3.3.90 \(\int \frac {\sin ^{\frac {5}{2}}(x)}{\sqrt {\cos (x)}} \, dx\) [290]
Optimal. Leaf size=143 \[ -\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \log \left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {3 \log \left (1+\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+\tan (x)\right )}{8 \sqrt {2}}-\frac {1}{2} \sqrt {\cos (x)} \sin ^{\frac {3}{2}}(x) \]
[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 = 0.07, 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 = {2648, 2654,
303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )}{4 \sqrt {2}}+\frac {3 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {\sin (x)}}{\sqrt {\cos (x)}}+1\right )}{4 \sqrt {2}}-\frac {1}{2} \sin ^{\frac {3}{2}}(x) \sqrt {\cos (x)}+\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}} \end {gather*}
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 210
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 303
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 631
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 642
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 1176
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 1179
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 2648
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 2654
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} \text {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} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\frac {3}{4} \text {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} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\frac {3}{8} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (x)}}{\sqrt {\cos (x)}}\right )+\frac {3 \text {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 \text {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 \text {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 \text {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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 38, normalized size = 0.27 \begin {gather*} \frac {2 \cos ^2(x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {7}{4};\frac {11}{4};\sin ^2(x)\right ) \sin ^{\frac {7}{2}}(x)}{7 \cos ^{\frac {3}{2}}(x)} \end {gather*}
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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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.12, size = 2595, normalized size = 18.15
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(x)^(5/2)/cos(x)^(1/2),x,method=_RETURNVERBOSE)
[Out]
1/32*sin(x)^(3/2)*(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))+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*((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))*sin(x)^2*cos(x)^2+6*((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))*sin(x)^2*cos(x)^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))*sin(x)^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)*Ellipt
icPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))*sin(x)^2*cos(x)-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))*sin(x)^4+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))*sin(x)^4-6*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))*sin(x)^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))*cos(x)
^4+6*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)*EllipticP
i(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2+1/2*I,1/2*2^(1/2))*sin(x)^2+3*I*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+c
os(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)^4+12*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))*cos(x)^3-12*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))*cos(x)^3-18*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))*cos(x)^2+18*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))*cos(x)^2+12*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))*cos(x)-12*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))*cos(x)+4*2^(1/2)*sin(x)^2*cos(x)^2-8*2^(
1/2)*sin(x)^2*cos(x)+4*2^(1/2)*sin(x)^2-4*2^(1/2)*cos(x)^4+16*2^(1/2)*cos(x)^3-24*2^(1/2)*cos(x)^2+16*2^(1/2)*
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)*Ellipti
cPi(((1-cos(x)+sin(x))/sin(x))^(1/2),1/2-1/2*I,1/2*2^(1/2))*sin(x)^4+3*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+c
os(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))*sin(x)^4+6*((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))*sin(x)^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))*cos(x)^4+6*((1-cos(x)+sin(x))/sin(x))^(1/2)*((-1+cos(x)+sin(x))/sin(x))^(1/2)*((-1+c
os(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))*sin(x)^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))*cos(x)^4-12*((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)^
3-12*((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+18*((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)^2+18*((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...
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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(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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 457 vs.
\(2 (97) = 194\).
time = 0.66, size = 457, normalized size = 3.20 \begin {gather*} -\frac {1}{2} \, \sqrt {\cos \left (x\right )} \sin \left (x\right )^{\frac {3}{2}} + \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {2 \, \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} \sin \left (x\right ) + \sqrt {2} \sqrt {2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - 2 \, \cos \left (x\right )}{2 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) + \frac {3}{16} \, \sqrt {2} \arctan \left (-\frac {2 \, \cos \left (x\right )^{3} - 2 \, \cos \left (x\right )^{2} \sin \left (x\right ) - \sqrt {2} \sqrt {-2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - 2 \, \cos \left (x\right )}{2 \, {\left (\cos \left (x\right )^{3} + \cos \left (x\right )^{2} \sin \left (x\right ) - \cos \left (x\right )\right )}}\right ) - \frac {3}{16} \, \sqrt {2} \arctan \left (-\frac {\sqrt {-2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} {\left (\sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + \cos \left (x\right ) + \sin \left (x\right )\right )} + \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac {3}{16} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1} {\left (\sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )\right )} + \sqrt {2} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )}}{\cos \left (x\right ) - \sin \left (x\right )}\right ) - \frac {3}{32} \, \sqrt {2} \log \left (2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) + \frac {3}{32} \, \sqrt {2} \log \left (-2 \, {\left (\sqrt {2} \cos \left (x\right ) + \sqrt {2} \sin \left (x\right )\right )} \sqrt {\cos \left (x\right )} \sqrt {\sin \left (x\right )} + 4 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \end {gather*}
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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(x)**(5/2)/cos(x)**(1/2),x)
[Out]
Exception raised: SystemError >> excessive stack use: stack is 3004 deep
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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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Mupad [B]
time = 0.81, size = 25, normalized size = 0.17 \begin {gather*} -\frac {2\,\sqrt {\cos \left (x\right )}\,{\sin \left (x\right )}^{7/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{4};\ \frac {5}{4};\ {\cos \left (x\right )}^2\right )}{{\left ({\sin \left (x\right )}^2\right )}^{7/4}} \end {gather*}
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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