3.290 \(\int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx\)
Optimal. Leaf size=143 \[ -\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{4 \sqrt {2}}+\frac {3 \log \left (\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{8 \sqrt {2}}-\frac {3 \log \left (\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{8 \sqrt {2}} \]
[Out]
-3/8*arctan(1-2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/2)+3/8*arctan(1+2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2))*2^(1/
2)+3/16*ln(1-2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2)+tan(x))*2^(1/2)-3/16*ln(1+2^(1/2)*sin(x)^(1/2)/cos(x)^(1/2)+tan
(x))*2^(1/2)-1/2*sin(x)^(3/2)*cos(x)^(1/2)
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Rubi [A] time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used
= {2568, 2574, 297, 1162, 617, 204, 1165, 628} \[ -\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{4 \sqrt {2}}+\frac {3 \log \left (\tan (x)-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{8 \sqrt {2}}-\frac {3 \log \left (\tan (x)+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
Int[Sin[x]^(5/2)/Sqrt[Cos[x]],x]
[Out]
(-3*ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]])/(4*Sqrt[2]) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[C
os[x]]])/(4*Sqrt[2]) + (3*Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (3*Log[1 + (Sqr
t[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]])/(8*Sqrt[2]) - (Sqrt[Cos[x]]*Sin[x]^(3/2))/2
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 297
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))
Rule 617
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 628
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 1162
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Rule 1165
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Rule 2568
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]
Rule 2574
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[(k*a*b)/f, Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos
[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]
Rubi steps
\begin {align*} \int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx &=-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x)+\frac {3}{4} \int \frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}} \, dx\\ &=-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x)+\frac {3}{2} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\\ &=-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x)-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )\\ &=-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x)+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{8 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{8 \sqrt {2}}\\ &=\frac {3 \log \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {3 \log \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x)+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}\\ &=-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \log \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {3 \log \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x)\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.27 \[ \frac {2 \sin ^{\frac {7}{2}}(x) \cos ^2(x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {7}{4};\frac {11}{4};\sin ^2(x)\right )}{7 \cos ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[x]^(5/2)/Sqrt[Cos[x]],x]
[Out]
(2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 7/4, 11/4, Sin[x]^2]*Sin[x]^(7/2))/(7*Cos[x]^(3/2))
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fricas [B] time = 0.83, size = 457, normalized size = 3.20 \[ -\frac {1}{2} \, \sqrt {\cos \relax (x)} \sin \relax (x)^{\frac {3}{2}} + \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {2 \, \cos \relax (x)^{3} - 2 \, \cos \relax (x)^{2} \sin \relax (x) + \sqrt {2} \sqrt {2 \, {\left (\sqrt {2} \cos \relax (x) + \sqrt {2} \sin \relax (x)\right )} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + 4 \, \cos \relax (x) \sin \relax (x) + 1} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} - \sqrt {2} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} - 2 \, \cos \relax (x)}{2 \, {\left (\cos \relax (x)^{3} + \cos \relax (x)^{2} \sin \relax (x) - \cos \relax (x)\right )}}\right ) + \frac {3}{16} \, \sqrt {2} \arctan \left (-\frac {2 \, \cos \relax (x)^{3} - 2 \, \cos \relax (x)^{2} \sin \relax (x) - \sqrt {2} \sqrt {-2 \, {\left (\sqrt {2} \cos \relax (x) + \sqrt {2} \sin \relax (x)\right )} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + 4 \, \cos \relax (x) \sin \relax (x) + 1} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + \sqrt {2} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} - 2 \, \cos \relax (x)}{2 \, {\left (\cos \relax (x)^{3} + \cos \relax (x)^{2} \sin \relax (x) - \cos \relax (x)\right )}}\right ) - \frac {3}{16} \, \sqrt {2} \arctan \left (-\frac {\sqrt {-2 \, {\left (\sqrt {2} \cos \relax (x) + \sqrt {2} \sin \relax (x)\right )} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + 4 \, \cos \relax (x) \sin \relax (x) + 1} {\left (\sqrt {2} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + \cos \relax (x) + \sin \relax (x)\right )} + \sqrt {2} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)}}{\cos \relax (x) - \sin \relax (x)}\right ) - \frac {3}{16} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2 \, {\left (\sqrt {2} \cos \relax (x) + \sqrt {2} \sin \relax (x)\right )} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + 4 \, \cos \relax (x) \sin \relax (x) + 1} {\left (\sqrt {2} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} - \cos \relax (x) - \sin \relax (x)\right )} + \sqrt {2} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)}}{\cos \relax (x) - \sin \relax (x)}\right ) - \frac {3}{32} \, \sqrt {2} \log \left (2 \, {\left (\sqrt {2} \cos \relax (x) + \sqrt {2} \sin \relax (x)\right )} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + 4 \, \cos \relax (x) \sin \relax (x) + 1\right ) + \frac {3}{32} \, \sqrt {2} \log \left (-2 \, {\left (\sqrt {2} \cos \relax (x) + \sqrt {2} \sin \relax (x)\right )} \sqrt {\cos \relax (x)} \sqrt {\sin \relax (x)} + 4 \, \cos \relax (x) \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)^(5/2)/cos(x)^(1/2),x, algorithm="fricas")
[Out]
-1/2*sqrt(cos(x))*sin(x)^(3/2) + 3/16*sqrt(2)*arctan(1/2*(2*cos(x)^3 - 2*cos(x)^2*sin(x) + sqrt(2)*sqrt(2*(sqr
t(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) + 4*cos(x)*sin(x) + 1)*sqrt(cos(x))*sqrt(sin(x)) - sqr
t(2)*sqrt(cos(x))*sqrt(sin(x)) - 2*cos(x))/(cos(x)^3 + cos(x)^2*sin(x) - cos(x))) + 3/16*sqrt(2)*arctan(-1/2*(
2*cos(x)^3 - 2*cos(x)^2*sin(x) - sqrt(2)*sqrt(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x)) +
