Integrand size = 23, antiderivative size = 189 \[ \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \text {arctanh}(\cosh (c+d x))}{256 d}+\frac {b^3 \cosh (c+d x)}{d}-\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{256 d}+\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^3(c+d x)}{128 d}-\frac {a^2 (21 a+80 b) \coth (c+d x) \text {csch}^5(c+d x)}{160 d}+\frac {9 a^3 \coth (c+d x) \text {csch}^7(c+d x)}{80 d}-\frac {a^3 \coth (c+d x) \text {csch}^9(c+d x)}{10 d} \]
3/256*a*(21*a^2+80*a*b+128*b^2)*arctanh(cosh(d*x+c))/d+b^3*cosh(d*x+c)/d-3 /256*a*(21*a^2+80*a*b+128*b^2)*coth(d*x+c)*csch(d*x+c)/d+1/128*a^2*(21*a+8 0*b)*coth(d*x+c)*csch(d*x+c)^3/d-1/160*a^2*(21*a+80*b)*coth(d*x+c)*csch(d* x+c)^5/d+9/80*a^3*coth(d*x+c)*csch(d*x+c)^7/d-1/10*a^3*coth(d*x+c)*csch(d* x+c)^9/d
Time = 2.71 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.48 \[ \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {b^3 \cosh (c+d x)}{d}-\frac {a \left (60 \left (21 a^2+80 a b+128 b^2\right ) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-40 a (7 a+24 b) \text {csch}^4\left (\frac {1}{2} (c+d x)\right )+10 a (7 a+16 b) \text {csch}^6\left (\frac {1}{2} (c+d x)\right )-15 a^2 \text {csch}^8\left (\frac {1}{2} (c+d x)\right )+2 a^2 \text {csch}^{10}\left (\frac {1}{2} (c+d x)\right )-240 \left (21 a^2+80 a b+128 b^2\right ) \left (\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+60 \left (21 a^2+80 a b+128 b^2\right ) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+40 a (7 a+24 b) \text {sech}^4\left (\frac {1}{2} (c+d x)\right )+10 a (7 a+16 b) \text {sech}^6\left (\frac {1}{2} (c+d x)\right )+15 a^2 \text {sech}^8\left (\frac {1}{2} (c+d x)\right )+2 a^2 \text {sech}^{10}\left (\frac {1}{2} (c+d x)\right )\right )}{20480 d} \]
(b^3*Cosh[c + d*x])/d - (a*(60*(21*a^2 + 80*a*b + 128*b^2)*Csch[(c + d*x)/ 2]^2 - 40*a*(7*a + 24*b)*Csch[(c + d*x)/2]^4 + 10*a*(7*a + 16*b)*Csch[(c + d*x)/2]^6 - 15*a^2*Csch[(c + d*x)/2]^8 + 2*a^2*Csch[(c + d*x)/2]^10 - 240 *(21*a^2 + 80*a*b + 128*b^2)*(Log[Cosh[(c + d*x)/2]] - Log[Sinh[(c + d*x)/ 2]]) + 60*(21*a^2 + 80*a*b + 128*b^2)*Sech[(c + d*x)/2]^2 + 40*a*(7*a + 24 *b)*Sech[(c + d*x)/2]^4 + 10*a*(7*a + 16*b)*Sech[(c + d*x)/2]^6 + 15*a^2*S ech[(c + d*x)/2]^8 + 2*a^2*Sech[(c + d*x)/2]^10))/(20480*d)
Time = 0.85 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.21, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {3042, 26, 3694, 1471, 25, 2345, 25, 2345, 27, 2345, 25, 1471, 25, 299, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+b \sin (i c+i d x)^4\right )^3}{\sin (i c+i d x)^{11}}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (b \sin (i c+i d x)^4+a\right )^3}{\sin (i c+i d x)^{11}}dx\) |
| \(\Big \downarrow \) 3694 |
