3.5.72 \(\int \frac {(a \sin (e+f x))^{9/2}}{(b \sec (e+f x))^{3/2}} \, dx\) [472]

Optimal. Leaf size=490 \[ -\frac {7 a^{9/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{128 \sqrt {2} b^{5/2} f}+\frac {7 a^{9/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{128 \sqrt {2} b^{5/2} f}+\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}} \]

[Out]

-7/192*a^3*(a*sin(f*x+e))^(3/2)/b/f/(b*sec(f*x+e))^(1/2)-1/48*a*(a*sin(f*x+e))^(7/2)/b/f/(b*sec(f*x+e))^(1/2)+
1/6*(a*sin(f*x+e))^(11/2)/a/b/f/(b*sec(f*x+e))^(1/2)-7/256*a^(9/2)*arctan(1-2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/
2)/a^(1/2)/(b*cos(f*x+e))^(1/2))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)/b^(5/2)/f*2^(1/2)+7/256*a^(9/2)*arc
tan(1+2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*cos(f*x+e))^(1/2))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(
1/2)/b^(5/2)/f*2^(1/2)+7/512*a^(9/2)*ln(a^(1/2)-2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/(b*cos(f*x+e))^(1/2)+a^(1
/2)*tan(f*x+e))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)/b^(5/2)/f*2^(1/2)-7/512*a^(9/2)*ln(a^(1/2)+2^(1/2)*b
^(1/2)*(a*sin(f*x+e))^(1/2)/(b*cos(f*x+e))^(1/2)+a^(1/2)*tan(f*x+e))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)
/b^(5/2)/f*2^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 490, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2662, 2663, 2665, 2654, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{128 \sqrt {2} b^{5/2} f}+\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{128 \sqrt {2} b^{5/2} f}+\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Sin[e + f*x])^(9/2)/(b*Sec[e + f*x])^(3/2),x]

[Out]

(-7*a^(9/2)*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f
*x]]*Sqrt[b*Sec[e + f*x]])/(128*Sqrt[2]*b^(5/2)*f) + (7*a^(9/2)*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x
]])/(Sqrt[a]*Sqrt[b*Cos[e + f*x]])]*Sqrt[b*Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(128*Sqrt[2]*b^(5/2)*f) + (7*a^
(9/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] - (Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]
*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(256*Sqrt[2]*b^(5/2)*f) - (7*a^(9/2)*Sqrt[b*Cos[e + f*x]]*Log[Sqrt[a] + (
Sqrt[2]*Sqrt[b]*Sqrt[a*Sin[e + f*x]])/Sqrt[b*Cos[e + f*x]] + Sqrt[a]*Tan[e + f*x]]*Sqrt[b*Sec[e + f*x]])/(256*
Sqrt[2]*b^(5/2)*f) - (7*a^3*(a*Sin[e + f*x])^(3/2))/(192*b*f*Sqrt[b*Sec[e + f*x]]) - (a*(a*Sin[e + f*x])^(7/2)
)/(48*b*f*Sqrt[b*Sec[e + f*x]]) + (a*Sin[e + f*x])^(11/2)/(6*a*b*f*Sqrt[b*Sec[e + f*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 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]

Rule 2662

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a*Sin[e + f*
x])^(m + 1)*((b*Sec[e + f*x])^(n + 1)/(a*b*f*(m - n))), x] - Dist[(n + 1)/(b^2*(m - n)), Int[(a*Sin[e + f*x])^
m*(b*Sec[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && NeQ[m - n, 0] && IntegersQ[2*
m, 2*n]

Rule 2663

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*b*(a*Sin
[e + f*x])^(m - 1)*((b*Sec[e + f*x])^(n - 1)/(f*(m - n))), x] + Dist[a^2*((m - 1)/(m - n)), Int[(a*Sin[e + f*x
])^(m - 2)*(b*Sec[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m - n, 0] && IntegersQ[
2*m, 2*n]

Rule 2665

Int[((b_.)*sec[(e_.) + (f_.)*(x_)])^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[(b*Cos[e + f*
x])^n*(b*Sec[e + f*x])^n, Int[(a*Sin[e + f*x])^m/(b*Cos[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&
IntegerQ[m - 1/2] && IntegerQ[n - 1/2]

