3.4.3 \(\int \frac {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx\) [303]

Optimal. Leaf size=226 \[ -\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}} \]

[Out]

-2/5*cos(b*x+a)^(5/2)/b/sin(b*x+a)^(5/2)-1/2*arctan(1-2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2))/b*2^(1/2)+1/2
*arctan(1+2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2))/b*2^(1/2)+1/4*ln(1-2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1
/2)+tan(b*x+a))/b*2^(1/2)-1/4*ln(1+2^(1/2)*sin(b*x+a)^(1/2)/cos(b*x+a)^(1/2)+tan(b*x+a))/b*2^(1/2)+2*cos(b*x+a
)^(1/2)/b/sin(b*x+a)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {2647, 2654, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{\sqrt {2} b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}+\frac {\log \left (\tan (a+b x)-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2} b}-\frac {\log \left (\tan (a+b x)+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+1\right )}{2 \sqrt {2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(7/2)/Sin[a + b*x]^(7/2),x]

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]]]/(Sqrt[2]*b)) + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[a +
b*x]])/Sqrt[Cos[a + b*x]]]/(Sqrt[2]*b) + Log[1 - (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x
]]/(2*Sqrt[2]*b) - Log[1 + (Sqrt[2]*Sqrt[Sin[a + b*x]])/Sqrt[Cos[a + b*x]] + Tan[a + b*x]]/(2*Sqrt[2]*b) - (2*
Cos[a + b*x]^(5/2))/(5*b*Sin[a + b*x]^(5/2)) + (2*Sqrt[Cos[a + b*x]])/(b*Sqrt[Sin[a + b*x]])

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 2647

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a*Cos[e +
f*x])^(m - 1)*((b*Sin[e + f*x])^(n + 1)/(b*f*(n + 1))), x] + Dist[a^2*((m - 1)/(b^2*(n + 1))), Int[(a*Cos[e +
f*x])^(m - 2)*(b*Sin[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && GtQ[m, 1] && LtQ[n, -1] && (Intege
rsQ[2*m, 2*n] || EqQ[m + n, 0])

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 {\cos ^{\frac {7}{2}}(a+b x)}{\sin ^{\frac {7}{2}}(a+b x)} \, dx &=-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}-\int \frac {\cos ^{\frac {3}{2}}(a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx\\ &=-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}+\int \frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}} \, dx\\ &=-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}+\frac {2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}\\ &=-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}-\frac {\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}+\frac {\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{b}\\ &=-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{2 \sqrt {2} b}+\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{2 \sqrt {2} b}\\ &=\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b x)}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}\\ &=-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}\right )}{\sqrt {2} b}+\frac {\log \left (1-\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {\log \left (1+\frac {\sqrt {2} \sqrt {\sin (a+b x)}}{\sqrt {\cos (a+b x)}}+\tan (a+b x)\right )}{2 \sqrt {2} b}-\frac {2 \cos ^{\frac {5}{2}}(a+b x)}{5 b \sin ^{\frac {5}{2}}(a+b x)}+\frac {2 \sqrt {\cos (a+b x)}}{b \sqrt {\sin (a+b 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 = 0.03, size = 57, normalized size = 0.25 \begin {gather*} -\frac {2 \cos ^2(a+b x)^{3/4} \, _2F_1\left (-\frac {5}{4},-\frac {5}{4};-\frac {1}{4};\sin ^2(a+b x)\right )}{5 b \cos ^{\frac {3}{2}}(a+b x) \sin ^{\frac {5}{2}}(a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(7/2)/Sin[a + b*x]^(7/2),x]

[Out]

(-2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, -5/4, -1/4, Sin[a + b*x]^2])/(5*b*Cos[a + b*x]^(3/2)*Sin[a
+ b*x]^(5/2))

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.12, size = 1934, normalized size = 8.56

method	result	size
default	\(\text {Expression too large to display}\)	\(1934\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x,method=_RETURNVERBOSE)

[Out]

