3.1438 \(\int \frac{\sqrt{d \sin (e+f x)}}{(g \cos (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\)

Optimal. Leaf size=374 \[ -\frac{2 a E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{f g^2 \left (a^2-b^2\right ) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a (d \sin (e+f x))^{3/2}}{d f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}-\frac{2 b \sqrt{d \sin (e+f x)}}{f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.911256, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27, Rules used = {2903, 2838, 2563, 2571, 2572, 2639, 2906, 2905, 490, 1218} \[ -\frac{2 a E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{f g^2 \left (a^2-b^2\right ) \sqrt{\sin (2 e+2 f x)}}+\frac{2 a (d \sin (e+f x))^{3/2}}{d f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}-\frac{2 b \sqrt{d \sin (e+f x)}}{f g \left (a^2-b^2\right ) \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left (-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}}+\frac{2 \sqrt{2} a b d \sqrt{\sin (e+f x)} \Pi \left (\frac{\sqrt{b-a}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{g \cos (e+f x)}}{\sqrt{g} \sqrt{\sin (e+f x)+1}}\right )\right |-1\right )}{f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt{d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2903

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

Rule 2838

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

Rule 2563

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

Rule 2571

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

Rule 2572

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 2639

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

Rule 2906

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 2905

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 490

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), In
t[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 1218

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

Rubi steps

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

Mathematica [C]  time = 23.0631, size = 1280, normalized size = 3.42 \[ \frac{a \sqrt{d \sin (e+f x)} \left (\frac{4 a \left (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)}{b^2-a^2}\right )-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)}{b^2-a^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{\cos (2 (e+f x)) \sqrt{\tan (e+f x)} \left (\sqrt{\tan ^2(e+f x)+1} a+b \tan (e+f x)\right ) \left (24 b \left (b^2-a^2\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\tan ^2(e+f x),\frac{\left (b^2-a^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac{7}{2}}(e+f x)+56 b \left (b^2-3 a^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(e+f x),\frac{\left (b^2-a^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac{3}{2}}(e+f x)+21 a^{3/2} \left (-\frac{4 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac{4 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right ) a^2}{\sqrt [4]{a^2-b^2}}-\frac{2 \sqrt{2} \log \left (-a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\sqrt{a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+\frac{2 \sqrt{2} \log \left (a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right ) a^2}{\sqrt [4]{a^2-b^2}}+4 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right ) a^{3/2}-4 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) a^{3/2}+2 \sqrt{2} \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) a^{3/2}-2 \sqrt{2} \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right ) a^{3/2}+\frac{8 b \tan ^{\frac{3}{2}}(e+f x) \sqrt{a}}{\sqrt{\tan ^2(e+f x)+1}}+\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{2 \sqrt{2} b^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)}}{\sqrt{a}}+1\right )}{\sqrt [4]{a^2-b^2}}+\frac{\sqrt{2} b^2 \log \left (-a+\sqrt{2} \sqrt [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}-\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 [4]{a^2-b^2} \sqrt{\tan (e+f x)} \sqrt{a}+\sqrt{a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}\right )\right )}{84 a^2 b \cos ^{\frac{3}{2}}(e+f x) (a+b \sin (e+f x)) \left (\tan ^2(e+f x)-1\right ) \sqrt{\tan ^2(e+f x)+1} \sqrt{\sin (e+f x)}}\right ) \cos ^{\frac{3}{2}}(e+f x)}{(a-b) (a+b) f (g \cos (e+f x))^{3/2} \sqrt{\sin (e+f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a \sin (e+f x)-b) \cos (e+f x)}{\left (a^2-b^2\right ) f (g \cos (e+f x))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Cos[e + f*x]*Sqrt[d*Sin[e + f*x]]*(-b + a*Sin[e + f*x]))/((a^2 - b^2)*f*(g*Cos[e + f*x])^(3/2)) + (a*Cos[e
+ f*x]^(3/2)*Sqrt[d*Sin[e + f*x]]*((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*AppellF1[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*S
in[e + f*x])) + (Cos[2*(e + f*x)]*Sqrt[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, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2) +
24*b*(-a^2 + b^2)*AppellF1[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, ((-a^2 + b^2)*Tan[e + f*x]^2)/a^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 + S
qrt[2]*Sqrt[Tan[e + f*x]]] - (4*Sqrt[2]*a^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]*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 + (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) +
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)*S
qrt[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 - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*S
qrt[Sin[e + f*x]])

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Maple [B]  time = 0.344, size = 2539, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*2^(1/2)*a/(a+b)/(-a^2+b^2)^(1/2)/(a-b+(-a^2+b^2)^(1/2))/(b+(-a^2+b^2)^(1/2)-a)*(2*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)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a+2*(-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)*EllipticF((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*cos(f*x+e)*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)*(-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/
(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*b+EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b
^2)^(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)*cos(f*x+e)*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/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*b^2-cos(f*x+e)*(-a^2+b^2)^(1/2)*Ellipt
icPi((-(-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))*(-(-1+cos(f*x+e)-si
n(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)*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))*cos(f*x+e)*a*b-cos(f*x+e)*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))*(-(-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)*b^2+2*(-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)*EllipticF((-(-1+cos(f*x+e)-sin(f
*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*a+2*(-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)*EllipticF((-(-1+cos(f*x+e)-sin(f*
x+e))/sin(f*x+e))^(1/2),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-(-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/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*b+EllipticPi((-(-1+cos(f*x+e)-sin(f*x+e))/sin(f*x+e
))^(1/2),a/(a-b+(-a^2+b^2)^(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*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/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*b^2-(-a^2+b^2)^(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))*(-(-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)*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))*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))*(-(-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)*b^2+2*cos(f*x+e)*(-a^2+b^2)^(1/2)*2^(1/2)*a+2*sin(f*x+e)*2^(1/2)*(-a^2+b^2)^(1/2)*b-2*2^(1/2)*(-a^2+b^2)^
(1/2)*a)*(d*sin(f*x+e))^(1/2)*cos(f*x+e)/(g*cos(f*x+e))^(3/2)/sin(f*x+e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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