3.85 \(\int \frac{\cosh (c+d x)}{(a+b \text{sech}^2(c+d x))^2} \, dx\)
Optimal. Leaf size=100 \[ \frac{b^2 \sinh (c+d x)}{2 a^2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{b (4 a+3 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 a^{5/2} d (a+b)^{3/2}}+\frac{\sinh (c+d x)}{a^2 d} \]
[Out]
-(b*(4*a + 3*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(5/2)*(a + b)^(3/2)*d) + Sinh[c + d*x]/(a^2*
d) + (b^2*Sinh[c + d*x])/(2*a^2*(a + b)*d*(a + b + a*Sinh[c + d*x]^2))
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Rubi [A] time = 0.131521, antiderivative size = 100, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used =
{4147, 390, 385, 205} \[ \frac{b^2 \sinh (c+d x)}{2 a^2 d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac{b (4 a+3 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 a^{5/2} d (a+b)^{3/2}}+\frac{\sinh (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
[In]
Int[Cosh[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]
[Out]
-(b*(4*a + 3*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(5/2)*(a + b)^(3/2)*d) + Sinh[c + d*x]/(a^2*
d) + (b^2*Sinh[c + d*x])/(2*a^2*(a + b)*d*(a + b + a*Sinh[c + d*x]^2))
Rule 4147
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/2] && IntegerQ[p]
Rule 390
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]
Rule 385
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{\left (a+b \text{sech}^2(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{b (2 a+b)+2 a b x^2}{a^2 \left (a+b+a x^2\right )^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\sinh (c+d x)}{a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{b (2 a+b)+2 a b x^2}{\left (a+b+a x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{a^2 d}\\ &=\frac{\sinh (c+d x)}{a^2 d}+\frac{b^2 \sinh (c+d x)}{2 a^2 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )}-\frac{(b (4 a+3 b)) \operatorname{Subst}\left (\int \frac{1}{a+b+a x^2} \, dx,x,\sinh (c+d x)\right )}{2 a^2 (a+b) d}\\ &=-\frac{b (4 a+3 b) \tan ^{-1}\left (\frac{\sqrt{a} \sinh (c+d x)}{\sqrt{a+b}}\right )}{2 a^{5/2} (a+b)^{3/2} d}+\frac{\sinh (c+d x)}{a^2 d}+\frac{b^2 \sinh (c+d x)}{2 a^2 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 1.83234, size = 234, normalized size = 2.34 \[ \frac{\text{sech}^3(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac{2 \sqrt{a} b^2 \tanh (c+d x)}{a+b}+2 \sqrt{a} \sinh (c) \cosh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+2 \sqrt{a} \cosh (c) \sinh (d x) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)+\frac{b (4 a+3 b) (\cosh (c)-\sinh (c)) \text{sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b) \tan ^{-1}\left (\frac{\sqrt{a+b} \sqrt{(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text{csch}(c+d x)}{\sqrt{a}}\right )}{(a+b)^{3/2} \sqrt{(\cosh (c)-\sinh (c))^2}}\right )}{8 a^{5/2} d \left (a+b \text{sech}^2(c+d x)\right )^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Cosh[c + d*x]/(a + b*Sech[c + d*x]^2)^2,x]
[Out]
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*((b*(4*a + 3*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[
c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*(Cosh[c] - Sinh[c
]))/((a + b)^(3/2)*Sqrt[(Cosh[c] - Sinh[c])^2]) + 2*Sqrt[a]*Cosh[d*x]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c +
d*x]*Sinh[c] + 2*Sqrt[a]*Cosh[c]*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]*Sinh[d*x] + (2*Sqrt[a]*b^2*Tan
h[c + d*x])/(a + b)))/(8*a^(5/2)*d*(a + b*Sech[c + d*x]^2)^2)
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Maple [B] time = 0.091, size = 385, normalized size = 3.9 \begin{align*} -{\frac{1}{d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{{b}^{2}}{d{a}^{2} \left ( a+b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+b \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) ^{-1}}+{\frac{{b}^{2}}{d{a}^{2} \left ( a+b \right ) }\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}a+b \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}+2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}b+a+b \right ) ^{-1}}-2\,{\frac{b}{d{a}^{3/2} \left ( a+b \right ) ^{3/2}}\arctan \left ( 1/2\,{\frac{2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b}}{\sqrt{a}}} \right ) }+2\,{\frac{b}{d{a}^{3/2} \left ( a+b \right ) ^{3/2}}\arctan \left ( 1/2\,{\frac{-2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b}}{\sqrt{a}}} \right ) }-{\frac{3\,{b}^{2}}{2\,d}\arctan \left ({\frac{1}{2} \left ( 2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}} \left ( a+b \right ) ^{-{\frac{3}{2}}}}+{\frac{3\,{b}^{2}}{2\,d}\arctan \left ({\frac{1}{2} \left ( -2\,\tanh \left ( 1/2\,dx+c/2 \right ) \sqrt{a+b}+2\,\sqrt{b} \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}} \left ( a+b \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x)
