\(\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx\) [450]

Optimal result

Integrand size = 25, antiderivative size = 449 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=-\frac {21 a^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}} \]

[Out]

-7/16*a^3*b*(a*sin(f*x+e))^(3/2)/f/(b*sec(f*x+e))^(1/2)-1/4*a*b*(a*sin(f*x+e))^(7/2)/f/(b*sec(f*x+e))^(1/2)-21
/64*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)/f*2^(1/2)/b^(1/2)+21/64*a^(9/2)*arctan(1+2^(1/2)*b^(1/2)*(a*sin(f*x+e))^(1/2)/a^(1/2)/(b*c
os(f*x+e))^(1/2))*(b*cos(f*x+e))^(1/2)*(b*sec(f*x+e))^(1/2)/f*2^(1/2)/b^(1/2)+21/128*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)/f*2^(1/2)/b^(1/2)-21/128*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)/f*2^(1/2)/b^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2663, 2665, 2654, 303, 1176, 631, 210, 1179, 642} \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{64 \sqrt {2} \sqrt {b} f}-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}} \]

[In]

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

[Out]

(-21*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]])/(32*Sqrt[2]*Sqrt[b]*f) + (21*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]])/(32*Sqrt[2]*Sqrt[b]*f) + (21*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]])/(64*Sqrt[2]*Sqrt[b]*f) - (21*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]])/(64*
Sqrt[2]*Sqrt[b]*f) - (7*a^3*b*(a*Sin[e + f*x])^(3/2))/(16*f*Sqrt[b*Sec[e + f*x]]) - (a*b*(a*Sin[e + f*x])^(7/2
))/(4*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 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*} \text {integral}& = -\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {1}{8} \left (7 a^2\right ) \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {1}{32} \left (21 a^4\right ) \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \, dx \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {1}{32} \left (21 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 \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {\left (21 a^5 b \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 )}{16 f} \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}-\frac {\left (21 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 )}{32 f}+\frac {\left (21 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 )}{32 f} \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {\left (21 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 )}{64 b f}+\frac {\left (21 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 )}{64 b f}+\frac {\left (21 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 )}{64 \sqrt {2} \sqrt {b} f}+\frac {\left (21 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 )}{64 \sqrt {2} \sqrt {b} f} \\ & = \frac {21 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)}}{64 \sqrt {2} \sqrt {b} f}-\frac {21 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)}}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {\left (21 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 )}{32 \sqrt {2} \sqrt {b} f}-\frac {\left (21 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 )}{32 \sqrt {2} \sqrt {b} f} \\ & = -\frac {21 a^{9/2} \arctan \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)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \arctan \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)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 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)}}{64 \sqrt {2} \sqrt {b} f}-\frac {21 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)}}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.10 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.38 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\frac {a^4 \cot (e+f x) \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \left (4 (-9+2 \cos (2 (e+f x))) \sin ^2(e+f x)+21 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}-21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}\right )}{64 f} \]

[In]

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

[Out]

(a^4*Cot[e + f*x]*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]]*(4*(-9 + 2*Cos[2*(e + f*x)])*Sin[e + f*x]^2 + 21*S
qrt[2]*ArcTan[(-1 + Sqrt[Tan[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)))/(64*f)

Maple [A] (verified)

Time = 3.49 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {\sqrt {2}\, \left (16 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+16 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-44 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {2}\, \sin \left (f x +e \right ) \cos \left (f x +e \right )-44 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+21 \ln \left (-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cot \left (f x +e \right )-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \csc \left (f x +e \right )+2-2 \cot \left (f x +e \right )\right )-21 \ln \left (2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cot \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \csc \left (f x +e \right )+2-2 \cot \left (f x +e \right )\right )+42 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )+42 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )\right ) \sqrt {b \sec \left (f x +e \right )}\, \sqrt {a \sin \left (f x +e \right )}\, a^{4} \cos \left (f x +e \right )}{128 f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\)	\(537\)

[In]

int((a*sin(f*x+e))^(9/2)*(b*sec(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/128/f*2^(1/2)*(16*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*2^(1/2)*cos(f*x+e)^3*sin(f*x+e)+16*(-sin(f
*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*2^(1/2)*cos(f*x+e)^2*sin(f*x+e)-44*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e
)+1)^2)^(1/2)*2^(1/2)*sin(f*x+e)*cos(f*x+e)-44*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x
+e)+21*ln(-2*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cot(f*x+e)-2*2^(1/2)*(-sin(f*x+e)*cos(f*x
+e)/(cos(f*x+e)+1)^2)^(1/2)*csc(f*x+e)+2-2*cot(f*x+e))-21*ln(2*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^
2)^(1/2)*cot(f*x+e)+2*2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*csc(f*x+e)+2-2*cot(f*x+e))+42*ar
ctan((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)-cos(f*x+e)+1)/(cos(f*x+e)-1))+42*arct
an((2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(cos(f*x+e)-1)))*(b*sec(f
*x+e))^(1/2)*(a*sin(f*x+e))^(1/2)*a^4*cos(f*x+e)/(cos(f*x+e)+1)/(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2
)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.55 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\text {Too large to display} \]