| \(\displaystyle \frac {\int \frac {\left (b \cosh ^4(c+d x)-2 b \cosh ^2(c+d x)+a+b\right )^3}{\left (1-\cosh ^2(c+d x)\right )^6}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}-\frac {1}{10} \int -\frac {-10 b^3 \cosh ^{10}(c+d x)+50 b^3 \cosh ^8(c+d x)-10 b^2 (3 a+10 b) \cosh ^6(c+d x)+10 b^2 (9 a+10 b) \cosh ^4(c+d x)-10 b \left (3 a^2+9 b a+5 b^2\right ) \cosh ^2(c+d x)+9 a^3+10 b^3+30 a b^2+30 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^5}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{10} \int \frac {-10 b^3 \cosh ^{10}(c+d x)+50 b^3 \cosh ^8(c+d x)-10 b^2 (3 a+10 b) \cosh ^6(c+d x)+10 b^2 (9 a+10 b) \cosh ^4(c+d x)-10 b \left (3 a^2+9 b a+5 b^2\right ) \cosh ^2(c+d x)+9 a^3+10 b^3+30 a b^2+30 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^5}d\cosh (c+d x)+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}-\frac {1}{8} \int -\frac {80 b^3 \cosh ^8(c+d x)-320 b^3 \cosh ^6(c+d x)+240 b^2 (a+2 b) \cosh ^4(c+d x)-160 b^2 (3 a+2 b) \cosh ^2(c+d x)+63 a^3+80 b^3+240 a b^2+240 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^4}d\cosh (c+d x)\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \int \frac {80 b^3 \cosh ^8(c+d x)-320 b^3 \cosh ^6(c+d x)+240 b^2 (a+2 b) \cosh ^4(c+d x)-160 b^2 (3 a+2 b) \cosh ^2(c+d x)+63 a^3+80 b^3+240 a b^2+240 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^4}d\cosh (c+d x)+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}-\frac {1}{6} \int -\frac {15 \left (-32 b^3 \cosh ^6(c+d x)+96 b^3 \cosh ^4(c+d x)-96 b^2 (a+b) \cosh ^2(c+d x)+21 a^3+32 b^3+96 a b^2+80 a^2 b\right )}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)\right )+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} \int \frac {-32 b^3 \cosh ^6(c+d x)+96 b^3 \cosh ^4(c+d x)-96 b^2 (a+b) \cosh ^2(c+d x)+21 a^3+32 b^3+96 a b^2+80 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^3}d\cosh (c+d x)+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}\right )+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} \left (\frac {a^2 (21 a+80 b) \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}-\frac {1}{4} \int -\frac {128 b^3 \cosh ^4(c+d x)-256 b^3 \cosh ^2(c+d x)+63 a^3+128 b^3+384 a b^2+240 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}\right )+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} \left (\frac {1}{4} \int \frac {128 b^3 \cosh ^4(c+d x)-256 b^3 \cosh ^2(c+d x)+63 a^3+128 b^3+384 a b^2+240 a^2 b}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}\right )+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} \left (\frac {1}{4} \left (\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}-\frac {1}{2} \int -\frac {63 a^3+240 b a^2+384 b^2 a+256 b^3-256 b^3 \cosh ^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}\right )+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \frac {63 a^3+240 b a^2+384 b^2 a+256 b^3-256 b^3 \cosh ^2(c+d x)}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}\right )+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 a \left (21 a^2+80 a b+128 b^2\right ) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)+256 b^3 \cosh (c+d x)\right )+\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}\right )+\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}\right )+\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {a^3 \cosh (c+d x)}{10 \left (1-\cosh ^2(c+d x)\right )^5}+\frac {1}{10} \left (\frac {9 a^3 \cosh (c+d x)}{8 \left (1-\cosh ^2(c+d x)\right )^4}+\frac {1}{8} \left (\frac {5}{2} \left (\frac {1}{4} \left (\frac {1}{2} \left (3 a \left (21 a^2+80 a b+128 b^2\right ) \text {arctanh}(\cosh (c+d x))+256 b^3 \cosh (c+d x)\right )+\frac {3 a \left (21 a^2+80 a b+128 b^2\right ) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{4 \left (1-\cosh ^2(c+d x)\right )^2}\right )+\frac {a^2 (21 a+80 b) \cosh (c+d x)}{2 \left (1-\cosh ^2(c+d x)\right )^3}\right )\right )}{d}\) |