[Out]
-1/d/a^2/(tanh(1/2*d*x+1/2*c)+1)-1/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1
/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)/(a+b)*tanh(1/2*d*x+1/2*c)^3+1/d*b^2/a^2/(tanh(1/2*d*x+1/2*c)^4*a+b*ta
nh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)/(a+b)*tanh(1/2*d*x+1/2*c)-2/d/a^(
3/2)*b/(a+b)^(3/2)*arctan(1/2*(2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))+2/d/a^(3/2)*b/(a+b)^(3/2)
*arctan(1/2*(-2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))-3/2/d*b^2/a^(5/2)/(a+b)^(3/2)*arctan(1/2*(
2*tanh(1/2*d*x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))+3/2/d/a^(5/2)*b^2/(a+b)^(3/2)*arctan(1/2*(-2*tanh(1/2*d*
x+1/2*c)*(a+b)^(1/2)+2*b^(1/2))/a^(1/2))-1/d/a^2/(tanh(1/2*d*x+1/2*c)-1)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{2} + a b -{\left (a^{2} e^{\left (6 \, c\right )} + a b e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} -{\left (a^{2} e^{\left (4 \, c\right )} + 5 \, a b e^{\left (4 \, c\right )} + 6 \, b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} +{\left (a^{2} e^{\left (2 \, c\right )} + 5 \, a b e^{\left (2 \, c\right )} + 6 \, b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{2 \,{\left ({\left (a^{4} d e^{\left (5 \, c\right )} + a^{3} b d e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + 2 \,{\left (a^{4} d e^{\left (3 \, c\right )} + 3 \, a^{3} b d e^{\left (3 \, c\right )} + 2 \, a^{2} b^{2} d e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (a^{4} d e^{c} + a^{3} b d e^{c}\right )} e^{\left (d x\right )}\right )}} - \frac{1}{2} \, \int \frac{2 \,{\left ({\left (4 \, a b e^{\left (3 \, c\right )} + 3 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (4 \, a b e^{c} + 3 \, b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{4} + a^{3} b +{\left (a^{4} e^{\left (4 \, c\right )} + a^{3} b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a^{4} e^{\left (2 \, c\right )} + 3 \, a^{3} b e^{\left (2 \, c\right )} + 2 \, a^{2} b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
-1/2*(a^2 + a*b - (a^2*e^(6*c) + a*b*e^(6*c))*e^(6*d*x) - (a^2*e^(4*c) + 5*a*b*e^(4*c) + 6*b^2*e^(4*c))*e^(4*d
*x) + (a^2*e^(2*c) + 5*a*b*e^(2*c) + 6*b^2*e^(2*c))*e^(2*d*x))/((a^4*d*e^(5*c) + a^3*b*d*e^(5*c))*e^(5*d*x) +
2*(a^4*d*e^(3*c) + 3*a^3*b*d*e^(3*c) + 2*a^2*b^2*d*e^(3*c))*e^(3*d*x) + (a^4*d*e^c + a^3*b*d*e^c)*e^(d*x)) - 1
/2*integrate(2*((4*a*b*e^(3*c) + 3*b^2*e^(3*c))*e^(3*d*x) + (4*a*b*e^c + 3*b^2*e^c)*e^(d*x))/(a^4 + a^3*b + (a
^4*e^(4*c) + a^3*b*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 3*a^3*b*e^(2*c) + 2*a^2*b^2*e^(2*c))*e^(2*d*x)), x)
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Fricas [B] time = 2.62117, size = 7525, normalized size = 75.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
[1/4*(2*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 12*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5
+ 2*(a^4 + 2*a^3*b + a^2*b^2)*sinh(d*x + c)^6 + 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^4 + 2*
(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3 + 15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 2*a^4
- 4*a^3*b - 2*a^2*b^2 + 8*(5*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3
)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2 + 2*(15*(a^4 + 2*a
^3*b + a^2*b^2)*cosh(d*x + c)^4 - a^4 - 6*a^3*b - 11*a^2*b^2 - 6*a*b^3 + 6*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b
^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a^2*b + 3*a*b^2)*cosh(d*x + c)^5 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x +
c)*sinh(d*x + c)^4 + (4*a^2*b + 3*a*b^2)*sinh(d*x + c)^5 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)^3 + 2
*(4*a^2*b + 11*a*b^2 + 6*b^3 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 3*a*b^
2)*cosh(d*x + c)^3 + 3*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 3*a*b^2)*cosh(
