3.1.34 \(\int \frac {\sinh (c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [34]
Optimal. Leaf size=40 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {b} d} \]
[Out]
arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))/d/(a-b)^(1/2)/b^(1/2)
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Rubi [A]
time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3265, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {b} d \sqrt {a-b}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^2),x]
[Out]
ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]]/(Sqrt[a - b]*Sqrt[b]*d)
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 3265
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-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]
Rubi steps
\begin {align*} \int \frac {\sinh (c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {b} d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 91, normalized size = 2.28 \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Sinh[c + d*x]/(a + b*Sinh[c + d*x]^2),x]
[Out]
(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/
Sqrt[a - b]])/(Sqrt[a - b]*Sqrt[b]*d)
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Maple [A]
time = 0.82, size = 51, normalized size = 1.28
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{d \sqrt {a b -b^{2}}}\) |
\(51\) |
| default |
\(\frac {\arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{d \sqrt {a b -b^{2}}}\) |
\(51\) |
| risch |
\(-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d}\) |
\(102\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(d*x+c)/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)
[Out]
1/d/(a*b-b^2)^(1/2)*arctan(1/4*(2*a*tanh(1/2*d*x+1/2*c)^2-2*a+4*b)/(a*b-b^2)^(1/2))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
[Out]
integrate(sinh(d*x + c)/(b*sinh(d*x + c)^2 + a), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 166 vs.
\(2 (32) = 64\).
time = 0.44, size = 502, normalized size = 12.55 \begin {gather*} \left [-\frac {\sqrt {-a b + b^{2}} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right )\right )} \sqrt {-a b + b^{2}} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right )}{2 \, {\left (a b - b^{2}\right )} d}, \frac {\sqrt {a b - b^{2}} \arctan \left (-\frac {b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (d x + c\right )}{2 \, \sqrt {a b - b^{2}}}\right ) - \sqrt {a b - b^{2}} \arctan \left (-\frac {\sqrt {a b - b^{2}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, {\left (a - b\right )}}\right )}{{\left (a b - b^{2}\right )} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
[Out]
[-1/2*sqrt(-a*b + b^2)*log((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(2*a
- 3*b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a -
3*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*b + b^2) + b)/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x
+ c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + 2*a - b)*sin
h(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b))/((a*b - b^2)*d), (sqrt(a*b
- b^2)*arctan(-1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3 + (4*a - 3*b)*co
sh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - 3*b)*sinh(d*x + c))/sqrt(a*b - b^2)) - sqrt(a*b - b^2)*arctan(-1/2*
sqrt(a*b - b^2)*(cosh(d*x + c) + sinh(d*x + c))/(a - b)))/((a*b - b^2)*d)]
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 367433 vs.
\(2 (32) = 64\).
time = 117.79, size = 367433, normalized size = 9185.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)**2),x)
[Out]
Piecewise((zoo*x/sinh(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (log(tanh(c/2 + d*x/2))/(b*d), Eq(a, 0)), (-2/(b*d*
tanh(c/2 + d*x/2)**2 + b*d), Eq(a, b)), (cosh(c + d*x)/(a*d), Eq(b, 0)), (x*sinh(c)/(a + b*sinh(c)**2), Eq(d,
0)), (-74*a**37*b*log(-sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**38*b*d - 5478*a**37*
b**2*d + 148*a**37*b*d*sqrt(-a*b + b**2) + 2502532*a**36*b**3*d - 135124*a**36*b**2*d*sqrt(-a*b + b**2) - 4569
61248*a**35*b**4*d + 36983424*a**35*b**3*d*sqrt(-a*b + b**2) + 44602414272*a**34*b**5*d - 4809599808*a**34*b**
4*d*sqrt(-a*b + b**2) - 2698911348224*a**33*b**6*d + 363524561920*a**33*b**5*d*sqrt(-a*b + b**2) + 11077603634
0736*a**32*b**7*d - 17891931206656*a**32*b**6*d*sqrt(-a*b + b**2) - 3275718126403584*a**31*b**8*d + 6168082597
80608*a**31*b**7*d*sqrt(-a*b + b**2) + 72854727629602816*a**30*b**9*d - 15666762815766528*a**30*b**8*d*sqrt(-a
*b + b**2) - 1258467596957384704*a**29*b**10*d + 304230303833522176*a**29*b**9*d*sqrt(-a*b + b**2) + 173061408
91880620032*a**28*b**11*d - 4645206174395269120*a**28*b**10*d*sqrt(-a*b + b**2) - 193199008739227598848*a**27*
b**12*d + 57001938802859573248*a**27*b**11*d*sqrt(-a*b + b**2) + 1778515685235870400512*a**26*b**13*d - 572029