((a^3*Cosh[c + d*x])/(10*(1 - Cosh[c + d*x]^2)^5) + ((9*a^3*Cosh[c + d*x]) /(8*(1 - Cosh[c + d*x]^2)^4) + ((a^2*(21*a + 80*b)*Cosh[c + d*x])/(2*(1 - Cosh[c + d*x]^2)^3) + (5*((a^2*(21*a + 80*b)*Cosh[c + d*x])/(4*(1 - Cosh[c + d*x]^2)^2) + ((3*a*(21*a^2 + 80*a*b + 128*b^2)*ArcTanh[Cosh[c + d*x]] + 256*b^3*Cosh[c + d*x])/2 + (3*a*(21*a^2 + 80*a*b + 128*b^2)*Cosh[c + d*x] )/(2*(1 - Cosh[c + d*x]^2)))/4))/2)/8)/10)/d
3.3.15.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(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] + Simp[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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[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] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 1.36 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{9}}{10}+\frac {9 \operatorname {csch}\left (d x +c \right )^{7}}{80}-\frac {21 \operatorname {csch}\left (d x +c \right )^{5}}{160}+\frac {21 \operatorname {csch}\left (d x +c \right )^{3}}{128}-\frac {63 \,\operatorname {csch}\left (d x +c \right )}{256}\right ) \coth \left (d x +c \right )+\frac {63 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{128}\right )+3 a^{2} b \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \cosh \left (d x +c \right )}{d}\) | \(166\) |
| default | \(\frac {a^{3} \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{9}}{10}+\frac {9 \operatorname {csch}\left (d x +c \right )^{7}}{80}-\frac {21 \operatorname {csch}\left (d x +c \right )^{5}}{160}+\frac {21 \operatorname {csch}\left (d x +c \right )^{3}}{128}-\frac {63 \,\operatorname {csch}\left (d x +c \right )}{256}\right ) \coth \left (d x +c \right )+\frac {63 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{128}\right )+3 a^{2} b \left (\left (-\frac {\operatorname {csch}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {csch}\left (d x +c \right )^{3}}{24}-\frac {5 \,\operatorname {csch}\left (d x +c \right )}{16}\right ) \coth \left (d x +c \right )+\frac {5 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )}{8}\right )+3 a \,b^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \cosh \left (d x +c \right )}{d}\) | \(166\) |
| parallelrisch | \(\frac {-\frac {63 \left (a^{2}+\frac {80}{21} a b +\frac {128}{21} b^{2}\right ) a \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}-\frac {3297 \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a^{3} \left (\cosh \left (5 d x +5 c \right )-\frac {1125 \cosh \left (6 d x +6 c \right )}{5024}-\frac {145 \cosh \left (7 d x +7 c \right )}{628}+\frac {125 \cosh \left (8 d x +8 c \right )}{2512}+\frac {15 \cosh \left (9 d x +9 c \right )}{628}-\frac {25 \cosh \left (10 d x +10 c \right )}{5024}+\frac {27985 \cosh \left (d x +c \right )}{6594}-\frac {2625 \cosh \left (2 d x +2 c \right )}{2512}-\frac {2805 \cosh \left (3 d x +3 c \right )}{1099}+\frac {375 \cosh \left (4 d x +4 c \right )}{628}+\frac {1575}{2512}\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{83886080}-\frac {15 \operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{2} b \left (\cosh \left (5 d x +5 c \right )-\frac {3 \cosh \left (6 d x +6 c \right )}{16}+\frac {66 \cosh \left (d x +c \right )}{5}-\frac {45 \cosh \left (2 d x +2 c \right )}{16}-\frac {17 \cosh \left (3 d x +3 c \right )}{3}+\frac {9 \cosh \left (4 d x +4 c \right )}{8}+\frac {15}{8}\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{16384}+\frac {3 b^{2} a \left (\operatorname {sech}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-3\right ) \operatorname {csch}\left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {3 \left (a \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {8 b \left (1+\cosh \left (d x +c \right )\right )}{3}\right ) b^{2}}{8}}{d}\) | \(318\) |
| risch | \(\frac {{\mathrm e}^{d x +c} b^{3}}{2 d}+\frac {{\mathrm e}^{-d x -c} b^{3}}{2 d}-\frac {a \,{\mathrm e}^{d x +c} \left (13188 a^{2} {\mathrm e}^{14 d x +14 c}+38400 b^{2} {\mathrm e}^{14 d x +14 c}-33660 a^{2} {\mathrm e}^{12 d x +12 c}-53760 b^{2} {\mathrm e}^{12 d x +12 c}-3045 a^{2} {\mathrm e}^{16 d x +16 c}-13440 b^{2} {\mathrm e}^{16 d x +16 c}+315 a^{2} {\mathrm e}^{18 d x +18 c}+1920 b^{2} {\mathrm e}^{18 d x +18 c}+55970 a^{2} {\mathrm e}^{10 d x +10 c}+26880 b^{2} {\mathrm e}^{10 d x +10 c}-33660 a^{2} {\mathrm e}^{6 d x +6 c}+55970 \,{\mathrm e}^{8 d x +8 c} a^{2}+53280 \,{\mathrm e}^{8 d x +8 c} a b +50240 \,{\mathrm e}^{4 d x +4 c} a b +1200 a b +315 a^{2}+1920 b^{2}-11600 \,{\mathrm e}^{2 d x +2 c} b a +1200 a b \,{\mathrm e}^{18 d x +18 c}-11600 a b \,{\mathrm e}^{16 d x +16 c}+50240 a b \,{\mathrm e}^{14 d x +14 c}-93120 a b \,{\mathrm e}^{12 d x +12 c}+53280 a b \,{\mathrm e}^{10 d x +10 c}-93120 \,{\mathrm e}^{6 d x +6 c} a b -13440 b^{2} {\mathrm e}^{2 d x +2 c}+38400 b^{2} {\mathrm e}^{4 d x +4 c}-3045 \,{\mathrm e}^{2 d x +2 c} a^{2}-53760 b^{2} {\mathrm e}^{6 d x +6 c}+13188 \,{\mathrm e}^{4 d x +4 c} a^{2}+26880 b^{2} {\mathrm e}^{8 d x +8 c}\right )}{640 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{10}}-\frac {63 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{256 d}-\frac {15 a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{16 d}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{2 d}+\frac {63 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{256 d}+\frac {15 a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{16 d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{2 d}\) | \(519\) |
1/d*(a^3*((-1/10*csch(d*x+c)^9+9/80*csch(d*x+c)^7-21/160*csch(d*x+c)^5+21/ 128*csch(d*x+c)^3-63/256*csch(d*x+c))*coth(d*x+c)+63/128*arctanh(exp(d*x+c )))+3*a^2*b*((-1/6*csch(d*x+c)^5+5/24*csch(d*x+c)^3-5/16*csch(d*x+c))*coth (d*x+c)+5/8*arctanh(exp(d*x+c)))+3*a*b^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arc tanh(exp(d*x+c)))+b^3*cosh(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 13503 vs. \(2 (177) = 354\).
Time = 0.40 (sec) , antiderivative size = 13503, normalized size of antiderivative = 71.44 \[ \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
Timed out. \[ \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (177) = 354\).