Rubi steps

\begin {align*} \int \frac {(a \sin (e+f x))^{9/2}}{(b \sec (e+f x))^{3/2}} \, dx &=\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}+\frac {\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx}{12 b^2}\\ &=-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (7 a^2\right ) \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx}{96 b^2}\\ &=-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (7 a^4\right ) \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \, dx}{128 b^2}\\ &=-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (7 a^4 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}} \, dx}{128 b^2}\\ &=-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (7 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{64 b f}\\ &=-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}-\frac {\left (7 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {a-b x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{128 b^2 f}+\frac {\left (7 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {a+b x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{128 b^2 f}\\ &=-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (7 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{b}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}+x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{256 b^3 f}+\frac {\left (7 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}+x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{256 b^3 f}+\frac {\left (7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {b}}+2 x}{-\frac {a}{b}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}-x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{256 \sqrt {2} b^{5/2} f}+\frac {\left (7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {b}}-2 x}{-\frac {a}{b}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}-x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{256 \sqrt {2} b^{5/2} f}\\ &=\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{128 \sqrt {2} b^{5/2} f}-\frac {\left (7 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{128 \sqrt {2} b^{5/2} f}\\ &=-\frac {7 a^{9/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{128 \sqrt {2} b^{5/2} f}+\frac {7 a^{9/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{128 \sqrt {2} b^{5/2} f}+\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{256 \sqrt {2} b^{5/2} f}-\frac {7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt {b \sec (e+f x)}}-\frac {a (a \sin (e+f x))^{7/2}}{48 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{11/2}}{6 a b f \sqrt {b \sec (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 0.98, size = 176, normalized size = 0.36 \begin {gather*} -\frac {a^5 \left (4 (-3+14 \cos (2 (e+f x))-4 \cos (4 (e+f x))) \sin ^2(e+f x)-21 \sqrt {2} \tan ^{-1}\left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}+21 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}\right )}{768 b f \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Sin[e + f*x])^(9/2)/(b*Sec[e + f*x])^(3/2),x]

[Out]

-1/768*(a^5*(4*(-3 + 14*Cos[2*(e + f*x)] - 4*Cos[4*(e + f*x)])*Sin[e + f*x]^2 - 21*Sqrt[2]*ArcTan[(-1 + Sqrt[T
an[e + f*x]^2])/(Sqrt[2]*(Tan[e + f*x]^2)^(1/4))]*(Tan[e + f*x]^2)^(1/4) + 21*Sqrt[2]*ArcTanh[(Sqrt[2]*(Tan[e
+ f*x]^2)^(1/4))/(1 + Sqrt[Tan[e + f*x]^2])]*(Tan[e + f*x]^2)^(1/4)))/(b*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e +
f*x]])

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.29, size = 572, normalized size = 1.17

method	result	size
default	\(-\frac {\left (-64 \sqrt {2}\, \left (\cos ^{6}\left (f x +e \right )\right )+64 \sqrt {2}\, \left (\cos ^{5}\left (f x +e \right )\right )+21 i \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-21 i \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+120 \left (\cos ^{4}\left (f x +e \right )\right ) \sqrt {2}-120 \left (\cos ^{3}\left (f x +e \right )\right ) \sqrt {2}-21 \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-42 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}+42 \cos \left (f x +e \right ) \sqrt {2}\right ) \left (a \sin \left (f x +e \right )\right )^{\frac {9}{2}} \sqrt {2}}{768 f \left (-1+\cos \left (f x +e \right )\right ) \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}\)	\(572\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(f*x+e))^(9/2)/(b*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/768/f*(-64*2^(1/2)*cos(f*x+e)^6+64*2^(1/2)*cos(f*x+e)^5+21*I*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*(
(sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi(((1-cos(f*x+e)+sin(f
*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))-21*I*((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((sin(f*x+e)
-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(
f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))+120*cos(f*x+e)^4*2^(1/2)-120*cos(f*x+e)^3*2^(1/2)-21*((1-cos(f*x+e)+sin(f
*x+e))/sin(f*x+e))^(1/2)*((sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*Ellip
ticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))-21*((1-cos(f*x+e)+sin(f*x+e))/sin(f*
x+e))^(1/2)*((sin(f*x+e)-1+cos(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticPi(((1-cos
(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))-42*cos(f*x+e)^2*2^(1/2)+42*cos(f*x+e)*2^(1/2))*(a
*sin(f*x+e))^(9/2)/(-1+cos(f*x+e))/sin(f*x+e)^3/cos(f*x+e)^2/(b/cos(f*x+e))^(3/2)*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((a*sin(f*x+e))^(9/2)/(b*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e))^(9/2)/(b*sec(f*x + e))^(3/2), x)

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Fricas [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((a*sin(f*x+e))^(9/2)/(b*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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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((a*sin(f*x+e))**(9/2)/(b*sec(f*x+e))**(3/2),x)

[Out]

Timed out

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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((a*sin(f*x+e))^(9/2)/(b*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e))^(9/2)/(b*sec(f*x + e))^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sin(e + f*x))^(9/2)/(b/cos(e + f*x))^(3/2),x)

[Out]

int((a*sin(e + f*x))^(9/2)/(b/cos(e + f*x))^(3/2), x)

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