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3.15.12 \(\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\) [1412]

Optimal. Leaf size=320 \[ -\frac {2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b \sqrt {g} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a+b} \sqrt {a+b} d f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b \sqrt {g} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a+b} \sqrt {a+b} d f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a d^2 f \sqrt {\sin (2 e+2 f x)}} \]

[Out]

-2*(g*cos(f*x+e))^(3/2)/a/d/f/g/(d*sin(f*x+e))^(1/2)-2*b*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e)
)^(1/2),-(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*g^(1/2)*sin(f*x+e)^(1/2)/a/d/f/(-a+b)^(1/2)/(a+b)^(1/2)/(d*sin(f*
x+e))^(1/2)+2*b*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/
2)*g^(1/2)*sin(f*x+e)^(1/2)/a/d/f/(-a+b)^(1/2)/(a+b)^(1/2)/(d*sin(f*x+e))^(1/2)+2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/
sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/a/d^2/f/sin(2
*f*x+2*e)^(1/2)

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Rubi [A]
time = 0.48, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {2989, 2650, 2652, 2719, 2985, 2984, 504, 1232} \begin {gather*} -\frac {2 \sqrt {2} b \sqrt {g} \sqrt {\sin (e+f x)} \Pi \left (-\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a d f \sqrt {b-a} \sqrt {a+b} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b \sqrt {g} \sqrt {\sin (e+f x)} \Pi \left (\frac {\sqrt {b-a}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right )\right |-1\right )}{a d f \sqrt {b-a} \sqrt {a+b} \sqrt {d \sin (e+f x)}}-\frac {2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a d^2 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt {d \sin (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]

[Out]

(-2*(g*Cos[e + f*x])^(3/2))/(a*d*f*g*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b*Sqrt[g]*EllipticPi[-(Sqrt[-a + b]/Sq
rt[a + b]), ArcSin[Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a
+ b]*Sqrt[a + b]*d*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b*Sqrt[g]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[
Sqrt[g*Cos[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*Sqrt[-a + b]*Sqrt[a + b]*d*
f*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*d^2*f*
Sqrt[Sin[2*e + 2*f*x]])

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
 = Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 2650

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

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
 x]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2984

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]))
, x_Symbol] :> Dist[-4*Sqrt[2]*(g/f), Subst[Int[x^2/(((a + b)*g^2 + (a - b)*x^4)*Sqrt[1 - x^4/g^2]), x], x, Sq
rt[g*Cos[e + f*x]]/Sqrt[1 + Sin[e + f*x]]], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2985

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x
_)])), x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]], Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[e + f*x]]
*(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2989

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[1/a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] - Dist[b/(a*d), Int[(g*Cos[
e + f*x])^p*((d*Sin[e + f*x])^(n + 1)/(a + b*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2
 - b^2, 0] && IntegersQ[2*n, 2*p] && LtQ[-1, p, 1] && LtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx &=\frac {\int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{a}-\frac {b \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a d}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt {d \sin (e+f x)}}-\frac {2 \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{a d^2}-\frac {\left (b \sqrt {\sin (e+f x)}\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a d \sqrt {d \sin (e+f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt {d \sin (e+f x)}}+\frac {\left (4 \sqrt {2} b g \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a d f \sqrt {d \sin (e+f x)}}-\frac {\left (2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{a d^2 \sqrt {\sin (2 e+2 f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a d^2 f \sqrt {\sin (2 e+2 f x)}}+\frac {\left (2 \sqrt {2} b g \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g-\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a \sqrt {-a+b} d f \sqrt {d \sin (e+f x)}}-\frac {\left (2 \sqrt {2} b g \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g+\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a \sqrt {-a+b} d f \sqrt {d \sin (e+f x)}}\\ &=-\frac {2 (g \cos (e+f x))^{3/2}}{a d f g \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b \sqrt {g} \Pi \left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a+b} \sqrt {a+b} d f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b \sqrt {g} \Pi \left (\frac {\sqrt {-a+b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right )\right |-1\right ) \sqrt {\sin (e+f x)}}{a \sqrt {-a+b} \sqrt {a+b} d f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a d^2 f \sqrt {\sin (2 e+2 f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 52.82, size = 1619, normalized size = 5.06 \begin {gather*} -\frac {2 \cos (e+f x) \sqrt {g \cos (e+f x)} \sin (e+f x)}{a f (d \sin (e+f x))^{3/2}}+\frac {\sqrt {g \cos (e+f x)} \sin ^{\frac {3}{2}}(e+f x) \left (\frac {4 a \left (-b F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+a F_1\left (\frac {3}{4};\frac {1}{4},1;\frac {7}{4};\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^{\frac {3}{2}}(e+f x) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \sin ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right ) \left (1-\cos ^2(e+f x)\right )^{3/4} (a+b \sin (e+f x))}-\frac {b \sqrt {\tan (e+f x)} \left (\frac {3 \sqrt {2} a^{3/2} \left (-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )+2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )-\log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )+\log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )\right )}{\sqrt [4]{a^2-b^2}}-8 b F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\tan ^2(e+f x),\frac {\left (-a^2+b^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac {3}{2}}(e+f x)\right ) \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right )}{6 a^2 \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (1+\tan ^2(e+f x)\right )^{3/2}}+\frac {\cos (2 (e+f x)) \sqrt {\tan (e+f x)} \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right ) \left (56 b \left (-3 a^2+b^2\right ) F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)+24 b \left (-a^2+b^2\right ) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)+21 a^{3/2} \left (4 \sqrt {2} a^{3/2} \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-4 \sqrt {2} a^{3/2} \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-\frac {4 \sqrt {2} a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} b^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {4 \sqrt {2} a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} b^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+2 \sqrt {2} a^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-2 \sqrt {2} a^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-\frac {2 \sqrt {2} a^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {\sqrt {2} b^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} a^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}-\frac {\sqrt {2} b^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {8 \sqrt {a} b \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {1+\tan ^2(e+f x)}}\right )\right )}{84 a^2 b \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (-1+\tan ^2(e+f x)\right ) \sqrt {1+\tan ^2(e+f x)}}\right )}{a f \sqrt {\cos (e+f x)} (d \sin (e+f x))^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[g*Cos[e + f*x]]/((d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]

