Integrand size = 23, antiderivative size = 88 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {b^{3/2} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a^2 \sqrt {a-b} d}+\frac {(a+2 b) \text {arctanh}(\cosh (c+d x))}{2 a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]
1/2*(a+2*b)*arctanh(cosh(d*x+c))/a^2/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d+b^( 3/2)*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))/a^2/d/(a-b)^(1/2)
Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.50 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \left (\frac {8 b^{3/2} \arctan \left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {8 b^{3/2} \arctan \left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+4 (a+2 b) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-4 (a+2 b) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{16 a^2 d \left (b+a \text {csch}^2(c+d x)\right )} \]
((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*((8*b^(3/2)*ArcTan[(Sqrt[ b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/Sqrt[a - b] + (8*b^(3/2)*A rcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]])/Sqrt[a - b] - a*Csch[(c + d*x)/2]^2 + 4*(a + 2*b)*Log[Cosh[(c + d*x)/2]] - 4*(a + 2*b)*L og[Sinh[(c + d*x)/2]] - a*Sech[(c + d*x)/2]^2))/(16*a^2*d*(b + a*Csch[c + d*x]^2))
Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 3665, 316, 397, 218, 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.
| \(\displaystyle \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\sin (i c+i d x)^3 \left (a-b \sin (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\sin (i c+i d x)^3 \left (a-b \sin (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 3665 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\cosh ^2(c+d x)\right )^2 \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\int \frac {b \cosh ^2(c+d x)+a+b}{\left (1-\cosh ^2(c+d x)\right ) \left (b \cosh ^2(c+d x)+a-b\right )}d\cosh (c+d x)}{2 a}+\frac {\cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {2 b^2 \int \frac {1}{b \cosh ^2(c+d x)+a-b}d\cosh (c+d x)}{a}+\frac {(a+2 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}}{2 a}+\frac {\cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {(a+2 b) \int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}}{2 a}+\frac {\cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a \sqrt {a-b}}+\frac {(a+2 b) \text {arctanh}(\cosh (c+d x))}{a}}{2 a}+\frac {\cosh (c+d x)}{2 a \left (1-\cosh ^2(c+d x)\right )}}{d}\) |
(((2*b^(3/2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a*Sqrt[a - b]) + ((a + 2*b)*ArcTanh[Cosh[c + d*x]])/a)/(2*a) + Cosh[c + d*x]/(2*a*(1 - Co sh[c + d*x]^2)))/d
3.1.38.3.1 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[((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)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (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 - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -4 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}+\frac {b^{2} \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a^{2} \sqrt {a b -b^{2}}}}{d}\) | \(113\) |
| default | \(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -4 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2}}+\frac {b^{2} \arctan \left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a^{2} \sqrt {a b -b^{2}}}}{d}\) | \(113\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d \,a^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a d}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d \,a^{2}}+\frac {\sqrt {-b \left (a -b \right )}\, b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d \,a^{2}}-\frac {\sqrt {-b \left (a -b \right )}\, b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d \,a^{2}}\) | \(227\) |
1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a-1/8/a/tanh(1/2*d*x+1/2*c)^2+1/4/a^2*(-2*a -4*b)*ln(tanh(1/2*d*x+1/2*c))+b^2/a^2/(a*b-b^2)^(1/2)*arctan(1/4*(2*tanh(1 /2*d*x+1/2*c)^2*a-2*a+4*b)/(a*b-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 865 vs. \(2 (76) = 152\).
Time = 0.31 (sec) , antiderivative size = 1837, normalized size of antiderivative = 20.88 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\text {Too large to display} \]
[-1/2*(2*a*cosh(d*x + c)^3 + 6*a*cosh(d*x + c)*sinh(d*x + c)^2 + 2*a*sinh( d*x + c)^3 - (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*si nh(d*x + c)^4 - 2*b*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - b*cosh(d*x + c))*sinh(d*x + c) + b)*sqrt( -b/(a - b))*log((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b *sinh(d*x + c)^4 - 2*(2*a - 3*b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + 4*((a - b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c )*sinh(d*x + c)^2 + (a - b)*sinh(d*x + c)^3 + (a - b)*cosh(d*x + c) + (3*( a - b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c))*sqrt(-b/(a - b)) + b)/(b*co sh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2* (2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c )^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b)) + 2*a*cosh(d*x + c) - ((a + 2*b)*cosh(d*x + c)^4 + 4*(a + 2*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2*b)*cosh(d*x + c) ^2 + 2*(3*(a + 2*b)*cosh(d*x + c)^2 - a - 2*b)*sinh(d*x + c)^2 + 4*((a + 2 *b)*cosh(d*x + c)^3 - (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a + 2*b)*lo g(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a + 2*b)*cosh(d*x + c)^4 + 4*(a + 2*b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + 2*b)*sinh(d*x + c)^4 - 2*(a + 2 *b)*cosh(d*x + c)^2 + 2*(3*(a + 2*b)*cosh(d*x + c)^2 - a - 2*b)*sinh(d*...
