Integrand size = 21, antiderivative size = 42 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {(a+b) \arctan (\sinh (c+d x))}{2 d}+\frac {(a-b) \text {sech}(c+d x) \tanh (c+d x)}{2 d} \] Output:
1/2*(a+b)*arctan(sinh(d*x+c))/d+1/2*(a-b)*sech(d*x+c)*tanh(d*x+c)/d
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.69 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {a \arctan (\sinh (c+d x))}{2 d}+\frac {b \arctan (\sinh (c+d x))}{2 d}+\frac {a \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {b \text {sech}(c+d x) \tanh (c+d x)}{2 d} \] Input:
Integrate[Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]
Output:
(a*ArcTan[Sinh[c + d*x]])/(2*d) + (b*ArcTan[Sinh[c + d*x]])/(2*d) + (a*Sec h[c + d*x]*Tanh[c + d*x])/(2*d) - (b*Sech[c + d*x]*Tanh[c + d*x])/(2*d)
Time = 0.24 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3669, 298, 216}
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.
\(\displaystyle \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a-b \sin (i c+i d x)^2}{\cos (i c+i d x)^3}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {b \sinh ^2(c+d x)+a}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {\frac {1}{2} (a+b) \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {(a-b) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {1}{2} (a+b) \arctan (\sinh (c+d x))+\frac {(a-b) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}}{d}\) |
Input:
Int[Sech[c + d*x]^3*(a + b*Sinh[c + d*x]^2),x]
Output:
(((a + b)*ArcTan[Sinh[c + d*x]])/2 + ((a - b)*Sinh[c + d*x])/(2*(1 + Sinh[ c + d*x]^2)))/d
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f S ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] /ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Time = 6.71 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.67
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(70\) |
default | \(\frac {a \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\) | \(70\) |
risch | \(\frac {{\mathrm e}^{d x +c} \left (a -b \right ) \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 d}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{2 d}\) | \(109\) |
Input:
int(sech(d*x+c)^3*(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+b*(-sinh(d*x+c)/co sh(d*x+c)^2+1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (38) = 76\).
Time = 0.10 (sec) , antiderivative size = 324, normalized size of antiderivative = 7.71 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {{\left (a - b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a - b\right )} \sinh \left (d x + c\right )^{3} + {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a + b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a + b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b\right )} \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (a - b\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} - a + b\right )} \sinh \left (d x + c\right )}{d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} + 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} + d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d} \] Input:
integrate(sech(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
Output:
((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)^4 + 4*(a + b)*cosh(d*x + c)*si nh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 + 2*(a + b)*cosh(d*x + c)^2 + 2*(3 *(a + b)*cosh(d*x + c)^2 + a + b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 + (a + b)*cosh(d*x + c))*sinh(d*x + c) + a + b)*arctan(cosh(d*x + c) + sinh(d*x + c)) - (a - b)*cosh(d*x + c) + (3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x + c)^3 + d*sinh(d*x + c)^4 + 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 + d)*s inh(d*x + c)^2 + 4*(d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c) + d )
\[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right ) \operatorname {sech}^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(sech(d*x+c)**3*(a+b*sinh(d*x+c)**2),x)
Output:
Integral((a + b*sinh(c + d*x)**2)*sech(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (38) = 76\).
Time = 0.14 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.24 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=-b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - a {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \] Input:
integrate(sech(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
Output:
-b*(arctan(e^(-d*x - c))/d + (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))) - a*(arctan(e^(-d*x - c))/d - (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. 105 vs. \(2 (38) = 76\).
Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.50 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a + b\right )} + \frac {4 \, {\left (a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{4 \, d} \] Input:
integrate(sech(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="giac")
Output:
1/4*((pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(a + b) + 4*( a*(e^(d*x + c) - e^(-d*x - c)) - b*(e^(d*x + c) - e^(-d*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4))/d
Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.02 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {d^2}+b\,\sqrt {d^2}\right )}{d\,\sqrt {a^2+2\,a\,b+b^2}}\right )\,\sqrt {a^2+2\,a\,b+b^2}}{\sqrt {d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a-b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a-b\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:
int((a + b*sinh(c + d*x)^2)/cosh(c + d*x)^3,x)
Output:
(atan((exp(d*x)*exp(c)*(a*(d^2)^(1/2) + b*(d^2)^(1/2)))/(d*(2*a*b + a^2 + b^2)^(1/2)))*(2*a*b + a^2 + b^2)^(1/2))/(d^2)^(1/2) + (exp(c + d*x)*(a - b ))/(d*(exp(2*c + 2*d*x) + 1)) - (2*exp(c + d*x)*(a - b))/(d*(2*exp(2*c + 2 *d*x) + exp(4*c + 4*d*x) + 1))
Time = 0.17 (sec) , antiderivative size = 177, normalized size of antiderivative = 4.21 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx=\frac {e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) a +e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) b +2 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) a +2 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) b +\mathit {atan} \left (e^{d x +c}\right ) a +\mathit {atan} \left (e^{d x +c}\right ) b +e^{3 d x +3 c} a -e^{3 d x +3 c} b -e^{d x +c} a +e^{d x +c} b}{d \left (e^{4 d x +4 c}+2 e^{2 d x +2 c}+1\right )} \] Input:
int(sech(d*x+c)^3*(a+b*sinh(d*x+c)^2),x)
Output:
(e**(4*c + 4*d*x)*atan(e**(c + d*x))*a + e**(4*c + 4*d*x)*atan(e**(c + d*x ))*b + 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a + 2*e**(2*c + 2*d*x)*atan(e **(c + d*x))*b + atan(e**(c + d*x))*a + atan(e**(c + d*x))*b + e**(3*c + 3 *d*x)*a - e**(3*c + 3*d*x)*b - e**(c + d*x)*a + e**(c + d*x)*b)/(d*(e**(4* c + 4*d*x) + 2*e**(2*c + 2*d*x) + 1))