 b^3 \cosh ^{11}(c+d x) \sinh (c+d x)}{168 d}+\frac {b^3 \cosh ^{13}(c+d x) \sinh (c+d x)}{14 d}\\ \end {align*}

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Mathematica [A]
time = 0.51, size = 189, normalized size = 0.74 \begin {gather*} \frac {-840 \left (1024 a^3+1920 a^2 b+1512 a b^2+429 b^3\right ) (c+d x)+105 \left (4096 a^3+11520 a^2 b+10080 a b^2+3003 b^3\right ) \sinh (2 (c+d x))-105 b \left (2304 a^2+2880 a b+1001 b^2\right ) \sinh (4 (c+d x))+35 b \left (768 a^2+2160 a b+1001 b^2\right ) \sinh (6 (c+d x))-105 b^2 (120 a+91 b) \sinh (8 (c+d x))+21 b^2 (48 a+91 b) \sinh (10 (c+d x))-245 b^3 \sinh (12 (c+d x))+15 b^3 \sinh (14 (c+d x))}{1720320 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^2*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-840*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*(c + d*x) + 105*(4096*a^3 + 11520*a^2*b + 10080*a*b^2 + 3
003*b^3)*Sinh[2*(c + d*x)] - 105*b*(2304*a^2 + 2880*a*b + 1001*b^2)*Sinh[4*(c + d*x)] + 35*b*(768*a^2 + 2160*a
*b + 1001*b^2)*Sinh[6*(c + d*x)] - 105*b^2*(120*a + 91*b)*Sinh[8*(c + d*x)] + 21*b^2*(48*a + 91*b)*Sinh[10*(c
+ d*x)] - 245*b^3*Sinh[12*(c + d*x)] + 15*b^3*Sinh[14*(c + d*x)])/(1720320*d)

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Maple [A]
time = 1.89, size = 215, normalized size = 0.84

method	result	size
default	\(\frac {\left (-\frac {91}{2048} b^{3}-\frac {15}{256} a \,b^{2}\right ) \sinh \left (8 d x +8 c \right )}{8 d}+\frac {\left (\frac {91}{8192} b^{3}+\frac {3}{512} a \,b^{2}\right ) \sinh \left (10 d x +10 c \right )}{10 d}+\frac {\left (-\frac {1001}{4096} b^{3}-\frac {45}{64} a \,b^{2}-\frac {9}{16} a^{2} b \right ) \sinh \left (4 d x +4 c \right )}{4 d}+\frac {\left (\frac {1001}{8192} b^{3}+\frac {135}{512} a \,b^{2}+\frac {3}{32} a^{2} b \right ) \sinh \left (6 d x +6 c \right )}{6 d}+\frac {\left (\frac {3003}{8192} b^{3}+\frac {315}{256} a \,b^{2}+\frac {45}{32} a^{2} b +\frac {1}{2} a^{3}\right ) \sinh \left (2 d x +2 c \right )}{2 d}-\frac {a^{3} x}{2}-\frac {429 b^{3} x}{2048}-\frac {189 a \,b^{2} x}{256}-\frac {15 a^{2} b x}{16}-\frac {7 b^{3} \sinh \left (12 d x +12 c \right )}{49152 d}+\frac {b^{3} \sinh \left (14 d x +14 c \right )}{114688 d}\)	\(215\)
risch	\(-\frac {15 a^{2} b x}{16}+\frac {3 b^{2} {\mathrm e}^{10 d x +10 c} a}{10240 d}-\frac {15 b^{2} {\mathrm e}^{8 d x +8 c} a}{4096 d}+\frac {45 b \,{\mathrm e}^{2 d x +2 c} a^{2}}{128 d}+\frac {b^{3} {\mathrm e}^{14 d x +14 c}}{229376 d}-\frac {7 b^{3} {\mathrm e}^{12 d x +12 c}}{98304 d}+\frac {7 b^{3} {\mathrm e}^{-12 d x -12 c}}{98304 d}-\frac {b^{3} {\mathrm e}^{-14 d x -14 c}}{229376 d}-\frac {3003 b^{3} {\mathrm e}^{-2 d x -2 c}}{32768 d}+\frac {1001 b^{3} {\mathrm e}^{-4 d x -4 c}}{32768 d}-\frac {429 b^{3} x}{2048}-\frac {315 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{1024 d}+\frac {9 b \,{\mathrm e}^{-4 d x -4 c} a^{2}}{128 d}+\frac {45 b^{2} {\mathrm e}^{-4 d x -4 c} a}{512 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c} a^{2}}{128 d}-\frac {45 b^{2} {\mathrm e}^{-6 d x -6 c} a}{2048 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}-\frac {a^{3} x}{2}-\frac {189 a \,b^{2} x}{256}-\frac {45 b \,{\mathrm e}^{-2 d x -2 c} a^{2}}{128 d}-\frac {1001 b^{3} {\mathrm e}^{-6 d x -6 c}}{98304 d}+\frac {91 b^{3} {\mathrm e}^{-8 d x -8 c}}{32768 d}+\frac {91 b^{3} {\mathrm e}^{10 d x +10 c}}{163840 d}-\frac {91 b^{3} {\mathrm e}^{8 d x +8 c}}{32768 d}+\frac {1001 b^{3} {\mathrm e}^{6 d x +6 c}}{98304 d}-\frac {91 b^{3} {\mathrm e}^{-10 d x -10 c}}{163840 d}-\frac {1001 b^{3} {\mathrm e}^{4 d x +4 c}}{32768 d}+\frac {3003 b^{3} {\mathrm e}^{2 d x +2 c}}{32768 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}+\frac {b \,{\mathrm e}^{6 d x +6 c} a^{2}}{128 d}+\frac {45 b^{2} {\mathrm e}^{6 d x +6 c} a}{2048 d}-\frac {9 b \,{\mathrm e}^{4 d x +4 c} a^{2}}{128 d}-\frac {45 b^{2} {\mathrm e}^{4 d x +4 c} a}{512 d}+\frac {315 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{1024 d}+\frac {15 b^{2} {\mathrm e}^{-8 d x -8 c} a}{4096 d}-\frac {3 b^{2} {\mathrm e}^{-10 d x -10 c} a}{10240 d}\)	\(588\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/8*(-91/2048*b^3-15/256*a*b^2)*sinh(8*d*x+8*c)/d+1/10*(91/8192*b^3+3/512*a*b^2)*sinh(10*d*x+10*c)/d+1/4*(-100
1/4096*b^3-45/64*a*b^2-9/16*a^2*b)*sinh(4*d*x+4*c)/d+1/6*(1001/8192*b^3+135/512*a*b^2+3/32*a^2*b)*sinh(6*d*x+6
*c)/d+1/2*(3003/8192*b^3+315/256*a*b^2+45/32*a^2*b+1/2*a^3)*sinh(2*d*x+2*c)/d-1/2*a^3*x-429/2048*b^3*x-189/256
*a*b^2*x-15/16*a^2*b*x-7/49152*b^3*sinh(12*d*x+12*c)/d+1/114688*b^3*sinh(14*d*x+14*c)/d

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Maxima [A]
time = 0.28, size = 442, normalized size = 1.73 \begin {gather*} -\frac {1}{8} \, a^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{3440640} \, b^{3} {\left (\frac {{\left (245 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1911 \, e^{\left (-4 \, d x - 4 \, c\right )} + 9555 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35035 \, e^{\left (-8 \, d x - 8 \, c\right )} + 105105 \, e^{\left (-10 \, d x - 10 \, c\right )} - 315315 \, e^{\left (-12 \, d x - 12 \, c\right )} - 15\right )} e^{\left (14 \, d x + 14 \, c\right )}}{d} + \frac {720720 \, {\left (d x + c\right )}}{d} + \frac {315315 \, e^{\left (-2 \, d x - 2 \, c\right )} - 105105 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35035 \, e^{\left (-6 \, d x - 6 \, c\right )} - 9555 \, e^{\left (-8 \, d x - 8 \, c\right )} + 1911 \, e^{\left (-10 \, d x - 10 \, c\right )} - 245 \, e^{\left (-12 \, d x - 12 \, c\right )} + 15 \, e^{\left (-14 \, d x - 14 \, c\right )}}{d}\right )} - \frac {3}{20480} \, a b^{2} {\left (\frac {{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac {5040 \, {\left (d x + c\right )}}{d} + \frac {2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac {1}{128} \, a^{2} b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/8*a^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/3440640*b^3*((245*e^(-2*d*x - 2*c) - 1911*e^(-4*d*
x - 4*c) + 9555*e^(-6*d*x - 6*c) - 35035*e^(-8*d*x - 8*c) + 105105*e^(-10*d*x - 10*c) - 315315*e^(-12*d*x - 12
*c) - 15)*e^(14*d*x + 14*c)/d + 720720*(d*x + c)/d + (315315*e^(-2*d*x - 2*c) - 105105*e^(-4*d*x - 4*c) + 3503
5*e^(-6*d*x - 6*c) - 9555*e^(-8*d*x - 8*c) + 1911*e^(-10*d*x - 10*c) - 245*e^(-12*d*x - 12*c) + 15*e^(-14*d*x
- 14*c))/d) - 3/20480*a*b^2*((25*e^(-2*d*x - 2*c) - 150*e^(-4*d*x - 4*c) + 600*e^(-6*d*x - 6*c) - 2100*e^(-8*d
*x - 8*c) - 2)*e^(10*d*x + 10*c)/d + 5040*(d*x + c)/d + (2100*e^(-2*d*x - 2*c) - 600*e^(-4*d*x - 4*c) + 150*e^
(-6*d*x - 6*c) - 25*e^(-8*d*x - 8*c) + 2*e^(-10*d*x - 10*c))/d) - 1/128*a^2*b*((9*e^(-2*d*x - 2*c) - 45*e^(-4*