d*x + c) + (5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 4*a^2*b + 3*a*b^2 + 6*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*
x + c)^2)*sinh(d*x + c))*sqrt(-a^2 - a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(
d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d
*x + c)^3 - (3*a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 +
sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + 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)) + 4*(3*(
a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 - (a^4 + 6
*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^5 + 5*
(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^6 + 2*a^5*b + a^4*b^2)*d*sinh(d*x + c)^5 + 2*(a
^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x + c)^3 + 2*(5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^2 + (
a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d)*sinh(d*x + c)^3 + (a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c) + 2*(5*(
a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^3 + 3*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x + c))*sinh(d
*x + c)^2 + (5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^4 + 6*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(
d*x + c)^2 + (a^6 + 2*a^5*b + a^4*b^2)*d)*sinh(d*x + c)), 1/2*((a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^6 + 6*(
a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4 + 2*a^3*b + a^2*b^2)*sinh(d*x + c)^6 + (a^4 + 6*
a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^4 + (a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3 + 15*(a^4 + 2*a^3*b + a^
2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - a^4 - 2*a^3*b - a^2*b^2 + 4*(5*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x +
c)^3 + (a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - (a^4 + 6*a^3*b + 11*a^2*b^2 + 6
*a*b^3)*cosh(d*x + c)^2 + (15*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^4 - a^4 - 6*a^3*b - 11*a^2*b^2 - 6*a*b^3
+ 6*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((4*a^2*b + 3*a*b^2)*cosh(d*x +
c)^5 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)*sinh(d*x + c)^4 + (4*a^2*b + 3*a*b^2)*sinh(d*x + c)^5 + 2*(4*a^2*b
+ 11*a*b^2 + 6*b^3)*cosh(d*x + c)^3 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*
sinh(d*x + c)^3 + 2*(5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c))*sin
h(d*x + c)^2 + (4*a^2*b + 3*a*b^2)*cosh(d*x + c) + (5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 4*a^2*b + 3*a*b^2
+ 6*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3
+ 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 +
3*a + 4*b)*sinh(d*x + c))/sqrt(a^2 + a*b)) - ((4*a^2*b + 3*a*b^2)*cosh(d*x + c)^5 + 5*(4*a^2*b + 3*a*b^2)*cosh
(d*x + c)*sinh(d*x + c)^4 + (4*a^2*b + 3*a*b^2)*sinh(d*x + c)^5 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c)
^3 + 2*(4*a^2*b + 11*a*b^2 + 6*b^3 + 5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 2*(5*(4*a^2*b +
3*a*b^2)*cosh(d*x + c)^3 + 3*(4*a^2*b + 11*a*b^2 + 6*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + (4*a^2*b + 3*a*b^2)
*cosh(d*x + c) + (5*(4*a^2*b + 3*a*b^2)*cosh(d*x + c)^4 + 4*a^2*b + 3*a*b^2 + 6*(4*a^2*b + 11*a*b^2 + 6*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a +
b)) + 2*(3*(a^4 + 2*a^3*b + a^2*b^2)*cosh(d*x + c)^5 + 2*(a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)
^3 - (a^4 + 6*a^3*b + 11*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*
x + c)^5 + 5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)*sinh(d*x + c)^4 + (a^6 + 2*a^5*b + a^4*b^2)*d*sinh(d*x
+ c)^5 + 2*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x + c)^3 + 2*(5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d
*x + c)^2 + (a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d)*sinh(d*x + c)^3 + (a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x
+ c) + 2*(5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^3 + 3*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3*b^3)*d*cosh(d*x
+ c))*sinh(d*x + c)^2 + (5*(a^6 + 2*a^5*b + a^4*b^2)*d*cosh(d*x + c)^4 + 6*(a^6 + 4*a^5*b + 5*a^4*b^2 + 2*a^3
*b^3)*d*cosh(d*x + c)^2 + (a^6 + 2*a^5*b + a^4*b^2)*d)*sinh(d*x + c))]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)**2)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cosh(d*x+c)/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError