\(\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx\) [456]
Optimal result
Integrand size = 25, antiderivative size = 128 \[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=-\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 a^4 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}{12 f \sqrt {a \sin (e+f x)}}
\]
[Out]
-1/3*a*b*(a*sin(f*x+e))^(5/2)/f/(b*sec(f*x+e))^(1/2)-5/6*a^3*b*(a*sin(f*x+e))^(1/2)/f/(b*sec(f*x+e))^(1/2)-5/1
2*a^4*(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)/f/(a*sin(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2663, 2665, 2653, 2720}
\[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\frac {5 a^4 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right ) \sqrt {b \sec (e+f x)}}{12 f \sqrt {a \sin (e+f x)}}-\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}
\]
[In]
Int[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2),x]
[Out]
(-5*a^3*b*Sqrt[a*Sin[e + f*x]])/(6*f*Sqrt[b*Sec[e + f*x]]) - (a*b*(a*Sin[e + f*x])^(5/2))/(3*f*Sqrt[b*Sec[e +
f*x]]) + (5*a^4*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[2*e + 2*f*x]])/(12*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 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 b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {1}{6} \left (5 a^2\right ) \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{3/2} \, dx \\ & = -\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {1}{12} \left (5 a^4\right ) \int \frac {\sqrt {b \sec (e+f x)}}{\sqrt {a \sin (e+f x)}} \, dx \\ & = -\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {1}{12} \left (5 a^4 \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 \\ & = -\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {\left (5 a^4 \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 \sqrt {a \sin (e+f x)}} \\ & = -\frac {5 a^3 b \sqrt {a \sin (e+f x)}}{6 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{5/2}}{3 f \sqrt {b \sec (e+f x)}}+\frac {5 a^4 \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 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 = 15.84 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70
\[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\frac {a^3 b \sqrt {a \sin (e+f x)} \left (2 (-6+\cos (2 (e+f x)))+5 \csc ^2(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},\sec ^2(e+f x)\right ) \left (-\tan ^2(e+f x)\right )^{3/4}\right )}{12 f \sqrt {b \sec (e+f x)}}
\]
[In]
Integrate[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2),x]
[Out]
(a^3*b*Sqrt[a*Sin[e + f*x]]*(2*(-6 + Cos[2*(e + f*x)]) + 5*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*f*Sqrt[b*Sec[e + f*x]])
Maple [C] (warning: unable to verify)
Result contains complex when optimal does not.
Time = 7.71 (sec) , antiderivative size = 1739, normalized size of antiderivative =
13.59
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\(1739\) |
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[In]
int((a*sin(f*x+e))^(7/2)*(b*sec(f*x+e))^(1/2),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))+8*2^(1/2)*cos(f*x+e)^3*sin(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)*E
llipticPi((-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*(-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*(-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)+32*(-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)+cs
c(f*x+e)+1)^(1/2),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))-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))+32*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(cot(f*x+e)-csc(f*x+e)+1)^(1/2)*(c
ot(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))-28*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)*(
-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)+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)*(b*sec(f*x+e))^(1
/2)*a^3*csc(f*x+e)
Fricas [F]
\[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x }
\]
[In]
integrate((a*sin(f*x+e))^(7/2)*(b*sec(f*x+e))^(1/2),x, algorithm="fricas")
[Out]
integral(-(a^3*cos(f*x + e)^2 - a^3)*sqrt(b*sec(f*x + e))*sqrt(a*sin(f*x + e))*sin(f*x + e), x)
Sympy [F(-1)]
Timed out. \[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\text {Timed out}
\]
[In]
integrate((a*sin(f*x+e))**(7/2)*(b*sec(f*x+e))**(1/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x }
\]
[In]
integrate((a*sin(f*x+e))^(7/2)*(b*sec(f*x+e))^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(b*sec(f*x + e))*(a*sin(f*x + e))^(7/2), x)
Giac [F]
\[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {7}{2}} \,d x }
\]
[In]
integrate((a*sin(f*x+e))^(7/2)*(b*sec(f*x+e))^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(b*sec(f*x + e))*(a*sin(f*x + e))^(7/2), x)
Mupad [F(-1)]
Timed out. \[
\int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{7/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x
\]
[In]
int((a*sin(e + f*x))^(7/2)*(b/cos(e + f*x))^(1/2),x)
[Out]
int((a*sin(e + f*x))^(7/2)*(b/cos(e + f*x))^(1/2), x)