[Out]

(-2*Cos[e + f*x]*Sqrt[g*Cos[e + f*x]]*Sin[e + f*x])/(a*f*(d*Sin[e + f*x])^(3/2)) + (Sqrt[g*Cos[e + f*x]]*Sin[e
 + f*x]^(3/2)*((4*a*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*A
ppellF1[3/4, 1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1
 - Cos[e + f*x]^2])*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) - (b*S
qrt[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] +
 2*ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(
1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e
 + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2,
 ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(6*a^2*
Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + (Cos[2*(e + f*x)]*Sqr
t[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4, 1/2, 1, 7/4,
-Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 1
1/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]*a^(3/2)*ArcTan
[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^2*A
rcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*b^2*ArcTan[1
 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*ArcTan[1 + (Sqr
t[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[1 + (Sqrt[2]*(a
^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[e
 + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - (2*Sqrt[2]*a
^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^
(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f
*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a
^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e
 + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/Sqrt[1 + Tan[e
+ f*x]^2])))/(84*a^2*b*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e + f*x]^2)*Sqrt[1
 + Tan[e + f*x]^2])))/(a*f*Sqrt[Cos[e + f*x]]*(d*Sin[e + f*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2496\) vs. \(2(291)=582\).
time = 0.36, size = 2497, normalized size = 7.80

method result size
default \(\text {Expression too large to display}\) \(2497\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/f*(a-b)*(2*cos(f*x+e)*(-a^2+b^2)^(1/2)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))
*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))
/sin(f*x+e))^(1/2)*a-2*cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/s
in(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1
/2*2^(1/2))*(-a^2+b^2)^(1/2)*b+cos(f*x+e)*(-a^2+b^2)^(1/2)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1
+cos(f*x+e)+sin(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),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b+cos(f*x+e)*(-a^2+b^2)^(1/2)*(-(-1+cos(f*x+e)
-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b-4*cos(f*x
+e)*(-a^2+b^2)^(1/2)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1
/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticE((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a
-cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(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),a/(-b+(-a^2+b^2)^(1/2)
+a),1/2*2^(1/2))*a*b-cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(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),a/
(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b^2+cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*
x+e)+sin(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a*b+cos(f*x+e)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e
))^(1/2)*((-1+cos(f*x+e)+sin(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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^2+2*(-a^2+b^2)^(1/2)*EllipticF(
(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-
1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*a-2*(-(-1+cos(f*x+e)-sin(f*x+e))
/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticF(
(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b+(-a^2+b^2)^(1/2)*(-(-1+cos(f*x+
e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(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),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b+(-a^2+b
^2)^(1/2)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+co
s(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),-a/(b+(-a^2+b^2)^(1/2)-a
),1/2*2^(1/2))*b-4*(-a^2+b^2)^(1/2)*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))
/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1/2)*EllipticE((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)
,1/2*2^(1/2))*a-(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(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),a/(-b+(-a^2+b^2)^(
1/2)+a),1/2*2^(1/2))*a*b-EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1
/2*2^(1/2))*b^2*(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*(
(-1+cos(f*x+e))/sin(f*x+e))^(1/2)+(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/s
in(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),
-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a*b+(-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(
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),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^2+2*2^(1/2)*cos(f*x+e)*(-a^2+b^2)^(1/2)*a)*sin(f*x+e)*(g*cos
(f*x+e))^(1/2)/(d*sin(f*x+e))^(3/2)/cos(f*x+e)*2^(1/2)/a/(-a^2+b^2)^(1/2)/(-b+(-a^2+b^2)^(1/2)+a)/(b+(-a^2+b^2
)^(1/2)-a)

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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((g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate(sqrt(g*cos(f*x + e))/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(3/2)), x)

________________________________________________________________________________________

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((g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {g \cos {\left (e + f x \right )}}}{\left (d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (a + b \sin {\left (e + f x \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))**(1/2)/(d*sin(f*x+e))**(3/2)/(a+b*sin(f*x+e)),x)

[Out]

Integral(sqrt(g*cos(e + f*x))/((d*sin(e + f*x))**(3/2)*(a + b*sin(e + f*x))), x)

________________________________________________________________________________________

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((g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(3/2)/(a+b*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(sqrt(g*cos(f*x + e))/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(3/2)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(e + f*x))^(1/2)/((d*sin(e + f*x))^(3/2)*(a + b*sin(e + f*x))),x)

[Out]

int((g*cos(e + f*x))^(1/2)/((d*sin(e + f*x))^(3/2)*(a + b*sin(e + f*x))), x)

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