8/5/b*cos(b*x+a)^(7/2)*(-1+cos(b*x+a))^4*(5*I*cos(b*x+a)^3*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+c
os(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a
))/sin(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))-5*I*cos(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+c
os(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a
))/sin(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))+5*cos(b*x+a)^3*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+c
os(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a
))/sin(b*x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))+5*cos(b*x+a)^3*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+c
os(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a
))/sin(b*x+a))^(1/2),1/2-1/2*I,1/2*2^(1/2))+5*I*cos(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+c
os(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a
))/sin(b*x+a))^(1/2),1/2-1/2*I,1/2*2^(1/2))+5*I*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+s
in(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+
a))^(1/2),1/2-1/2*I,1/2*2^(1/2))+5*cos(b*x+a)^2*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+s
in(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+
a))^(1/2),1/2+1/2*I,1/2*2^(1/2))+5*cos(b*x+a)^2*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+s
in(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+
a))^(1/2),1/2-1/2*I,1/2*2^(1/2))-5*I*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/
sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1
/2+1/2*I,1/2*2^(1/2))-5*I*cos(b*x+a)^2*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a)
)/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)
,1/2-1/2*I,1/2*2^(1/2))-5*cos(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/
sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1
/2+1/2*I,1/2*2^(1/2))-5*cos(b*x+a)*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/si
n(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1/2
-1/2*I,1/2*2^(1/2))+5*I*cos(b*x+a)^2*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/
sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1
/2+1/2*I,1/2*2^(1/2))-5*I*cos(b*x+a)^3*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a)
)/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)
,1/2-1/2*I,1/2*2^(1/2))-12*cos(b*x+a)^3*2^(1/2)-5*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)
+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*
x+a))^(1/2),1/2+1/2*I,1/2*2^(1/2))-5*((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2)*((-1+cos(b*x+a)+sin(b*x+a))/
sin(b*x+a))^(1/2)*((-1+cos(b*x+a))/sin(b*x+a))^(1/2)*EllipticPi(((1-cos(b*x+a)+sin(b*x+a))/sin(b*x+a))^(1/2),1
/2-1/2*I,1/2*2^(1/2))+10*cos(b*x+a)*2^(1/2))/sin(b*x+a)^(5/2)/(-1+cos(b*x+a)+sin(b*x+a))^4/(-1+cos(b*x+a)-sin(
b*x+a))^4*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(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x, algorithm="maxima")

[Out]

integrate(cos(b*x + a)^(7/2)/sin(b*x + a)^(7/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1670 vs. \(2 (181) = 362\).
time = 27.37, size = 1670, normalized size = 7.39 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x, algorithm="fricas")

[Out]