907419376123904*a**26*b**12*d*sqrt(-a*b + b**2) - 13673782930644613988352*a**25*b**14*d + 47610201097691253964
80*a**25*b**13*d*sqrt(-a*b + b**2) + 88722183139577965838336*a**24*b**15*d - 33244276082712682430464*a**24*b**
14*d*sqrt(-a*b + b**2) - 490030319626953299066880*a**23*b**16*d + 196589323247525507891200*a**23*b**15*d*sqrt(
-a*b + b**2) + 2320264659880999460536320*a**22*b**17*d - 992185245208510642257920*a**22*b**16*d*sqrt(-a*b + b*
*2) - 9473237423050314565550080*a**21*b**18*d + 4301031135604201236725760*a**21*b**17*d*sqrt(-a*b + b**2) + 33
508135815008970573086720*a**20*b**19*d - 16096759227611109665013760*a**20*b**18*d*sqrt(-a*b + b**2) - 10306600
1007281297455841280*a**19*b**20*d + 52224655042483407940485120*a**19*b**19*d*sqrt(-a*b + b**2) + 2764606599494
10743463444480*a**18*b**21*d - 147353756231010598099353600*a**18*b**20*d*sqrt(-a*b + b**2) - 64801707291816239
5858206720*a**17*b**22*d + 362406002426292494841937920*a**17*b**21*d*sqrt(-a*b + b**2) + 132896736702920448874
1191680*a**16*b**23*d - 778070309276565523251855360*a**16*b**22*d*sqrt(-a*b + b**2) - 238570463131696852308590
5920*a**15*b**24*d + 1459214048234341782006005760*a**15*b**23*d*sqrt(-a*b + b**2) + 37475296552465658698058956
80*a**14*b**25*d - 2390143076492438985108357120*a**14*b**24*d*sqrt(-a*b + b**2) - 5144960127422757831513735168
*a**13*b**26*d + 3415726795297830225523507200*a**13*b**25*d*sqrt(-a*b + b**2) + 6160343368926179873261617152*a
**12*b**27*d - 4250434573627220170723295232*a**12*b**26*d*sqrt(-a*b + b**2) - 6412553052386662194178162688*a**
11*b**28*d + 4591390200361817020475375616*a**11*b**27*d*sqrt(-a*b + b**2) + 5777443964131114181336236032*a**10
*b**29*d - 4286847917414518496444809216*a**10*b**28*d*sqrt(-a*b + b**2) - 4478521132381256779508482048*a**9*b*
*30*d + 3439316943952955273874767872*a**9*b**29*d*sqrt(-a*b + b**2) + 2963524367539447964941418496*a**8*b**31*
d - 2352691778556737395705774080*a**8*b**30*d*sqrt(-a*b + b**2) - 1656691644402709111026745344*a**7*b**32*d +
1358109508879153777946394624*a**7*b**31*d*sqrt(-a*b + b**2) + 771639221199942160467623936*a**6*b**33*d - 65251
7637350019802209452032*a**6*b**32*d*sqrt(-a*b + b**2) - 293857629218373137558667264*a**5*b**34*d + 25608013620
1453725194125312*a**5*b**33*d*sqrt(-a*b + b**2) + 89102865177381478141526016*a**4*b**35*d - 799453111233816980
13691904*a**4*b**34*d*sqrt(-a*b + b**2) - 20683374899158687330336768*a**3*b**36*d + 19090166507000540776366080
*a**3*b**35*d*sqrt(-a*b + b**2) + 3450869307356993239908352*a**2*b**37*d - 3273780564249381544394752*a**2*b**3
6*d*sqrt(-a*b + b**2) - 368344585663832326668288*a*b**38*d + 358899852698093036240896*a*b**37*d*sqrt(-a*b + b*
*2) + 18889465931478580854784*b**39*d - 18889465931478580854784*b**38*d*sqrt(-a*b + b**2)) - 74*a**37*b*log(sq
rt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**38*b*d - 5478*a**37*b**2*d + 148*a**37*b*d*sq
rt(-a*b + b**2) + 2502532*a**36*b**3*d - 135124*a**36*b**2*d*sqrt(-a*b + b**2) - 456961248*a**35*b**4*d + 3698
3424*a**35*b**3*d*sqrt(-a*b + b**2) + 44602414272*a**34*b**5*d - 4809599808*a**34*b**4*d*sqrt(-a*b + b**2) - 2
698911348224*a**33*b**6*d + 363524561920*a**33*b**5*d*sqrt(-a*b + b**2) + 110776036340736*a**32*b**7*d - 17891
931206656*a**32*b**6*d*sqrt(-a*b + b**2) - 3275718126403584*a**31*b**8*d + 616808259780608*a**31*b**7*d*sqrt(-
a*b + b**2) + 72854727629602816*a**30*b**9*d - 15666762815766528*a**30*b**8*d*sqrt(-a*b + b**2) - 125846759695
7384704*a**29*b**10*d + 304230303833522176*a**29*b**9*d*sqrt(-a*b + b**2) + 17306140891880620032*a**28*b**11*d
- 4645206174395269120*a**28*b**10*d*sqrt(-a*b + b**2) - 193199008739227598848*a**27*b**12*d + 570019388028595
73248*a**27*b**11*d*sqrt(-a*b + b**2) + 1778515685235870400512*a**26*b**13*d - 572029907419376123904*a**26*b**
12*d*sqrt(-a*b + b**2) - 1367378293064461398835...
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sinh(d*x+c)/(a+b*sinh(d*x+c)^2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done
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Mupad [B]
time = 0.99, size = 116, normalized size = 2.90 \begin {gather*} \frac {\ln \left (-\frac {4\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^2\,\left (a-b\right )}-\frac {8\,a\,{\mathrm {e}}^{c+d\,x}}{{\left (-b\right )}^{5/2}\,\sqrt {a-b}}\right )-\ln \left (\frac {8\,a\,{\mathrm {e}}^{c+d\,x}}{{\left (-b\right )}^{5/2}\,\sqrt {a-b}}-\frac {4\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{b^2\,\left (a-b\right )}\right )}{2\,\sqrt {-b}\,d\,\sqrt {a-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sinh(c + d*x)/(a + b*sinh(c + d*x)^2),x)
[Out]
(log(- (4*(a + a*exp(2*c + 2*d*x)))/(b^2*(a - b)) - (8*a*exp(c + d*x))/((-b)^(5/2)*(a - b)^(1/2))) - log((8*a*
exp(c + d*x))/((-b)^(5/2)*(a - b)^(1/2)) - (4*(a + a*exp(2*c + 2*d*x)))/(b^2*(a - b))))/(2*(-b)^(1/2)*d*(a - b
)^(1/2))
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