3.590 \(\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2} \, dx\)
Optimal. Leaf size=429 \[ \frac{b \sqrt{e \cos (c+d x)}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d \sqrt{e} \left (b^2-a^2\right )^{7/4}}+\frac{3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d \sqrt{e} \left (b^2-a^2\right )^{7/4}}-\frac{\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \left (a^2-b^2\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 d \left (a^2-b^2\right ) \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 d \left (a^2-b^2\right ) \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}} \]
[Out]
(3*a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt
[e]) + (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4
)*d*Sqrt[e]) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/((a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sq
rt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-
a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c +
d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]]
)/((a^2 - b^2)*d*e*(a + b*Sin[c + d*x]))
________________________________________________________________________________________
Rubi [A] time = 0.898004, antiderivative size = 429, normalized size of antiderivative =
1., number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.44, Rules used = {2694, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}
\[ \frac{b \sqrt{e \cos (c+d x)}}{d e \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac{3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d \sqrt{e} \left (b^2-a^2\right )^{7/4}}+\frac{3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 d \sqrt{e} \left (b^2-a^2\right )^{7/4}}-\frac{\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \left (a^2-b^2\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 d \left (a^2-b^2\right ) \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 d \left (a^2-b^2\right ) \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2),x]
[Out]
(3*a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt
[e]) + (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4
)*d*Sqrt[e]) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/((a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sq
rt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-
a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c +
d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]]
)/((a^2 - b^2)*d*e*(a + b*Sin[c + d*x]))
Rule 2694
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m +
1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; F
reeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Rule 2867
Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]))/((a_) + (b_.)*sin[(e_.) + (
f_.)*(x_)]), x_Symbol] :> Dist[d/b, Int[(g*Cos[e + f*x])^p, x], x] + Dist[(b*c - a*d)/b, Int[(g*Cos[e + f*x])^
p/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Rule 2642
Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]
Rule 2641
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]
Rule 2702
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[
-a^2 + b^2, 2]}, -Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Dist[(b*g)/f, Sub
st[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e
+ f*x]]*(q - b*Cos[e + f*x])), x], x])] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]
Rule 2807
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] && !GtQ[c + d, 0]
Rule 2805
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 212
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2} \, dx &=\frac{b \sqrt{e \cos (c+d x)}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac{\int \frac{-a+\frac{1}{2} b \sin (c+d x)}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{-a^2+b^2}\\ &=\frac{b \sqrt{e \cos (c+d x)}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}-\frac{\int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{2 \left (a^2-b^2\right )}+\frac{(3 a) \int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac{b \sqrt{e \cos (c+d x)}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac{\left (3 a^2\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{4 \left (-a^2+b^2\right )^{3/2}}+\frac{\left (3 a^2\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{4 \left (-a^2+b^2\right )^{3/2}}+\frac{(3 a b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \cos (c+d x)\right )}{2 \left (a^2-b^2\right ) d}-\frac{\sqrt{\cos (c+d x)} \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 \left (a^2-b^2\right ) \sqrt{e \cos (c+d x)}}\\ &=-\frac{\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt{e \cos (c+d x)}}+\frac{b \sqrt{e \cos (c+d x)}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac{(3 a b e) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac{\left (3 a^2 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{4 \left (-a^2+b^2\right )^{3/2} \sqrt{e \cos (c+d x)}}+\frac{\left (3 a^2 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{4 \left (-a^2+b^2\right )^{3/2} \sqrt{e \cos (c+d x)}}\\ &=-\frac{\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt{e \cos (c+d x)}}-\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 \left (-a^2+b^2\right )^{3/2} \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 \left (-a^2+b^2\right )^{3/2} \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{b \sqrt{e \cos (c+d x)}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}+\frac{(3 a b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e-b x^2} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{2 \left (-a^2+b^2\right )^{3/2} d}+\frac{(3 a b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e+b x^2} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{2 \left (-a^2+b^2\right )^{3/2} d}\\ &=\frac{3 a \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 \left (-a^2+b^2\right )^{7/4} d \sqrt{e}}+\frac{3 a \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 \left (-a^2+b^2\right )^{7/4} d \sqrt{e}}-\frac{\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{\left (a^2-b^2\right ) d \sqrt{e \cos (c+d x)}}-\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 \left (-a^2+b^2\right )^{3/2} \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 \left (-a^2+b^2\right )^{3/2} \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{b \sqrt{e \cos (c+d x)}}{\left (a^2-b^2\right ) d e (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 24.19, size = 1181, normalized size = 2.75 \[ \frac{b \cos (c+d x)}{\left (a^2-b^2\right ) d \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{\left (\frac{2 b \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (\frac{5 b \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )}{\left (2 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )\right ) \cos ^2(c+d x)-5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )\right ) \left (a^2+b^2 \left (\cos ^2(c+d x)-1\right )\right )}+\frac{a \left (-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right )+\log \left (b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right )\right )}{4 \sqrt{2} \sqrt{b} \left (a^2-b^2\right )^{3/4}}\right ) \sin ^2(c+d x)}{\left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac{4 a \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (\frac{5 a \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )-2 \left (2 F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (b^2-a^2\right ) F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )\right ) \cos ^2(c+d x)\right ) \left (a^2+b^2 \left (\cos ^2(c+d x)-1\right )\right )}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{b} \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )+\log \left (i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right )-\log \left (i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right )\right )}{\left (b^2-a^2\right )^{3/4}}\right ) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right ) \sqrt{\cos (c+d x)}}{2 (a-b) (a+b) d \sqrt{e \cos (c+d x)}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2),x]
[Out]
(b*Cos[c + d*x])/((a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) + (Sqrt[Cos[c + d*x]]*((-4*a*(a + b
*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(
-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d
*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d
*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^
2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*
Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)
] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqr
t[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4)
)*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + (2*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*
(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*
Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-
a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 -
b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^
2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcT
an[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 -
b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt
[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2
)*(a + b*Sin[c + d*x]))))/(2*(a - b)*(a + b)*d*Sqrt[e*Cos[c + d*x]])
________________________________________________________________________________________
Maple [C] time = 6.516, size = 4457, normalized size = 10.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2),x)
[Out]
-512/d*a*b*e^(7/2)/(1024*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*
x+1/2*c)^6-1792*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c
)+256*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*
sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1
/2*d*x+1/2*c)^8-192*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*si
n(1/2*d*x+1/2*c)^6+512*a^2*e^4*sin(1/2*d*x+1/2*c)^4+4160*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+
1/2*c)^2-1088*b^2*e^4*sin(1/2*d*x+1/2*c)^2+272*e^4*a^2)/(a^2-b^2)*2^(1/2)*cos(1/2*d*x+1/2*c)^5+160/d*a*b*e^3/(
1024*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1792*(-2*
sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+256*(-2*sin(1/2*d*
x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*sin(1/2*d*x+1/2*c)^2
*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1/2*d*x+1/2*c)^8-192*
(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*sin(1/2*d*x+1/2*c)^6+5
12*a^2*e^4*sin(1/2*d*x+1/2*c)^4+4160*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+1/2*c)^2-1088*b^2*e^
4*sin(1/2*d*x+1/2*c)^2+272*e^4*a^2)/(a^2-b^2)*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)*cos(1/2*d*x+1/2*c)^4+384/d*
a*b*e^(7/2)/(1024*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c
)^6-1792*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+256*(
-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*sin(1/2
*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1/2*d*x+
1/2*c)^8-192*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*sin(1/2*d
*x+1/2*c)^6+512*a^2*e^4*sin(1/2*d*x+1/2*c)^4+4160*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+1/2*c)^
2-1088*b^2*e^4*sin(1/2*d*x+1/2*c)^2+272*e^4*a^2)/(a^2-b^2)*2^(1/2)*cos(1/2*d*x+1/2*c)^3+160/d*a*b*e^2/(1024*(-
2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1792*(-2*sin(1/2
*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+256*(-2*sin(1/2*d*x+1/2*c
)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(
1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1/2*d*x+1/2*c)^8-192*(-2*sin
(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*sin(1/2*d*x+1/2*c)^6+512*a^2*