4*cos(x)*sin(x) + 1)*sqrt(cos(x))*sqrt(sin(x)) + sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) - 2*cos(x))/(cos(x)^3 + co
s(x)^2*sin(x) - cos(x))) - 3/16*sqrt(2)*arctan(-(sqrt(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(s
in(x)) + 4*cos(x)*sin(x) + 1)*(sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) + cos(x) + sin(x)) + sqrt(2)*sqrt(cos(x))*sqr
t(sin(x)))/(cos(x) - sin(x))) - 3/16*sqrt(2)*arctan(-(sqrt(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sq
rt(sin(x)) + 4*cos(x)*sin(x) + 1)*(sqrt(2)*sqrt(cos(x))*sqrt(sin(x)) - cos(x) - sin(x)) + sqrt(2)*sqrt(cos(x))
*sqrt(sin(x)))/(cos(x) - sin(x))) - 3/32*sqrt(2)*log(2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin
(x)) + 4*cos(x)*sin(x) + 1) + 3/32*sqrt(2)*log(-2*(sqrt(2)*cos(x) + sqrt(2)*sin(x))*sqrt(cos(x))*sqrt(sin(x))
+ 4*cos(x)*sin(x) + 1)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \relax (x)^{\frac {5}{2}}}{\sqrt {\cos \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)^(5/2)/cos(x)^(1/2),x, algorithm="giac")
[Out]
integrate(sin(x)^(5/2)/sqrt(cos(x)), x)
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maple [C] time = 0.15, size = 2595, normalized size = 18.15 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(x)^(5/2)/cos(x)^(1/2),x)
[Out]
-1/32*sin(x)^(3/2)*(-6*cos(x)^2*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((
-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-4*sin(x)^2*2^(1/2)
-16*cos(x)^3*2^(1/2)+4*cos(x)^4*2^(1/2)+24*cos(x)^2*2^(1/2)-16*cos(x)*2^(1/2)+4*2^(1/2)-3*((1-cos(x)+sin(x))/s
in(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x)
)^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x
))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-6*cos(x)^2*sin(x)^2*((1-co
s(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+
sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+12*cos(x)*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+s
in(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1
/2))+12*cos(x)*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x)
)^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*sin(x)^4*((1-cos(x)+sin(x))/sin
(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^
(1/2),1/2-1/2*I,1/2*2^(1/2))-3*I*sin(x)^4*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*(
(-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*cos(x)^4*((1-
cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x
)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*I*cos(x)^4*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x
))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))
+6*I*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*El
lipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+8*cos(x)*sin(x)^2*2^(1/2)-4*cos(x)^2*sin(x)^2
*2^(1/2)+6*I*cos(x)^2*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))
/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-6*I*cos(x)^2*sin(x)^2*((1-co
s(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+
sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-12*I*cos(x)*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)
+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^
(1/2))+12*I*cos(x)*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/si
n(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+12*((1-cos(x)+sin(x))/sin(x))^(
1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),
1/2-1/2*I,1/2*2^(1/2))*cos(x)+12*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x
))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(x)-3*sin(x)^4*((1-cos(
x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+si
n(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*sin(x)^4*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin
(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-3*cos
(x)^4*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi
(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*cos(x)^4*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(
x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*
2^(1/2))-6*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1
/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-6*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(
1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),
1/2+1/2*I,1/2*2^(1/2))+12*cos(x)^3*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos
(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+12*cos(x)^3*((1-cos(x)+s
in(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x)
)/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-18*cos(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x)
)^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-18*cos(x
)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((
(1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x)
)/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-
3*I*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi((
(1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-6*I*sin(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(
x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*
2^(1/2))-12*I*cos(x)^3*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))
^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))+12*I*cos(x)^3*((1-cos(x)+sin(x))/sin
(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^
(1/2),1/2+1/2*I,1/2*2^(1/2))+18*I*cos(x)^2*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*
((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))-18*I*cos(x)^2*((
1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos
(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))-12*I*cos(x)*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(
x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*EllipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2)
)+12*I*cos(x)*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x))/sin(x))^(1/2)*El
lipticPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2)))/(-1+cos(x))^3/cos(x)^(1/2)*2^(1/2)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \relax (x)^{\frac {5}{2}}}{\sqrt {\cos \relax (x)}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)^(5/2)/cos(x)^(1/2),x, algorithm="maxima")
[Out]
integrate(sin(x)^(5/2)/sqrt(cos(x)), x)
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mupad [B] time = 0.81, size = 25, normalized size = 0.17 \[ -\frac {2\,\sqrt {\cos \relax (x)}\,{\sin \relax (x)}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \relax (x)}^2\right )}{{\left ({\sin \relax (x)}^2\right )}^{7/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(x)^(5/2)/cos(x)^(1/2),x)
[Out]
-(2*cos(x)^(1/2)*sin(x)^(7/2)*hypergeom([-3/4, 1/4], 5/4, cos(x)^2))/(sin(x)^2)^(7/4)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)**(5/2)/cos(x)**(1/2),x)
[Out]
Timed out
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