d*x - 4*c) - 1)*e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x -
6*c))/d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (239) = 478\).
time = 0.39, size = 627, normalized size = 2.46 \begin {gather*} \frac {105 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{13} + 210 \, {\left (13 \, b^{3} \cosh \left (d x + c\right )^{3} - 7 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{11} + 35 \, {\left (429 \, b^{3} \cosh \left (d x + c\right )^{5} - 770 \, b^{3} \cosh \left (d x + c\right )^{3} + 3 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{9} + 60 \, {\left (429 \, b^{3} \cosh \left (d x + c\right )^{7} - 1617 \, b^{3} \cosh \left (d x + c\right )^{5} + 21 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 7 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 21 \, {\left (715 \, b^{3} \cosh \left (d x + c\right )^{9} - 4620 \, b^{3} \cosh \left (d x + c\right )^{7} + 126 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 140 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 70 \, {\left (39 \, b^{3} \cosh \left (d x + c\right )^{11} - 385 \, b^{3} \cosh \left (d x + c\right )^{9} + 18 \, {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 42 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 420 \, {\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} d x + 105 \, {\left (b^{3} \cosh \left (d x + c\right )^{13} - 14 \, b^{3} \cosh \left (d x + c\right )^{11} + {\left (48 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} - 4 \, {\left (120 \, a b^{2} + 91 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + {\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 2 \, {\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + {\left (4096 \, a^{3} + 11520 \, a^{2} b + 10080 \, a b^{2} + 3003 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{860160 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/860160*(105*b^3*cosh(d*x + c)*sinh(d*x + c)^13 + 210*(13*b^3*cosh(d*x + c)^3 - 7*b^3*cosh(d*x + c))*sinh(d*x
+ c)^11 + 35*(429*b^3*cosh(d*x + c)^5 - 770*b^3*cosh(d*x + c)^3 + 3*(48*a*b^2 + 91*b^3)*cosh(d*x + c))*sinh(d
*x + c)^9 + 60*(429*b^3*cosh(d*x + c)^7 - 1617*b^3*cosh(d*x + c)^5 + 21*(48*a*b^2 + 91*b^3)*cosh(d*x + c)^3 -
7*(120*a*b^2 + 91*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 21*(715*b^3*cosh(d*x + c)^9 - 4620*b^3*cosh(d*x + c)^7
+ 126*(48*a*b^2 + 91*b^3)*cosh(d*x + c)^5 - 140*(120*a*b^2 + 91*b^3)*cosh(d*x + c)^3 + 5*(768*a^2*b + 2160*a*
b^2 + 1001*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 70*(39*b^3*cosh(d*x + c)^11 - 385*b^3*cosh(d*x + c)^9 + 18*(4
8*a*b^2 + 91*b^3)*cosh(d*x + c)^7 - 42*(120*a*b^2 + 91*b^3)*cosh(d*x + c)^5 + 5*(768*a^2*b + 2160*a*b^2 + 1001
*b^3)*cosh(d*x + c)^3 - 3*(2304*a^2*b + 2880*a*b^2 + 1001*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 420*(1024*a^3
+ 1920*a^2*b + 1512*a*b^2 + 429*b^3)*d*x + 105*(b^3*cosh(d*x + c)^13 - 14*b^3*cosh(d*x + c)^11 + (48*a*b^2 + 9
1*b^3)*cosh(d*x + c)^9 - 4*(120*a*b^2 + 91*b^3)*cosh(d*x + c)^7 + (768*a^2*b + 2160*a*b^2 + 1001*b^3)*cosh(d*x
+ c)^5 - 2*(2304*a^2*b + 2880*a*b^2 + 1001*b^3)*cosh(d*x + c)^3 + (4096*a^3 + 11520*a^2*b + 10080*a*b^2 + 300
3*b^3)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 877 vs. \(2 (250) = 500\).
