3.3.86 \(\int \frac {\text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [286]

Optimal. Leaf size=91 \[ \frac {3 \text {ArcTan}(\sinh (c+d x))}{8 a d}-\frac {i}{8 d (a-i a \sinh (c+d x))}+\frac {i a}{8 d (a+i a \sinh (c+d x))^2}+\frac {i}{4 d (a+i a \sinh (c+d x))} \]

[Out]

3/8*arctan(sinh(d*x+c))/a/d-1/8*I/d/(a-I*a*sinh(d*x+c))+1/8*I*a/d/(a+I*a*sinh(d*x+c))^2+1/4*I/d/(a+I*a*sinh(d*
x+c))

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2746, 46, 212} \begin {gather*} \frac {3 \text {ArcTan}(\sinh (c+d x))}{8 a d}+\frac {i a}{8 d (a+i a \sinh (c+d x))^2}-\frac {i}{8 d (a-i a \sinh (c+d x))}+\frac {i}{4 d (a+i a \sinh (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sech[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]

[Out]

(3*ArcTan[Sinh[c + d*x]])/(8*a*d) - (I/8)/(d*(a - I*a*Sinh[c + d*x])) + ((I/8)*a)/(d*(a + I*a*Sinh[c + d*x])^2
) + (I/4)/(d*(a + I*a*Sinh[c + d*x]))

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rubi steps

\begin {align*} \int \frac {\text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx &=-\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^3} \, dx,x,i a \sinh (c+d x)\right )}{d}\\ &=-\frac {\left (i a^3\right ) \text {Subst}\left (\int \left (\frac {1}{8 a^3 (a-x)^2}+\frac {1}{4 a^2 (a+x)^3}+\frac {1}{4 a^3 (a+x)^2}+\frac {3}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,i a \sinh (c+d x)\right )}{d}\\ &=-\frac {i}{8 d (a-i a \sinh (c+d x))}+\frac {i a}{8 d (a+i a \sinh (c+d x))^2}+\frac {i}{4 d (a+i a \sinh (c+d x))}-\frac {(3 i) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,i a \sinh (c+d x)\right )}{8 d}\\ &=\frac {3 \tan ^{-1}(\sinh (c+d x))}{8 a d}-\frac {i}{8 d (a-i a \sinh (c+d x))}+\frac {i a}{8 d (a+i a \sinh (c+d x))^2}+\frac {i}{4 d (a+i a \sinh (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 101, normalized size = 1.11 \begin {gather*} \frac {\text {sech}^2(c+d x) \left (2-3 i \text {ArcTan}(\sinh (c+d x))+3 (-i+\text {ArcTan}(\sinh (c+d x))) \sinh (c+d x)+(3-3 i \text {ArcTan}(\sinh (c+d x))) \sinh ^2(c+d x)+3 \text {ArcTan}(\sinh (c+d x)) \sinh ^3(c+d x)\right )}{8 a d (-i+\sinh (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sech[c + d*x]^3/(a + I*a*Sinh[c + d*x]),x]

[Out]

(Sech[c + d*x]^2*(2 - (3*I)*ArcTan[Sinh[c + d*x]] + 3*(-I + ArcTan[Sinh[c + d*x]])*Sinh[c + d*x] + (3 - (3*I)*
ArcTan[Sinh[c + d*x]])*Sinh[c + d*x]^2 + 3*ArcTan[Sinh[c + d*x]]*Sinh[c + d*x]^3))/(8*a*d*(-I + Sinh[c + d*x])
)

________________________________________________________________________________________

Maple [A]
time = 1.55, size = 141, normalized size = 1.55

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{3 d x +3 c}+6 i {\mathrm e}^{2 d x +2 c}+3 \,{\mathrm e}^{d x +c}+3 \,{\mathrm e}^{5 d x +5 c}-6 i {\mathrm e}^{4 d x +4 c}}{4 \left ({\mathrm e}^{d x +c}+i\right )^{2} \left ({\mathrm e}^{d x +c}-i\right )^{4} d a}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 a d}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 a d}\)	\(125\)
derivativedivides	\(\frac {\frac {i}{4 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}{8}-\frac {1}{4 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}+\frac {i}{2 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 i \ln \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {3 i}{2 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {1}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a d}\)	\(141\)
default	\(\frac {\frac {i}{4 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {3 i \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}{8}-\frac {1}{4 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}+\frac {i}{2 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {3 i \ln \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {3 i}{2 \left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {1}{\left (-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {1}{-i+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a d}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

