Integrand size = 23, antiderivative size = 71 \[ \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\sqrt {b} (a+b) \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{5/2} d}-\frac {(a+b) \cosh (c+d x)}{a^2 d}+\frac {\cosh ^3(c+d x)}{3 a d} \]
-(a+b)*cosh(d*x+c)/a^2/d+1/3*cosh(d*x+c)^3/a/d+(a+b)*arctan(cosh(d*x+c)*a^ (1/2)/b^(1/2))*b^(1/2)/a^(5/2)/d
Result contains complex when optimal does not.
Time = 3.01 (sec) , antiderivative size = 372, normalized size of antiderivative = 5.24 \[ \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \left (3 \left (a^2+8 a b+8 b^2\right ) \arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )+3 \left (a^2+8 a b+8 b^2\right ) \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )-3 a^2 \left (\arctan \left (\frac {\sqrt {a}-i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )+\arctan \left (\frac {\sqrt {a}+i \sqrt {a+b} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )\right )-6 \sqrt {a} \sqrt {b} (3 a+4 b) \cosh (c+d x)+2 a^{3/2} \sqrt {b} \cosh (3 (c+d x))\right )}{48 a^{5/2} \sqrt {b} d \left (b+a \cosh ^2(c+d x)\right )} \]
((a + 2*b + a*Cosh[2*(c + d*x)])*(3*(a^2 + 8*a*b + 8*b^2)*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh [c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/S qrt[b]] + 3*(a^2 + 8*a*b + 8*b^2)*ArcTan[((Sqrt[a] + I*Sqrt[a + b]*Sqrt[(C osh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]] - 3*a^2*(ArcTan[ (Sqrt[a] - I*Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]] + ArcTan[(Sqrt[a] + I *Sqrt[a + b]*Tanh[(c + d*x)/2])/Sqrt[b]]) - 6*Sqrt[a]*Sqrt[b]*(3*a + 4*b)* Cosh[c + d*x] + 2*a^(3/2)*Sqrt[b]*Cosh[3*(c + d*x)]))/(48*a^(5/2)*Sqrt[b]* d*(b + a*Cosh[c + d*x]^2))
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4621, 363, 262, 218}
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.
| \(\displaystyle \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i c+i d x)^3}{a+b \sec (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i c+i d x)^3}{b \sec (i c+i d x)^2+a}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle -\frac {\int \frac {\cosh ^2(c+d x) \left (1-\cosh ^2(c+d x)\right )}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle -\frac {\frac {(a+b) \int \frac {\cosh ^2(c+d x)}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a}-\frac {\cosh ^3(c+d x)}{3 a}}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle -\frac {\frac {(a+b) \left (\frac {\cosh (c+d x)}{a}-\frac {b \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a}\right )}{a}-\frac {\cosh ^3(c+d x)}{3 a}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {(a+b) \left (\frac {\cosh (c+d x)}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{a^{3/2}}\right )}{a}-\frac {\cosh ^3(c+d x)}{3 a}}{d}\) |
-((-1/3*Cosh[c + d*x]^3/a + ((a + b)*(-((Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/a^(3/2)) + Cosh[c + d*x]/a))/a)/d)
3.1.26.3.1 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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(169\) vs. \(2(61)=122\).
Time = 37.00 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.39
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{3 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a +2 b}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +b \right ) b \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{a^{2} \sqrt {a b}}-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -2 b}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(170\) |
| default | \(\frac {\frac {1}{3 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {1}{2 a \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a +2 b}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (a +b \right ) b \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{a^{2} \sqrt {a b}}-\frac {1}{3 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a -2 b}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(170\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 d a}-\frac {3 \,{\mathrm e}^{d x +c}}{8 a d}-\frac {{\mathrm e}^{d x +c} b}{2 a^{2} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 a d}-\frac {{\mathrm e}^{-d x -c} b}{2 a^{2} d}+\frac {{\mathrm e}^{-3 d x -3 c}}{24 d a}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{2} d}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{2 a^{3} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a^{2} d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right ) b}{2 a^{3} d}\) | \(274\) |
1/d*(1/3/a/(1+tanh(1/2*d*x+1/2*c))^3-1/2/a/(1+tanh(1/2*d*x+1/2*c))^2-1/2*( a+2*b)/a^2/(1+tanh(1/2*d*x+1/2*c))+(a+b)*b/a^2/(a*b)^(1/2)*arctan(1/4*(2*( a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b)/(a*b)^(1/2))-1/3/a/(tanh(1/2*d*x+1/2*c )-1)^3-1/2/a/(tanh(1/2*d*x+1/2*c)-1)^2-1/2/a^2*(-a-2*b)/(tanh(1/2*d*x+1/2* c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (61) = 122\).
