\(\int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx\) [74]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=i \text {Int}\left (-\frac {i \sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)},x\right ) \]

[Out]

I*Unintegrable(-I*sinh(d*x+c)^3/(a+b*tanh(d*x+c)^3),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]

[In]

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

[Out]

I*Defer[Int][((-I)*Sinh[c + d*x]^3)/(a + b*Tanh[c + d*x]^3), x]

Rubi steps \begin{align*} \text {integral}& = i \int -\frac {i \sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(826\) vs. \(2(33)=66\).

Time = 1.06 (sec) , antiderivative size = 826, normalized size of antiderivative = 35.91 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\frac {-9 a \left (a^2+3 b^2\right ) \cosh (c+d x)+a^3 \cosh (3 (c+d x))-a b^2 \cosh (3 (c+d x))-2 a b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\&,\frac {3 a^2 c+3 a b c+3 b^2 c+3 a^2 d x+3 a b d x+3 b^2 d x+6 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 a^2 c \text {$\#$1}^2-2 b^2 c \text {$\#$1}^2+2 a^2 d x \text {$\#$1}^2-2 b^2 d x \text {$\#$1}^2+4 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-4 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+3 a^2 c \text {$\#$1}^4-3 a b c \text {$\#$1}^4+3 b^2 c \text {$\#$1}^4+3 a^2 d x \text {$\#$1}^4-3 a b d x \text {$\#$1}^4+3 b^2 d x \text {$\#$1}^4+6 a^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-6 a b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+6 b^2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\&\right ]+27 a^2 b \sinh (c+d x)+9 b^3 \sinh (c+d x)-a^2 b \sinh (3 (c+d x))+b^3 \sinh (3 (c+d x))}{12 (a-b)^2 (a+b)^2 d} \]

[In]

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

[Out]

(-9*a*(a^2 + 3*b^2)*Cosh[c + d*x] + a^3*Cosh[3*(c + d*x)] - a*b^2*Cosh[3*(c + d*x)] - 2*a*b*RootSum[a - b + 3*
a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (3*a^2*c + 3*a*b*c + 3*b^2*c + 3*a^2*d*x + 3*a*b
*d*x + 3*b^2*d*x + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]
*#1] + 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 6*b^2
*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*a^2*c*#1^2 - 2*
b^2*c*#1^2 + 2*a^2*d*x*#1^2 - 2*b^2*d*x*#1^2 + 4*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*
x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 4*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#
1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 3*a^2*c*#1^4 - 3*a*b*c*#1^4 + 3*b^2*c*#1^4 + 3*a^2*d*x*#1^4 - 3*a*b*d*x*#1^4
+ 3*b^2*d*x*#1^4 + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]
*#1]*#1^4 - 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1
^4 + 6*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*
#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ] + 27*a^2*b*Sinh[c + d*x] + 9*b^3*Sinh[c + d*x] - a^2*b*
Sinh[3*(c + d*x)] + b^3*Sinh[3*(c + d*x)])/(12*(a - b)^2*(a + b)^2*d)

Maple [N/A] (verified)

Time = 7.25 (sec) , antiderivative size = 289, normalized size of antiderivative = 12.57

method	result	size
derivativedivides	\(\frac {-\frac {a b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 a b \,\textit {\_R}^{3}+2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a b \textit {\_R} +2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +2 b}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8}{\left (16 a -16 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {16}{3 \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (16 a -16 b \right )}-\frac {a +2 b}{2 \left (a -b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\)	\(289\)
default	\(\frac {-\frac {a b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 a b \,\textit {\_R}^{3}+2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a b \textit {\_R} +2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a +2 b}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8}{\left (16 a -16 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {16}{3 \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (16 a -16 b \right )}-\frac {a +2 b}{2 \left (a -b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\)	\(289\)
risch	\(\text {Expression too large to display}\)	\(2825\)

[In]

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

[Out]

1/d*(-1/3*a*b/(a+b)^2/(a-b)^2*sum(((2*a^2+b^2)*_R^4-6*a*b*_R^3+2*(4*a^2+5*b^2)*_R^2-6*a*b*_R+2*a^2+b^2)/(_R^5*
a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))-16/3/(tan
h(1/2*d*x+1/2*c)-1)^3/(16*a+16*b)-8/(16*a+16*b)/(tanh(1/2*d*x+1/2*c)-1)^2-1/2/(a+b)^2*(-a+2*b)/(tanh(1/2*d*x+1
/2*c)-1)-8/(16*a-16*b)/(1+tanh(1/2*d*x+1/2*c))^2+16/3/(1+tanh(1/2*d*x+1/2*c))^3/(16*a-16*b)-1/2*(a+2*b)/(a-b)^
2/(1+tanh(1/2*d*x+1/2*c)))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 4.17 (sec) , antiderivative size = 62017, normalized size of antiderivative = 2696.39 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(sinh(d*x+c)**3/(a+b*tanh(d*x+c)**3),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 533, normalized size of antiderivative = 23.17 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \]

[In]

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

[Out]

1/24*(a^3 + a^2*b - a*b^2 - b^3 + (a^3*e^(6*c) - a^2*b*e^(6*c) - a*b^2*e^(6*c) + b^3*e^(6*c))*e^(6*d*x) - 9*(a
^3*e^(4*c) - 3*a^2*b*e^(4*c) + 3*a*b^2*e^(4*c) - b^3*e^(4*c))*e^(4*d*x) - 9*(a^3*e^(2*c) + 3*a^2*b*e^(2*c) + 3
*a*b^2*e^(2*c) + b^3*e^(2*c))*e^(2*d*x))*e^(-3*d*x)/(a^4*d*e^(3*c) - 2*a^2*b^2*d*e^(3*c) + b^4*d*e^(3*c)) - 1/
8*integrate(16*(3*(a^3*b*e^(5*c) - a^2*b^2*e^(5*c) + a*b^3*e^(5*c))*e^(5*d*x) + 2*(a^3*b*e^(3*c) - a*b^3*e^(3*
c))*e^(3*d*x) + 3*(a^3*b*e^c + a^2*b^2*e^c + a*b^3*e^c)*e^(d*x))/(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4
- b^5 + (a^5*e^(6*c) + a^4*b*e^(6*c) - 2*a^3*b^2*e^(6*c) - 2*a^2*b^3*e^(6*c) + a*b^4*e^(6*c) + b^5*e^(6*c))*e^
(6*d*x) + 3*(a^5*e^(4*c) - a^4*b*e^(4*c) - 2*a^3*b^2*e^(4*c) + 2*a^2*b^3*e^(4*c) + a*b^4*e^(4*c) - b^5*e^(4*c)
)*e^(4*d*x) + 3*(a^5*e^(2*c) + a^4*b*e^(2*c) - 2*a^3*b^2*e^(2*c) - 2*a^2*b^3*e^(2*c) + a*b^4*e^(2*c) + b^5*e^(
2*c))*e^(2*d*x)), x)

Giac [N/A]

Not integrable

Time = 3.88 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{3}}{b \tanh \left (d x + c\right )^{3} + a} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx=\text {Hanged} \]

[In]

int(sinh(c + d*x)^3/(a + b*tanh(c + d*x)^3),x)

[Out]

\text{Hanged}