Time = 0.20 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.03 \[ \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {1}{2} \, b^{3} {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {1}{1280} \, a^{3} {\left (\frac {315 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {315 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (315 \, e^{\left (-d x - c\right )} - 3045 \, e^{\left (-3 \, d x - 3 \, c\right )} + 13188 \, e^{\left (-5 \, d x - 5 \, c\right )} - 33660 \, e^{\left (-7 \, d x - 7 \, c\right )} + 55970 \, e^{\left (-9 \, d x - 9 \, c\right )} + 55970 \, e^{\left (-11 \, d x - 11 \, c\right )} - 33660 \, e^{\left (-13 \, d x - 13 \, c\right )} + 13188 \, e^{\left (-15 \, d x - 15 \, c\right )} - 3045 \, e^{\left (-17 \, d x - 17 \, c\right )} + 315 \, e^{\left (-19 \, d x - 19 \, c\right )}\right )}}{d {\left (10 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} + 120 \, e^{\left (-6 \, d x - 6 \, c\right )} - 210 \, e^{\left (-8 \, d x - 8 \, c\right )} + 252 \, e^{\left (-10 \, d x - 10 \, c\right )} - 210 \, e^{\left (-12 \, d x - 12 \, c\right )} + 120 \, e^{\left (-14 \, d x - 14 \, c\right )} - 45 \, e^{\left (-16 \, d x - 16 \, c\right )} + 10 \, e^{\left (-18 \, d x - 18 \, c\right )} - e^{\left (-20 \, d x - 20 \, c\right )} - 1\right )}}\right )} + \frac {1}{16} \, a^{2} b {\left (\frac {15 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {15 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (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 )}\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}{2} \, a b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
1/2*b^3*(e^(d*x + c)/d + e^(-d*x - c)/d) + 1/1280*a^3*(315*log(e^(-d*x - c ) + 1)/d - 315*log(e^(-d*x - c) - 1)/d + 2*(315*e^(-d*x - c) - 3045*e^(-3* d*x - 3*c) + 13188*e^(-5*d*x - 5*c) - 33660*e^(-7*d*x - 7*c) + 55970*e^(-9 *d*x - 9*c) + 55970*e^(-11*d*x - 11*c) - 33660*e^(-13*d*x - 13*c) + 13188* e^(-15*d*x - 15*c) - 3045*e^(-17*d*x - 17*c) + 315*e^(-19*d*x - 19*c))/(d* (10*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) + 120*e^(-6*d*x - 6*c) - 210*e^ (-8*d*x - 8*c) + 252*e^(-10*d*x - 10*c) - 210*e^(-12*d*x - 12*c) + 120*e^( -14*d*x - 14*c) - 45*e^(-16*d*x - 16*c) + 10*e^(-18*d*x - 18*c) - e^(-20*d *x - 20*c) - 1))) + 1/16*a^2*b*(15*log(e^(-d*x - c) + 1)/d - 15*log(e^(-d* x - c) - 1)/d + 2*(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))) + 3/2* a*b^2*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(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)))
Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (177) = 354\).
Time = 0.56 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.52 \[ \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\frac {1280 \, b^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 15 \, {\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (21 \, a^{3} + 80 \, a^{2} b + 128 \, a b^{2}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (315 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1200 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 1920 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 5880 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 22400 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 30720 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 43008 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 163840 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 184320 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 151680 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 542720 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 491520 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 247040 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 675840 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 491520 \, a b^{2} {\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 )}^{5}}}{2560 \, d} \]
1/2560*(1280*b^3*(e^(d*x + c) + e^(-d*x - c)) + 15*(21*a^3 + 80*a^2*b + 12 8*a*b^2)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 15*(21*a^3 + 80*a^2*b + 128 *a*b^2)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(315*a^3*(e^(d*x + c) + e^ (-d*x - c))^9 + 1200*a^2*b*(e^(d*x + c) + e^(-d*x - c))^9 + 1920*a*b^2*(e^ (d*x + c) + e^(-d*x - c))^9 - 5880*a^3*(e^(d*x + c) + e^(-d*x - c))^7 - 22 400*a^2*b*(e^(d*x + c) + e^(-d*x - c))^7 - 30720*a*b^2*(e^(d*x + c) + e^(- d*x - c))^7 + 43008*a^3*(e^(d*x + c) + e^(-d*x - c))^5 + 163840*a^2*b*(e^( d*x + c) + e^(-d*x - c))^5 + 184320*a*b^2*(e^(d*x + c) + e^(-d*x - c))^5 - 151680*a^3*(e^(d*x + c) + e^(-d*x - c))^3 - 542720*a^2*b*(e^(d*x + c) + e ^(-d*x - c))^3 - 491520*a*b^2*(e^(d*x + c) + e^(-d*x - c))^3 + 247040*a^3* (e^(d*x + c) + e^(-d*x - c)) + 675840*a^2*b*(e^(d*x + c) + e^(-d*x - c)) + 491520*a*b^2*(e^(d*x + c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^ 2 - 4)^5)/d
Time = 1.83 (sec) , antiderivative size = 1194, normalized size of antiderivative = 6.32 \[ \int \text {csch}^{11}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx=\text {Too large to display} \]
(b^3*exp(c + d*x))/(2*d) - ((24*exp(5*c + 5*d*x)*(7*a*b^2 + 4*a^2*b))/(5*d ) - (48*exp(7*c + 7*d*x)*(7*a*b^2 + 8*a^2*b))/(5*d) - (48*exp(11*c + 11*d* x)*(7*a*b^2 + 8*a^2*b))/(5*d) + (24*exp(13*c + 13*d*x)*(7*a*b^2 + 4*a^2*b) )/(5*d) + (4*exp(9*c + 9*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(5*d) - ( 48*a*b^2*exp(3*c + 3*d*x))/(5*d) - (48*a*b^2*exp(15*c + 15*d*x))/(5*d) + ( 6*a*b^2*exp(17*c + 17*d*x))/(5*d) + (6*a*b^2*exp(c + d*x))/(5*d))/(45*exp( 4*c + 4*d*x) - 10*exp(2*c + 2*d*x) - 120*exp(6*c + 6*d*x) + 210*exp(8*c + 8*d*x) - 252*exp(10*c + 10*d*x) + 210*exp(12*c + 12*d*x) - 120*exp(14*c + 14*d*x) + 45*exp(16*c + 16*d*x) - 10*exp(18*c + 18*d*x) + exp(20*c + 20*d* x) + 1) + (b^3*exp(- c - d*x))/(2*d) + (3*atan((exp(d*x)*exp(c)*(21*a^3*(- d^2)^(1/2) + 128*a*b^2*(-d^2)^(1/2) + 80*a^2*b*(-d^2)^(1/2)))/(d*(3360*a^5 *b + 441*a^6 + 16384*a^2*b^4 + 20480*a^3*b^3 + 11776*a^4*b^2)^(1/2)))*(336 0*a^5*b + 441*a^6 + 16384*a^2*b^4 + 20480*a^3*b^3 + 11776*a^4*b^2)^(1/2))/ (128*(-d^2)^(1/2)) - (exp(c + d*x)*(208*a^2*b + a^3))/(5*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) + ex p(10*c + 10*d*x) - 1)) - (exp(c + d*x)*(80*a^2*b + 21*a^3))/(80*d*(3*exp(2 *c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (3*exp(c + d*x )*(464*a^2*b - 3*a^3))/(40*d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4* exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (3*exp(c + d*x)*(128*a*b^2 + 8 0*a^2*b + 21*a^3))/(128*d*(exp(2*c + 2*d*x) - 1)) - (1032*a^3*exp(c + d...