Integrand size = 21, antiderivative size = 52 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {a^2 \text {arctanh}(\cosh (c+d x))}{d}+\frac {(2 a-b) b \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d} \]
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.00 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {2 a b \cosh (c) \cosh (d x)}{d}-\frac {3 b^2 \cosh (c+d x)}{4 d}+\frac {b^2 \cosh (3 (c+d x))}{12 d}-\frac {a^2 \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a^2 \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {2 a b \sinh (c) \sinh (d x)}{d} \]
(2*a*b*Cosh[c]*Cosh[d*x])/d - (3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) - (a^2*Log[Cosh[c/2 + (d*x)/2]])/d + (a^2*Log[Sinh[c/2 + (d*x)/2]])/d + (2*a*b*Sinh[c]*Sinh[d*x])/d
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 26, 3665, 300, 2009}
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.
\(\displaystyle \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \left (a-b \sin (i c+i d x)^2\right )^2}{\sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\left (a-b \sin (i c+i d x)^2\right )^2}{\sin (i c+i d x)}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle -\frac {\int \frac {\left (b \cosh ^2(c+d x)+a-b\right )^2}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 300 |
\(\displaystyle -\frac {\int \left (\frac {a^2}{1-\cosh ^2(c+d x)}-b^2 \cosh ^2(c+d x)-(2 a-b) b\right )d\cosh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 \text {arctanh}(\cosh (c+d x))-b (2 a-b) \cosh (c+d x)-\frac {1}{3} b^2 \cosh ^3(c+d x)}{d}\) |
3.1.15.3.1 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_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
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 = 0.22 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {-2 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+2 a b \cosh \left (d x +c \right )+b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\) | \(50\) |
default | \(\frac {-2 a^{2} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+2 a b \cosh \left (d x +c \right )+b^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )}{d}\) | \(50\) |
parallelrisch | \(\frac {a^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\frac {b \cosh \left (3 d x +3 c \right )}{24}+\left (a -\frac {3 b}{8}\right ) \cosh \left (d x +c \right )+a -\frac {b}{3}\right ) b}{d}\) | \(52\) |
risch | \(\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}+\frac {{\mathrm e}^{d x +c} a b}{d}-\frac {3 \,{\mathrm e}^{d x +c} b^{2}}{8 d}+\frac {{\mathrm e}^{-d x -c} a b}{d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\) | \(127\) |
Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (50) = 100\).
Time = 0.30 (sec) , antiderivative size = 492, normalized size of antiderivative = 9.46 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} - 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \, {\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \]
1/24*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh (d*x + c)^6 + 3*(8*a*b - 3*b^2)*cosh(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + 8*a*b - 3*b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*(8*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(8*a*b - 3*b^2)*cosh(d*x + c)^2 + 3*(5*b^2*cosh(d*x + c)^4 + 6*(8*a*b - 3*b^2)*cosh(d*x + c)^2 + 8*a*b - 3 *b^2)*sinh(d*x + c)^2 + b^2 - 24*(a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c )^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*sinh(d*x + c )^3)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 24*(a^2*cosh(d*x + c)^3 + 3* a^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*sinh(d*x + c)^3)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(b^2*cosh( d*x + c)^5 + 2*(8*a*b - 3*b^2)*cosh(d*x + c)^3 + (8*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)^2*sinh(d*x + c ) + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + d*sinh(d*x + c)^3)
\[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (50) = 100\).
Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.96 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {1}{24} \, b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \]
1/24*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d *x - 3*c)/d) + a*b*(e^(d*x + c)/d + e^(-d*x - c)/d) + a^2*log(tanh(1/2*d*x + 1/2*c))/d
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (50) = 100\).
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 2.12 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b e^{\left (d x + c\right )} - 9 \, b^{2} e^{\left (d x + c\right )} - 24 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 24 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + {\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \]
1/24*(b^2*e^(3*d*x + 3*c) + 24*a*b*e^(d*x + c) - 9*b^2*e^(d*x + c) - 24*a^ 2*log(e^(d*x + c) + 1) + 24*a^2*log(abs(e^(d*x + c) - 1)) + (24*a*b*e^(2*d *x + 2*c) - 9*b^2*e^(2*d*x + 2*c) + b^2)*e^(-3*d*x - 3*c))/d
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.23 \[ \int \text {csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}+\frac {b\,{\mathrm {e}}^{-c-d\,x}\,\left (8\,a-3\,b\right )}{8\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a-3\,b\right )}{8\,d} \]