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \sinh ^{2}{\left (c + d x \right )}}\, dx \]
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
-(e^(3*d*x + 3*c) + e^(d*x + c))/(a*d*e^(4*d*x + 4*c) - 2*a*d*e^(2*d*x + 2 *c) + a*d) + 1/2*(a + 2*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^2*d) - 1/2*(a + 2*b)*log((e^(d*x + c) - 1)*e^(-c))/(a^2*d) + 8*integrate(1/4*(b^2*e^(3*d *x + 3*c) - b^2*e^(d*x + c))/(a^2*b*e^(4*d*x + 4*c) + a^2*b + 2*(2*a^3*e^( 2*c) - a^2*b*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right )^{2} + a} \,d x } \]
Time = 2.18 (sec) , antiderivative size = 571, normalized size of antiderivative = 6.49 \[ \int \frac {\text {csch}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^7\,\sqrt {-a^4\,d^2}+18\,b^7\,\sqrt {-a^4\,d^2}-36\,a^2\,b^5\,\sqrt {-a^4\,d^2}-30\,a^3\,b^4\,\sqrt {-a^4\,d^2}+12\,a^4\,b^3\,\sqrt {-a^4\,d^2}+21\,a^5\,b^2\,\sqrt {-a^4\,d^2}+9\,a\,b^6\,\sqrt {-a^4\,d^2}+8\,a^6\,b\,\sqrt {-a^4\,d^2}\right )}{a^8\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+9\,a^2\,b^6\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}-18\,a^4\,b^4\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}-6\,a^5\,b^3\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+9\,a^6\,b^2\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}+6\,a^7\,b\,d\,\sqrt {a^2+4\,a\,b+4\,b^2}}\right )\,\sqrt {a^2+4\,a\,b+4\,b^2}}{\sqrt {-a^4\,d^2}}-\frac {{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\left (-b\right )}^{3/2}\,\ln \left (\frac {64\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a^3+3\,a^2\,b-3\,b^3\right )}{a^5\,{\left (a-b\right )}^2}-\frac {128\,{\mathrm {e}}^{c+d\,x}\,\left (a^3+3\,a^2\,b-3\,b^3\right )}{a^5\,\sqrt {-b}\,{\left (a-b\right )}^{3/2}}\right )}{2\,a^2\,d\,\sqrt {a-b}}+\frac {{\left (-b\right )}^{3/2}\,\ln \left (\frac {64\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a^3+3\,a^2\,b-3\,b^3\right )}{a^5\,{\left (a-b\right )}^2}+\frac {128\,{\mathrm {e}}^{c+d\,x}\,\left (a^3+3\,a^2\,b-3\,b^3\right )}{a^5\,\sqrt {-b}\,{\left (a-b\right )}^{3/2}}\right )}{2\,a^2\,d\,\sqrt {a-b}} \]
(atan((exp(d*x)*exp(c)*(a^7*(-a^4*d^2)^(1/2) + 18*b^7*(-a^4*d^2)^(1/2) - 3 6*a^2*b^5*(-a^4*d^2)^(1/2) - 30*a^3*b^4*(-a^4*d^2)^(1/2) + 12*a^4*b^3*(-a^ 4*d^2)^(1/2) + 21*a^5*b^2*(-a^4*d^2)^(1/2) + 9*a*b^6*(-a^4*d^2)^(1/2) + 8* a^6*b*(-a^4*d^2)^(1/2)))/(a^8*d*(4*a*b + a^2 + 4*b^2)^(1/2) + 9*a^2*b^6*d* (4*a*b + a^2 + 4*b^2)^(1/2) - 18*a^4*b^4*d*(4*a*b + a^2 + 4*b^2)^(1/2) - 6 *a^5*b^3*d*(4*a*b + a^2 + 4*b^2)^(1/2) + 9*a^6*b^2*d*(4*a*b + a^2 + 4*b^2) ^(1/2) + 6*a^7*b*d*(4*a*b + a^2 + 4*b^2)^(1/2)))*(4*a*b + a^2 + 4*b^2)^(1/ 2))/(-a^4*d^2)^(1/2) - exp(c + d*x)/(a*d*(exp(2*c + 2*d*x) - 1)) - (2*exp( c + d*x))/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - ((-b)^(3/2)* log((64*(exp(2*c + 2*d*x) + 1)*(3*a^2*b + a^3 - 3*b^3))/(a^5*(a - b)^2) - (128*exp(c + d*x)*(3*a^2*b + a^3 - 3*b^3))/(a^5*(-b)^(1/2)*(a - b)^(3/2))) )/(2*a^2*d*(a - b)^(1/2)) + ((-b)^(3/2)*log((64*(exp(2*c + 2*d*x) + 1)*(3* a^2*b + a^3 - 3*b^3))/(a^5*(a - b)^2) + (128*exp(c + d*x)*(3*a^2*b + a^3 - 3*b^3))/(a^5*(-b)^(1/2)*(a - b)^(3/2))))/(2*a^2*d*(a - b)^(1/2))