Time = 0.28 (sec) , antiderivative size = 1246, normalized size of antiderivative = 17.55 \[ \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Too large to display} \]
[1/24*(a*cosh(d*x + c)^6 + 6*a*cosh(d*x + c)*sinh(d*x + c)^5 + a*sinh(d*x + c)^6 - 3*(3*a + 4*b)*cosh(d*x + c)^4 + 3*(5*a*cosh(d*x + c)^2 - 3*a - 4* b)*sinh(d*x + c)^4 + 4*(5*a*cosh(d*x + c)^3 - 3*(3*a + 4*b)*cosh(d*x + c)) *sinh(d*x + c)^3 - 3*(3*a + 4*b)*cosh(d*x + c)^2 + 3*(5*a*cosh(d*x + c)^4 - 6*(3*a + 4*b)*cosh(d*x + c)^2 - 3*a - 4*b)*sinh(d*x + c)^2 + 12*((a + b) *cosh(d*x + c)^3 + 3*(a + b)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a + b)*cos h(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3)*sqrt(-b/a)*log((a*co sh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2* (a - 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a - 2*b)*sinh(d*x + c )^2 + 4*(a*cosh(d*x + c)^3 + (a - 2*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(a *cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a)*sinh(d*x + c))*sqrt(-b/a) + a )/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c) ^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh( d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 6*(a*cosh(d*x + c)^5 - 2*(3*a + 4*b)*cosh(d*x + c)^3 - (3*a + 4*b )*cosh(d*x + c))*sinh(d*x + c) + a)/(a^2*d*cosh(d*x + c)^3 + 3*a^2*d*cosh( d*x + c)^2*sinh(d*x + c) + 3*a^2*d*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*d*s inh(d*x + c)^3), 1/24*(a*cosh(d*x + c)^6 + 6*a*cosh(d*x + c)*sinh(d*x + c) ^5 + a*sinh(d*x + c)^6 - 3*(3*a + 4*b)*cosh(d*x + c)^4 + 3*(5*a*cosh(d*...
\[ \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\sinh ^{3}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
-1/24*(3*(3*a*e^(4*c) + 4*b*e^(4*c))*e^(4*d*x) + 3*(3*a*e^(2*c) + 4*b*e^(2 *c))*e^(2*d*x) - a*e^(6*d*x + 6*c) - a)*e^(-3*d*x - 3*c)/(a^2*d) + 1/8*int egrate(16*((a*b*e^(3*c) + b^2*e^(3*c))*e^(3*d*x) - (a*b*e^c + b^2*e^c)*e^( d*x))/(a^3*e^(4*d*x + 4*c) + a^3 + 2*(a^3*e^(2*c) + 2*a^2*b*e^(2*c))*e^(2* d*x)), x)
\[ \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.56 (sec) , antiderivative size = 473, normalized size of antiderivative = 6.66 \[ \int \frac {\sinh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,a\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (2\,a^4\,b\,d\,\sqrt {a^2\,b+2\,a\,b^2+b^3}+2\,a^2\,b^3\,d\,\sqrt {a^2\,b+2\,a\,b^2+b^3}+4\,a^3\,b^2\,d\,\sqrt {a^2\,b+2\,a\,b^2+b^3}\right )}{a^{11}\,d^2\,\left (a+b\right )}+\frac {2\,\left (b^4\,\sqrt {a^5\,d^2}+3\,a^2\,b^2\,\sqrt {a^5\,d^2}+3\,a\,b^3\,\sqrt {a^5\,d^2}+a^3\,b\,\sqrt {a^5\,d^2}\right )}{a^8\,d\,\sqrt {b\,{\left (a+b\right )}^2}\,\sqrt {a^5\,d^2}}\right )\,\sqrt {a^5\,d^2}}{4\,a^2\,b+8\,a\,b^2+4\,b^3}+\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (b^4\,\sqrt {a^5\,d^2}+3\,a^2\,b^2\,\sqrt {a^5\,d^2}+3\,a\,b^3\,\sqrt {a^5\,d^2}+a^3\,b\,\sqrt {a^5\,d^2}\right )}{a^2\,d\,\sqrt {b\,{\left (a+b\right )}^2}\,\left (4\,a^2\,b+8\,a\,b^2+4\,b^3\right )}\right )-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a+b\right )\,\sqrt {a^5\,d^2}}{2\,a^2\,d\,\sqrt {b\,{\left (a+b\right )}^2}}\right )\right )\,\sqrt {a^2\,b+2\,a\,b^2+b^3}}{2\,\sqrt {a^5\,d^2}}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,a\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a+4\,b\right )}{8\,a^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a+4\,b\right )}{8\,a^2\,d} \]
exp(- 3*c - 3*d*x)/(24*a*d) - ((2*atan((a^6*exp(d*x)*exp(c)*((4*(2*a^4*b*d *(2*a*b^2 + a^2*b + b^3)^(1/2) + 2*a^2*b^3*d*(2*a*b^2 + a^2*b + b^3)^(1/2) + 4*a^3*b^2*d*(2*a*b^2 + a^2*b + b^3)^(1/2)))/(a^11*d^2*(a + b)) + (2*(b^ 4*(a^5*d^2)^(1/2) + 3*a^2*b^2*(a^5*d^2)^(1/2) + 3*a*b^3*(a^5*d^2)^(1/2) + a^3*b*(a^5*d^2)^(1/2)))/(a^8*d*(b*(a + b)^2)^(1/2)*(a^5*d^2)^(1/2)))*(a^5* d^2)^(1/2))/(8*a*b^2 + 4*a^2*b + 4*b^3) + (2*exp(3*c)*exp(3*d*x)*(b^4*(a^5 *d^2)^(1/2) + 3*a^2*b^2*(a^5*d^2)^(1/2) + 3*a*b^3*(a^5*d^2)^(1/2) + a^3*b* (a^5*d^2)^(1/2)))/(a^2*d*(b*(a + b)^2)^(1/2)*(8*a*b^2 + 4*a^2*b + 4*b^3))) - 2*atan((exp(d*x)*exp(c)*(a + b)*(a^5*d^2)^(1/2))/(2*a^2*d*(b*(a + b)^2) ^(1/2))))*(2*a*b^2 + a^2*b + b^3)^(1/2))/(2*(a^5*d^2)^(1/2)) + exp(3*c + 3 *d*x)/(24*a*d) - (exp(c + d*x)*(3*a + 4*b))/(8*a^2*d) - (exp(- c - d*x)*(3 *a + 4*b))/(8*a^2*d)