\(\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx\) [478]
Optimal result
Integrand size = 25, antiderivative size = 135 \[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=-\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {a^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{12 b^2 f \sqrt {a \sin (e+f x)}}
\]
[Out]
1/3*(a*sin(f*x+e))^(5/2)/a/b/f/(b*sec(f*x+e))^(1/2)-1/6*a*(a*sin(f*x+e))^(1/2)/b/f/(b*sec(f*x+e))^(1/2)-1/12*a
^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^(1/2))*(b*sec(f*x+e))^(1/2)*sin
(2*f*x+2*e)^(1/2)/b^2/f/(a*sin(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 0.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2662, 2663, 2665, 2653, 2720}
\[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=\frac {a^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {b \sec (e+f x)}}{12 b^2 f \sqrt {a \sin (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}-\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}
\]
[In]
Int[(a*Sin[e + f*x])^(3/2)/(b*Sec[e + f*x])^(3/2),x]
[Out]
-1/6*(a*Sqrt[a*Sin[e + f*x]])/(b*f*Sqrt[b*Sec[e + f*x]]) + (a*Sin[e + f*x])^(5/2)/(3*a*b*f*Sqrt[b*Sec[e + f*x]
]) + (a^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(12*b^2*f*Sqrt[a*Sin[e + f
*x]])
Rule 2653
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[Sqrt[Sin[2*
e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b*Cos[e + f*x]]), Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b,
e, f}, x]
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]
Rule 2720
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]
Rubi steps \begin{align*}
\text {integral}& = \frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2} \, dx}{6 b^2} \\ & = -\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {a^2 \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx}{12 b^2} \\ & = -\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (a^2 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {b \cos (e+f x)} \sqrt {a \sin (e+f x)}} \, dx}{12 b^2} \\ & = -\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {\left (a^2 \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}\right ) \int \frac {1}{\sqrt {\sin (2 e+2 f x)}} \, dx}{12 b^2 \sqrt {a \sin (e+f x)}} \\ & = -\frac {a \sqrt {a \sin (e+f x)}}{6 b f \sqrt {b \sec (e+f x)}}+\frac {(a \sin (e+f x))^{5/2}}{3 a b f \sqrt {b \sec (e+f x)}}+\frac {a^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{12 b^2 f \sqrt {a \sin (e+f x)}} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.48 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.64
\[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=\frac {a \sqrt {a \sin (e+f x)} \left (-2 \cos (2 (e+f x))+\csc ^2(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{12 b f \sqrt {b \sec (e+f x)}}
\]
[In]
Integrate[(a*Sin[e + f*x])^(3/2)/(b*Sec[e + f*x])^(3/2),x]
[Out]
(a*Sqrt[a*Sin[e + f*x]]*(-2*Cos[2*(e + f*x)] + Csc[e + f*x]^2*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]
*(-Tan[e + f*x]^2)^(3/4)))/(12*b*f*Sqrt[b*Sec[e + f*x]])
Maple [C] (warning: unable to verify)
Result contains complex when optimal does not.
Time = 1.97 (sec) , antiderivative size = 1746, normalized size of antiderivative =
12.93
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\(1746\) |
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[In]
int((a*sin(f*x+e))^(3/2)/(b*sec(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
[Out]
-1/48/f*2^(1/2)*(-6*I*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))
^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(f*x+e)-6*I*(-cot(f*x+e)+csc(f*x+
e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1
)^(1/2),1/2-1/2*I,1/2*2^(1/2))+6*I*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e
)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-6*(-cot(f*x+e)+csc(f*x+
e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1
)^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(f*x+e)+8*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(
cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*cos(f*x+e)+6*I*(-cot(f*x+
e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+c
sc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(f*x+e)-6*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)
+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos
(f*x+e)+8*2^(1/2)*cos(f*x+e)^3*sin(f*x+e)-6*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(
cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+8*(-cot(f*x+e)
+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(
f*x+e)+1)^(1/2),1/2*2^(1/2))-6*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-cs
c(f*x+e))^(1/2)*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))-4*2^(1/2)*cos(f*x+e)*sin(f*
x+e)-3*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))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x
+e)+3*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))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e
)+6*arctan((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))*(
-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)+6*arctan((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))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x
+e)-3*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))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+3*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))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)+6*arctan((2^(1/2)*(-s
in(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))*(-sin(f*x+e)*cos(f*x+e)/
(cos(f*x+e)+1)^2)^(1/2)+6*arctan((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))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2))*(a*sin(f*x+e))^(1/2)*a/(b*sec(f*x+e))^(1
/2)/b*sec(f*x+e)*csc(f*x+e)
Fricas [F]
\[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x }
\]
[In]
integrate((a*sin(f*x+e))^(3/2)/(b*sec(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
integral(sqrt(b*sec(f*x + e))*sqrt(a*sin(f*x + e))*a*sin(f*x + e)/(b^2*sec(f*x + e)^2), x)
Sympy [F(-1)]
Timed out. \[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=\text {Timed out}
\]
[In]
integrate((a*sin(f*x+e))**(3/2)/(b*sec(f*x+e))**(3/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x }
\]
[In]
integrate((a*sin(f*x+e))^(3/2)/(b*sec(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
integrate((a*sin(f*x + e))^(3/2)/(b*sec(f*x + e))^(3/2), x)
Giac [F]
\[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int { \frac {\left (a \sin \left (f x + e\right )\right )^{\frac {3}{2}}}{\left (b \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x }
\]
[In]
integrate((a*sin(f*x+e))^(3/2)/(b*sec(f*x+e))^(3/2),x, algorithm="giac")
[Out]
integrate((a*sin(f*x + e))^(3/2)/(b*sec(f*x + e))^(3/2), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x
\]
[In]
int((a*sin(e + f*x))^(3/2)/(b/cos(e + f*x))^(3/2),x)
[Out]
int((a*sin(e + f*x))^(3/2)/(b/cos(e + f*x))^(3/2), x)