[In]

integrate((a*sin(f*x+e))^(9/2)*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/256*(16*(4*a^4*cos(f*x + e)^3 - 11*a^4*cos(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))*sin(f*x + e)
- 21*(-a^18*b^2/f^4)^(1/4)*f*log(9261/2*a^14*b^2*cos(f*x + e)*sin(f*x + e) + 9261/2*((-a^18*b^2/f^4)^(1/4)*a^9
*b*f*cos(f*x + e)*sin(f*x + e) - (-a^18*b^2/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x
+ e)) - 9261/4*sqrt(-a^18*b^2/f^4)*(2*a^5*b*f^2*cos(f*x + e)^2 - a^5*b*f^2)) + 21*(-a^18*b^2/f^4)^(1/4)*f*log
(9261/2*a^14*b^2*cos(f*x + e)*sin(f*x + e) - 9261/2*((-a^18*b^2/f^4)^(1/4)*a^9*b*f*cos(f*x + e)*sin(f*x + e) -
(-a^18*b^2/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)) - 9261/4*sqrt(-a^18*b^2/f
^4)*(2*a^5*b*f^2*cos(f*x + e)^2 - a^5*b*f^2)) + 21*I*(-a^18*b^2/f^4)^(1/4)*f*log(9261/2*a^14*b^2*cos(f*x + e)*
sin(f*x + e) - 9261/2*(I*(-a^18*b^2/f^4)^(1/4)*a^9*b*f*cos(f*x + e)*sin(f*x + e) + I*(-a^18*b^2/f^4)^(3/4)*f^3
*cos(f*x + e)^2)*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e)) + 9261/4*sqrt(-a^18*b^2/f^4)*(2*a^5*b*f^2*cos(f*x +
e)^2 - a^5*b*f^2)) - 21*I*(-a^18*b^2/f^4)^(1/4)*f*log(9261/2*a^14*b^2*cos(f*x + e)*sin(f*x + e) - 9261/2*(-I*
(-a^18*b^2/f^4)^(1/4)*a^9*b*f*cos(f*x + e)*sin(f*x + e) - I*(-a^18*b^2/f^4)^(3/4)*f^3*cos(f*x + e)^2)*sqrt(a*s
in(f*x + e))*sqrt(b/cos(f*x + e)) + 9261/4*sqrt(-a^18*b^2/f^4)*(2*a^5*b*f^2*cos(f*x + e)^2 - a^5*b*f^2)) - 21*
(-a^18*b^2/f^4)^(1/4)*f*log(9261*a^14*b^2 + 18522*((-a^18*b^2/f^4)^(1/4)*a^9*b*f*cos(f*x + e)^2 - (-a^18*b^2/f
^4)^(3/4)*f^3*cos(f*x + e)*sin(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))) + 21*(-a^18*b^2/f^4)^(1/4)
*f*log(9261*a^14*b^2 - 18522*((-a^18*b^2/f^4)^(1/4)*a^9*b*f*cos(f*x + e)^2 - (-a^18*b^2/f^4)^(3/4)*f^3*cos(f*x
+ e)*sin(f*x + e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))) + 21*I*(-a^18*b^2/f^4)^(1/4)*f*log(9261*a^14*b^
2 - 18522*(I*(-a^18*b^2/f^4)^(1/4)*a^9*b*f*cos(f*x + e)^2 + I*(-a^18*b^2/f^4)^(3/4)*f^3*cos(f*x + e)*sin(f*x +
e))*sqrt(a*sin(f*x + e))*sqrt(b/cos(f*x + e))) - 21*I*(-a^18*b^2/f^4)^(1/4)*f*log(9261*a^14*b^2 - 18522*(-I*(
-a^18*b^2/f^4)^(1/4)*a^9*b*f*cos(f*x + e)^2 - I*(-a^18*b^2/f^4)^(3/4)*f^3*cos(f*x + e)*sin(f*x + e))*sqrt(a*si
n(f*x + e))*sqrt(b/cos(f*x + e))))/f

Sympy [F(-1)]

Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\text {Timed out} \]

[In]

integrate((a*sin(f*x+e))**(9/2)*(b*sec(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate((a*sin(f*x+e))^(9/2)*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}} \,d x } \]

[In]

integrate((a*sin(f*x+e))^(9/2)*(b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]

[In]

int((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(1/2),x)

[Out]

int((a*sin(e + f*x))^(9/2)*(b/cos(e + f*x))^(1/2), x)