-1/80*(32*(6*cos(b*x + a)^2 - 5)*sqrt(cos(b*x + a))*sin(b*x + a)^(3/2) - 20*(sqrt(2)*b*cos(b*x + a)^4 - 2*sqrt
(2)*b*cos(b*x + a)^2 + sqrt(2)*b)*(b^(-4))^(1/4)*arctan(((sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt(2)*b*
(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + sqrt(4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin(
b*x + a) - 2*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x +
a))*sqrt(sin(b*x + a)) + 1)*(b^2*sqrt(b^(-4)) + (sqrt(2)*b^3*(b^(-4))^(3/4)*cos(b*x + a) + sqrt(2)*b*(b^(-4))^
(1/4)*sin(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 2*cos(b*x + a)*sin(b*x + a)))/(2*cos(b*x + a)^2 -
1)) - 20*(sqrt(2)*b*cos(b*x + a)^4 - 2*sqrt(2)*b*cos(b*x + a)^2 + sqrt(2)*b)*(b^(-4))^(1/4)*arctan(((sqrt(2)*b
^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a))
- sqrt(4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin(b*x + a) + 2*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt(2)*b*(
b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1)*(b^2*sqrt(b^(-4)) - (sqrt(2)*b^3*(b^(-4
))^(3/4)*cos(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*sin(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 2*cos(b
*x + a)*sin(b*x + a)))/(2*cos(b*x + a)^2 - 1)) - 20*(sqrt(2)*b*cos(b*x + a)^4 - 2*sqrt(2)*b*cos(b*x + a)^2 + s
qrt(2)*b)*(b^(-4))^(1/4)*arctan(1/2*((sqrt(2)*b^3*(b^(-4))^(3/4)*cos(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*sin(b
*x + a))*sqrt(4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin(b*x + a) + 2*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt
(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1)*sqrt(cos(b*x + a))*sqrt(sin(b*x
+ a)) - (sqrt(2)*b^3*(b^(-4))^(3/4)*cos(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*sin(b*x + a))*sqrt(cos(b*x + a))*s
qrt(sin(b*x + a)) - 2*cos(b*x + a)*sin(b*x + a) + 4*(b^2*cos(b*x + a)^4 - b^2*cos(b*x + a)^2)*sqrt(b^(-4)))/((
2*cos(b*x + a)^3 - cos(b*x + a))*sin(b*x + a))) - 20*(sqrt(2)*b*cos(b*x + a)^4 - 2*sqrt(2)*b*cos(b*x + a)^2 +
sqrt(2)*b)*(b^(-4))^(1/4)*arctan(1/2*((sqrt(2)*b^3*(b^(-4))^(3/4)*cos(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*sin(
b*x + a))*sqrt(4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin(b*x + a) - 2*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqr
t(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1)*sqrt(cos(b*x + a))*sqrt(sin(b*x
+ a)) - (sqrt(2)*b^3*(b^(-4))^(3/4)*cos(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*sin(b*x + a))*sqrt(cos(b*x + a))*
sqrt(sin(b*x + a)) + 2*cos(b*x + a)*sin(b*x + a) - 4*(b^2*cos(b*x + a)^4 - b^2*cos(b*x + a)^2)*sqrt(b^(-4)))/(
(2*cos(b*x + a)^3 - cos(b*x + a))*sin(b*x + a))) + 5*(sqrt(2)*b*cos(b*x + a)^4 - 2*sqrt(2)*b*cos(b*x + a)^2 +
sqrt(2)*b)*(b^(-4))^(1/4)*log(4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin(b*x + a) + 2*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin
(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1) - 5*(sqrt(2)*b*c
os(b*x + a)^4 - 2*sqrt(2)*b*cos(b*x + a)^2 + sqrt(2)*b)*(b^(-4))^(1/4)*log(4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin
(b*x + a) - 2*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x +
a))*sqrt(sin(b*x + a)) + 1) + 5*(sqrt(2)*b*cos(b*x + a)^4 - 2*sqrt(2)*b*cos(b*x + a)^2 + sqrt(2)*b)*(b^(-4))^
(1/4)*log(1/4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin(b*x + a) + 1/8*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt
(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt(sin(b*x + a)) + 1/16) - 5*(sqrt(2)*b*cos(b*x + a)^4
- 2*sqrt(2)*b*cos(b*x + a)^2 + sqrt(2)*b)*(b^(-4))^(1/4)*log(1/4*b^2*sqrt(b^(-4))*cos(b*x + a)*sin(b*x + a) -
1/8*(sqrt(2)*b^3*(b^(-4))^(3/4)*sin(b*x + a) + sqrt(2)*b*(b^(-4))^(1/4)*cos(b*x + a))*sqrt(cos(b*x + a))*sqrt
(sin(b*x + a)) + 1/16))/(b*cos(b*x + a)^4 - 2*b*cos(b*x + a)^2 + b)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**(7/2)/sin(b*x+a)**(7/2),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(7/2)/sin(b*x+a)^(7/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 1.88, size = 44, normalized size = 0.19 \begin {gather*} -\frac {2\,{\cos \left (a+b\,x\right )}^{9/2}\,{\left ({\sin \left (a+b\,x\right )}^2\right )}^{5/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {9}{4},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (a+b\,x\right )}^2\right )}{9\,b\,{\sin \left (a+b\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^(7/2)/sin(a + b*x)^(7/2),x)

[Out]

-(2*cos(a + b*x)^(9/2)*(sin(a + b*x)^2)^(5/4)*hypergeom([9/4, 9/4], 13/4, cos(a + b*x)^2))/(9*b*sin(a + b*x)^(
5/2))

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