e^4*sin(1/2*d*x+1/2*c)^4+4160*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+1/2*c)^2-1088*b^2*e^4*sin(1
/2*d*x+1/2*c)^2+272*e^4*a^2)/(a^2-b^2)*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(3/2)*cos(1/2*d*x+1/2*c)^2-64/d*a*b*e^(7
/2)/(1024*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1792
*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+256*(-2*sin(1
/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*sin(1/2*d*x+1/2
*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1/2*d*x+1/2*c)^8
-192*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*sin(1/2*d*x+1/2*c
)^6+512*a^2*e^4*sin(1/2*d*x+1/2*c)^4+4160*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+1/2*c)^2-1088*b
^2*e^4*sin(1/2*d*x+1/2*c)^2+272*e^4*a^2)/(a^2-b^2)*2^(1/2)*cos(1/2*d*x+1/2*c)+8/d*a*b*e/(1024*(-2*sin(1/2*d*x+
1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1792*(-2*sin(1/2*d*x+1/2*c)^2*
e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+256*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)
*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2
^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1/2*d*x+1/2*c)^8-192*(-2*sin(1/2*d*x+1/2*c
)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*sin(1/2*d*x+1/2*c)^6+512*a^2*e^4*sin(1/2*d*
x+1/2*c)^4+4160*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+1/2*c)^2-1088*b^2*e^4*sin(1/2*d*x+1/2*c)^
2+272*e^4*a^2)/(a^2-b^2)*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(5/2)-48/d*a*b*e^3/(1024*(-2*sin(1/2*d*x+1/2*c)^2*e+e)
^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1792*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^
(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+256*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1
/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*co
s(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1/2*d*x+1/2*c)^8-192*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2
)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*sin(1/2*d*x+1/2*c)^6+512*a^2*e^4*sin(1/2*d*x+1/2*c)^4+41
60*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+1/2*c)^2-1088*b^2*e^4*sin(1/2*d*x+1/2*c)^2+272*e^4*a^2
)/(a^2-b^2)*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)*cos(1/2*d*x+1/2*c)^2-8/d*a*b*e^2/(1024*(-2*sin(1/2*d*x+1/2*c)
^2*e+e)^(1/2)*e^(7/2)*2^(1/2)*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-1792*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(
1/2)*e^(7/2)*2^(1/2)*b^2*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+256*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/
2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+768*(-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)*e^(7/2)*2^(1/2)
*b^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2+2048*b^2*e^4*sin(1/2*d*x+1/2*c)^8-192*(-2*sin(1/2*d*x+1/2*c)^2*e+
e)^(1/2)*e^(7/2)*2^(1/2)*a^2*cos(1/2*d*x+1/2*c)-5120*b^2*e^4*sin(1/2*d*x+1/2*c)^6+512*a^2*e^4*sin(1/2*d*x+1/2*
c)^4+4160*b^2*e^4*sin(1/2*d*x+1/2*c)^4-768*a^2*e^4*sin(1/2*d*x+1/2*c)^2-1088*b^2*e^4*sin(1/2*d*x+1/2*c)^2+272*
e^4*a^2)/(a^2-b^2)*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(3/2)+3/d*a*b*e/(a^2-b^2)*sum((_R^4+_R^2*e)/(_R^7*b^2-3*_R^5
*b^2*e+8*_R^3*a^2*e^2-5*_R^3*b^2*e^2-_R*b^2*e^3)*ln((-2*sin(1/2*d*x+1/2*c)^2*e+e)^(1/2)-e^(1/2)*cos(1/2*d*x+1/
2*c)*2^(1/2)-_R),_R=RootOf(b^2*_Z^8-4*b^2*e*_Z^6+(16*a^2*e^2-10*b^2*e^2)*_Z^4-4*b^2*e^3*_Z^2+b^2*e^4))-2/d*(e*
(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)
*b^2/(a^2-b^2)/e*cos(1/2*d*x+1/2*c)*(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/(4*b^2*cos(1/2*d*
x+1/2*c)^4-4*b^2*cos(1/2*d*x+1/2*c)^2+a^2)+1/d*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1
/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*
c)^2+1)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1
/16/d*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-
1))^(1/2)/b^2*sum((-5*a^2+2*b^2)/(a+b)/(a-b)/(2*_alpha^2-1)/_alpha*(2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)
^(1/2)*arctanh(1/2*e*(4*_alpha^2-3)/(4*a^2-3*b^2)*(4*cos(1/2*d*x+1/2*c)^2*a^2-3*b^2*cos(1/2*d*x+1/2*c)^2+b^2*_
alpha^2-3*a^2+2*b^2)*2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+
1/2*c)^2))^(1/2))+8*b^2/a^2*_alpha*(_alpha^2-1)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)
/(-sin(1/2*d*x+1/2*c)^2*e*(2*sin(1/2*d*x+1/2*c)^2-1))^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-4*b^2/a^2*(_alpha^2
-1),2^(1/2))),_alpha=RootOf(4*_Z^4*b^2-4*_Z^2*b^2+a^2))+1/8/d*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)
^2)^(1/2)/sin(1/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/b^2*sum(1/_alpha/(2*_alpha^2-1)*(2^(1/2)/(e*
(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)*arctanh(1/2*e*(4*_alpha^2-3)/(4*a^2-3*b^2)*(4*cos(1/2*d*x+1/2*c)^2*a^2-3
*b^2*cos(1/2*d*x+1/2*c)^2+b^2*_alpha^2-3*a^2+2*b^2)*2^(1/2)/(e*(2*_alpha^2*b^2+a^2-2*b^2)/b^2)^(1/2)/(-e*(2*si
n(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2))+8*b^2/a^2*_alpha*(_alpha^2-1)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-
2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-sin(1/2*d*x+1/2*c)^2*e*(2*sin(1/2*d*x+1/2*c)^2-1))^(1/2)*EllipticPi(cos(1/2*
d*x+1/2*c),-4*b^2/a^2*(_alpha^2-1),2^(1/2))),_alpha=RootOf(4*_Z^4*b^2-4*_Z^2*b^2+a^2))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate(1/(sqrt(e*cos(d*x + c))*(b*sin(d*x + c) + a)^2), 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(1/(a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2),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(1/(a+b*sin(d*x+c))**2/(e*cos(d*x+c))**(1/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{e \cos \left (d x + c\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sin(d*x+c))^2/(e*cos(d*x+c))^(1/2),x, algorithm="giac")
[Out]
integrate(1/(sqrt(e*cos(d*x + c))*(b*sin(d*x + c) + a)^2), x)