2/d/a*(1/8*I/(tanh(1/2*d*x+1/2*c)+I)^2+3/16*I*ln(tanh(1/2*d*x+1/2*c)+I)-1/8/(tanh(1/2*d*x+1/2*c)+I)+1/4*I/(-I+
tanh(1/2*d*x+1/2*c))^4-3/16*I*ln(-I+tanh(1/2*d*x+1/2*c))-3/4*I/(-I+tanh(1/2*d*x+1/2*c))^2+1/2/(-I+tanh(1/2*d*x
+1/2*c))^3-1/2/(-I+tanh(1/2*d*x+1/2*c)))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (71) = 142\).
time = 0.43, size = 287, normalized size = 3.15 \begin {gather*} -\frac {3 \, {\left (-i \, e^{\left (6 \, d x + 6 \, c\right )} - 2 \, e^{\left (5 \, d x + 5 \, c\right )} - i \, e^{\left (4 \, d x + 4 \, c\right )} - 4 \, e^{\left (3 \, d x + 3 \, c\right )} + i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + i\right )} \log \left (e^{\left (d x + c\right )} + i\right ) + 3 \, {\left (i \, e^{\left (6 \, d x + 6 \, c\right )} + 2 \, e^{\left (5 \, d x + 5 \, c\right )} + i \, e^{\left (4 \, d x + 4 \, c\right )} + 4 \, e^{\left (3 \, d x + 3 \, c\right )} - i \, e^{\left (2 \, d x + 2 \, c\right )} + 2 \, e^{\left (d x + c\right )} - i\right )} \log \left (e^{\left (d x + c\right )} - i\right ) - 6 \, e^{\left (5 \, d x + 5 \, c\right )} + 12 i \, e^{\left (4 \, d x + 4 \, c\right )} - 4 \, e^{\left (3 \, d x + 3 \, c\right )} - 12 i \, e^{\left (2 \, d x + 2 \, c\right )} - 6 \, e^{\left (d x + c\right )}}{8 \, {\left (a d e^{\left (6 \, d x + 6 \, c\right )} - 2 i \, a d e^{\left (5 \, d x + 5 \, c\right )} + a d e^{\left (4 \, d x + 4 \, c\right )} - 4 i \, a d e^{\left (3 \, d x + 3 \, c\right )} - a d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d e^{\left (d x + c\right )} - a d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/8*(3*(-I*e^(6*d*x + 6*c) - 2*e^(5*d*x + 5*c) - I*e^(4*d*x + 4*c) - 4*e^(3*d*x + 3*c) + I*e^(2*d*x + 2*c) -
2*e^(d*x + c) + I)*log(e^(d*x + c) + I) + 3*(I*e^(6*d*x + 6*c) + 2*e^(5*d*x + 5*c) + I*e^(4*d*x + 4*c) + 4*e^(
3*d*x + 3*c) - I*e^(2*d*x + 2*c) + 2*e^(d*x + c) - I)*log(e^(d*x + c) - I) - 6*e^(5*d*x + 5*c) + 12*I*e^(4*d*x
+ 4*c) - 4*e^(3*d*x + 3*c) - 12*I*e^(2*d*x + 2*c) - 6*e^(d*x + c))/(a*d*e^(6*d*x + 6*c) - 2*I*a*d*e^(5*d*x +
5*c) + a*d*e^(4*d*x + 4*c) - 4*I*a*d*e^(3*d*x + 3*c) - a*d*e^(2*d*x + 2*c) - 2*I*a*d*e^(d*x + c) - a*d)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)

[Out]

-I*Integral(sech(c + d*x)**3/(sinh(c + d*x) - I), x)/a

________________________________________________________________________________________

Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (71) = 142\).
time = 0.43, size = 177, normalized size = 1.95 \begin {gather*} -\frac {-\frac {6 i \, \log \left (-i \, e^{\left (d x + c\right )} + i \, e^{\left (-d x - c\right )} + 2\right )}{a} + \frac {6 i \, \log \left (-i \, e^{\left (d x + c\right )} + i \, e^{\left (-d x - c\right )} - 2\right )}{a} - \frac {2 \, {\left (3 \, e^{\left (d x + c\right )} - 3 \, e^{\left (-d x - c\right )} + 10 i\right )}}{a {\left (i \, e^{\left (d x + c\right )} - i \, e^{\left (-d x - c\right )} - 2\right )}} + \frac {-9 i \, {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 52 \, e^{\left (d x + c\right )} + 52 \, e^{\left (-d x - c\right )} + 84 i}{a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} - 2 i\right )}^{2}}}{32 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

-1/32*(-6*I*log(-I*e^(d*x + c) + I*e^(-d*x - c) + 2)/a + 6*I*log(-I*e^(d*x + c) + I*e^(-d*x - c) - 2)/a - 2*(3
*e^(d*x + c) - 3*e^(-d*x - c) + 10*I)/(a*(I*e^(d*x + c) - I*e^(-d*x - c) - 2)) + (-9*I*(e^(d*x + c) - e^(-d*x
- c))^2 - 52*e^(d*x + c) + 52*e^(-d*x - c) + 84*I)/(a*(e^(d*x + c) - e^(-d*x - c) - 2*I)^2))/d

________________________________________________________________________________________

Mupad [B]
time = 0.99, size = 137, normalized size = 1.51 \begin {gather*} \frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2\,d^2}}{a\,d}\right )}{4\,\sqrt {a^2\,d^2}}+\frac {1}{2\,a\,d\,\left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )}+\frac {1}{4\,a\,d\,\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )}-\frac {1{}\mathrm {i}}{4\,a\,d\,{\left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )}^2}-\frac {1{}\mathrm {i}}{a\,d\,{\left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^3}+\frac {1{}\mathrm {i}}{2\,a\,d\,{\left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)

[Out]

(3*atan((exp(d*x)*exp(c)*(a^2*d^2)^(1/2))/(a*d)))/(4*(a^2*d^2)^(1/2)) + 1/(2*a*d*(exp(c + d*x) - 1i)) + 1/(4*a
*d*(exp(c + d*x) + 1i)) - 1i/(4*a*d*(exp(c + d*x) + 1i)^2) - 1i/(a*d*(exp(c + d*x)*1i + 1)^3) + 1i/(2*a*d*(exp
(c + d*x)*1i + 1)^4)

________________________________________________________________________________________