time = 7.78, size = 877, normalized size = 3.44 \begin {gather*} \begin {cases} \frac {a^{3} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a^{3} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a^{3} \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {15 a^{2} b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {45 a^{2} b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {45 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {15 a^{2} b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {33 a^{2} b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{2 d} + \frac {15 a^{2} b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} + \frac {189 a b^{2} x \sinh ^{10}{\left (c + d x \right )}}{256} - \frac {945 a b^{2} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{256} + \frac {945 a b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{128} - \frac {945 a b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{128} + \frac {945 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{256} - \frac {189 a b^{2} x \cosh ^{10}{\left (c + d x \right )}}{256} + \frac {579 a b^{2} \sinh ^{9}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{256 d} - \frac {711 a b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac {63 a b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{10 d} - \frac {441 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac {189 a b^{2} \sinh {\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{256 d} + \frac {429 b^{3} x \sinh ^{14}{\left (c + d x \right )}}{2048} - \frac {3003 b^{3} x \sinh ^{12}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{2048} + \frac {9009 b^{3} x \sinh ^{10}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{2048} - \frac {15015 b^{3} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{2048} + \frac {15015 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{2048} - \frac {9009 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{10}{\left (c + d x \right )}}{2048} + \frac {3003 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{12}{\left (c + d x \right )}}{2048} - \frac {429 b^{3} x \cosh ^{14}{\left (c + d x \right )}}{2048} + \frac {1619 b^{3} \sinh ^{13}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2048 d} - \frac {4511 b^{3} \sinh ^{11}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{1536 d} + \frac {171457 b^{3} \sinh ^{9}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{30720 d} - \frac {429 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{70 d} + \frac {40469 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{10240 d} - \frac {715 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{11}{\left (c + d x \right )}}{512 d} + \frac {429 b^{3} \sinh {\left (c + d x \right )} \cosh ^{13}{\left (c + d x \right )}}{2048 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{3} \sinh ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**2*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Piecewise((a**3*x*sinh(c + d*x)**2/2 - a**3*x*cosh(c + d*x)**2/2 + a**3*sinh(c + d*x)*cosh(c + d*x)/(2*d) + 15
*a**2*b*x*sinh(c + d*x)**6/16 - 45*a**2*b*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 45*a**2*b*x*sinh(c + d*x)**
2*cosh(c + d*x)**4/16 - 15*a**2*b*x*cosh(c + d*x)**6/16 + 33*a**2*b*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*
a**2*b*sinh(c + d*x)**3*cosh(c + d*x)**3/(2*d) + 15*a**2*b*sinh(c + d*x)*cosh(c + d*x)**5/(16*d) + 189*a*b**2*
x*sinh(c + d*x)**10/256 - 945*a*b**2*x*sinh(c + d*x)**8*cosh(c + d*x)**2/256 + 945*a*b**2*x*sinh(c + d*x)**6*c
osh(c + d*x)**4/128 - 945*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**6/128 + 945*a*b**2*x*sinh(c + d*x)**2*cosh(
c + d*x)**8/256 - 189*a*b**2*x*cosh(c + d*x)**10/256 + 579*a*b**2*sinh(c + d*x)**9*cosh(c + d*x)/(256*d) - 711
*a*b**2*sinh(c + d*x)**7*cosh(c + d*x)**3/(128*d) + 63*a*b**2*sinh(c + d*x)**5*cosh(c + d*x)**5/(10*d) - 441*a
*b**2*sinh(c + d*x)**3*cosh(c + d*x)**7/(128*d) + 189*a*b**2*sinh(c + d*x)*cosh(c + d*x)**9/(256*d) + 429*b**3
*x*sinh(c + d*x)**14/2048 - 3003*b**3*x*sinh(c + d*x)**12*cosh(c + d*x)**2/2048 + 9009*b**3*x*sinh(c + d*x)**1
0*cosh(c + d*x)**4/2048 - 15015*b**3*x*sinh(c + d*x)**8*cosh(c + d*x)**6/2048 + 15015*b**3*x*sinh(c + d*x)**6*
cosh(c + d*x)**8/2048 - 9009*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**10/2048 + 3003*b**3*x*sinh(c + d*x)**2*cos
h(c + d*x)**12/2048 - 429*b**3*x*cosh(c + d*x)**14/2048 + 1619*b**3*sinh(c + d*x)**13*cosh(c + d*x)/(2048*d) -
4511*b**3*sinh(c + d*x)**11*cosh(c + d*x)**3/(1536*d) + 171457*b**3*sinh(c + d*x)**9*cosh(c + d*x)**5/(30720*
d) - 429*b**3*sinh(c + d*x)**7*cosh(c + d*x)**7/(70*d) + 40469*b**3*sinh(c + d*x)**5*cosh(c + d*x)**9/(10240*d
) - 715*b**3*sinh(c + d*x)**3*cosh(c + d*x)**11/(512*d) + 429*b**3*sinh(c + d*x)*cosh(c + d*x)**13/(2048*d), N
e(d, 0)), (x*(a + b*sinh(c)**4)**3*sinh(c)**2, True))

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Giac [A]
time = 0.57, size = 401, normalized size = 1.57 \begin {gather*} \frac {b^{3} e^{\left (14 \, d x + 14 \, c\right )}}{229376 \, d} - \frac {7 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )}}{98304 \, d} + \frac {7 \, b^{3} e^{\left (-12 \, d x - 12 \, c\right )}}{98304 \, d} - \frac {b^{3} e^{\left (-14 \, d x - 14 \, c\right )}}{229376 \, d} - \frac {1}{2048} \, {\left (1024 \, a^{3} + 1920 \, a^{2} b + 1512 \, a b^{2} + 429 \, b^{3}\right )} x + \frac {{\left (48 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (10 \, d x + 10 \, c\right )}}{163840 \, d} - \frac {{\left (120 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )}}{32768 \, d} + \frac {{\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{98304 \, d} - \frac {{\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{32768 \, d} + \frac {{\left (4096 \, a^{3} + 11520 \, a^{2} b + 10080 \, a b^{2} + 3003 \, b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{32768 \, d} - \frac {{\left (4096 \, a^{3} + 11520 \, a^{2} b + 10080 \, a b^{2} + 3003 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{32768 \, d} + \frac {{\left (2304 \, a^{2} b + 2880 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{32768 \, d} - \frac {{\left (768 \, a^{2} b + 2160 \, a b^{2} + 1001 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{98304 \, d} + \frac {{\left (120 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{32768 \, d} - \frac {{\left (48 \, a b^{2} + 91 \, b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^2*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

1/229376*b^3*e^(14*d*x + 14*c)/d - 7/98304*b^3*e^(12*d*x + 12*c)/d + 7/98304*b^3*e^(-12*d*x - 12*c)/d - 1/2293
76*b^3*e^(-14*d*x - 14*c)/d - 1/2048*(1024*a^3 + 1920*a^2*b + 1512*a*b^2 + 429*b^3)*x + 1/163840*(48*a*b^2 + 9
1*b^3)*e^(10*d*x + 10*c)/d - 1/32768*(120*a*b^2 + 91*b^3)*e^(8*d*x + 8*c)/d + 1/98304*(768*a^2*b + 2160*a*b^2
+ 1001*b^3)*e^(6*d*x + 6*c)/d - 1/32768*(2304*a^2*b + 2880*a*b^2 + 1001*b^3)*e^(4*d*x + 4*c)/d + 1/32768*(4096
*a^3 + 11520*a^2*b + 10080*a*b^2 + 3003*b^3)*e^(2*d*x + 2*c)/d - 1/32768*(4096*a^3 + 11520*a^2*b + 10080*a*b^2
+ 3003*b^3)*e^(-2*d*x - 2*c)/d + 1/32768*(2304*a^2*b + 2880*a*b^2 + 1001*b^3)*e^(-4*d*x - 4*c)/d - 1/98304*(7
68*a^2*b + 2160*a*b^2 + 1001*b^3)*e^(-6*d*x - 6*c)/d + 1/32768*(120*a*b^2 + 91*b^3)*e^(-8*d*x - 8*c)/d - 1/163
840*(48*a*b^2 + 91*b^3)*e^(-10*d*x - 10*c)/d

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Mupad [B]
time = 0.79, size = 393, normalized size = 1.54 \begin {gather*} \frac {{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (768\,a^2\,b+2160\,a\,b^2+1001\,b^3\right )}{98304\,d}-\frac {{\mathrm {e}}^{-6\,c-6\,d\,x}\,\left (768\,a^2\,b+2160\,a\,b^2+1001\,b^3\right )}{98304\,d}-x\,\left (\frac {a^3}{2}+\frac {15\,a^2\,b}{16}+\frac {189\,a\,b^2}{256}+\frac {429\,b^3}{2048}\right )+\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,\left (2304\,a^2\,b+2880\,a\,b^2+1001\,b^3\right )}{32768\,d}-\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (2304\,a^2\,b+2880\,a\,b^2+1001\,b^3\right )}{32768\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (4096\,a^3+11520\,a^2\,b+10080\,a\,b^2+3003\,b^3\right )}{32768\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (4096\,a^3+11520\,a^2\,b+10080\,a\,b^2+3003\,b^3\right )}{32768\,d}+\frac {7\,b^3\,{\mathrm {e}}^{-12\,c-12\,d\,x}}{98304\,d}-\frac {7\,b^3\,{\mathrm {e}}^{12\,c+12\,d\,x}}{98304\,d}-\frac {b^3\,{\mathrm {e}}^{-14\,c-14\,d\,x}}{229376\,d}+\frac {b^3\,{\mathrm {e}}^{14\,c+14\,d\,x}}{229376\,d}-\frac {b^2\,{\mathrm {e}}^{-10\,c-10\,d\,x}\,\left (48\,a+91\,b\right )}{163840\,d}+\frac {b^2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (48\,a+91\,b\right )}{163840\,d}+\frac {b^2\,{\mathrm {e}}^{-8\,c-8\,d\,x}\,\left (120\,a+91\,b\right )}{32768\,d}-\frac {b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (120\,a+91\,b\right )}{32768\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^2*(a + b*sinh(c + d*x)^4)^3,x)

[Out]

(exp(6*c + 6*d*x)*(2160*a*b^2 + 768*a^2*b + 1001*b^3))/(98304*d) - (exp(- 6*c - 6*d*x)*(2160*a*b^2 + 768*a^2*b
+ 1001*b^3))/(98304*d) - x*((189*a*b^2)/256 + (15*a^2*b)/16 + a^3/2 + (429*b^3)/2048) + (exp(- 4*c - 4*d*x)*(
2880*a*b^2 + 2304*a^2*b + 1001*b^3))/(32768*d) - (exp(4*c + 4*d*x)*(2880*a*b^2 + 2304*a^2*b + 1001*b^3))/(3276
8*d) - (exp(- 2*c - 2*d*x)*(10080*a*b^2 + 11520*a^2*b + 4096*a^3 + 3003*b^3))/(32768*d) + (exp(2*c + 2*d*x)*(1
0080*a*b^2 + 11520*a^2*b + 4096*a^3 + 3003*b^3))/(32768*d) + (7*b^3*exp(- 12*c - 12*d*x))/(98304*d) - (7*b^3*e
xp(12*c + 12*d*x))/(98304*d) - (b^3*exp(- 14*c - 14*d*x))/(229376*d) + (b^3*exp(14*c + 14*d*x))/(229376*d) - (
b^2*exp(- 10*c - 10*d*x)*(48*a + 91*b))/(163840*d) + (b^2*exp(10*c + 10*d*x)*(48*a + 91*b))/(163840*d) + (b^2*
exp(- 8*c - 8*d*x)*(120*a + 91*b))/(32768*d) - (b^2*exp(8*c + 8*d*x)*(120*a + 91*b))/(32768*d)

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