3.288 \(\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx\)
Optimal. Leaf size=240 \[ -\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sec (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{3 a^2 e \sin (c+d x) \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}-\frac{5 a^2 e \sqrt{\sin (c+d x)} E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \csc (c+d x)}}{d} \]
[Out]
(-4*a^2*e*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Sqrt[e*Csc[c + d*
x]]*Sec[c + d*x])/d - (2*a^2*e*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*
e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d - (5*a^2*e*Sqrt[e*Csc[c + d*x]]*Ellip
ticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d + (3*a^2*e*Sqrt[e*Csc[c + d*x]]*Sin[c + d*x]*Tan[c + d*x])/d
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Rubi [A] time = 0.330845, antiderivative size = 240, normalized size of antiderivative = 1.,
number of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52,
Rules used = {3878, 3872, 2873, 2636, 2639, 2564, 325, 329, 298, 203, 206, 2570, 2571}
\[ -\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sec (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}+\frac{3 a^2 e \sin (c+d x) \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right )}{d}-\frac{5 a^2 e \sqrt{\sin (c+d x)} E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \csc (c+d x)}}{d} \]
Antiderivative was successfully verified.
[In]
Int[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]
[Out]
(-4*a^2*e*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Cos[c + d*x]*Sqrt[e*Csc[c + d*x]])/d - (2*a^2*e*Sqrt[e*Csc[c + d*
x]]*Sec[c + d*x])/d - (2*a^2*e*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d + (2*a^2*
e*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])/d - (5*a^2*e*Sqrt[e*Csc[c + d*x]]*Ellip
ticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/d + (3*a^2*e*Sqrt[e*Csc[c + d*x]]*Sin[c + d*x]*Tan[c + d*x])/d
Rule 3878
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*(x_)])^(p_), x_Symbol] :> Dist[g^Int
Part[p]*(g*Sec[e + f*x])^FracPart[p]*Cos[e + f*x]^FracPart[p], Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x],
x] /; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Rule 3872
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Rule 2873
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Rule 2636
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]
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 2564
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])
Rule 325
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]
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 298
Int[(x_)^2/((a_) + (b_.)*(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), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
& !GtQ[a/b, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 2570
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 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]
Rubi steps
\begin{align*} \int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx &=\left (e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{(a+a \sec (c+d x))^2}{\sin ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\left (e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{\sin ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\left (e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \left (\frac{a^2}{\sin ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sec (c+d x)}{\sin ^{\frac{3}{2}}(c+d x)}+\frac{a^2 \sec ^2(c+d x)}{\sin ^{\frac{3}{2}}(c+d x)}\right ) \, dx\\ &=\left (a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sin ^{\frac{3}{2}}(c+d x)} \, dx+\left (a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\sec ^2(c+d x)}{\sin ^{\frac{3}{2}}(c+d x)} \, dx+\left (2 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \frac{\sec (c+d x)}{\sin ^{\frac{3}{2}}(c+d x)} \, dx\\ &=-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{e \csc (c+d x)} \sec (c+d x)}{d}-\left (a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx+\left (3 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \sec ^2(c+d x) \sqrt{\sin (c+d x)} \, dx+\frac{\left (2 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{e \csc (c+d x)} \sec (c+d x)}{d}-\frac{2 a^2 e \sqrt{e \csc (c+d x)} E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}+\frac{3 a^2 e \sqrt{e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}-\frac{1}{2} \left (3 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx+\frac{\left (2 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{e \csc (c+d x)} \sec (c+d x)}{d}-\frac{5 a^2 e \sqrt{e \csc (c+d x)} E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}+\frac{3 a^2 e \sqrt{e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}+\frac{\left (4 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}\\ &=-\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{e \csc (c+d x)} \sec (c+d x)}{d}-\frac{5 a^2 e \sqrt{e \csc (c+d x)} E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}+\frac{3 a^2 e \sqrt{e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}+\frac{\left (2 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}-\frac{\left (2 a^2 e \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\sin (c+d x)}\right )}{d}\\ &=-\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{e \csc (c+d x)} \sec (c+d x)}{d}-\frac{2 a^2 e \tan ^{-1}\left (\sqrt{\sin (c+d x)}\right ) \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}{d}+\frac{2 a^2 e \tanh ^{-1}\left (\sqrt{\sin (c+d x)}\right ) \sqrt{e \csc (c+d x)} \sqrt{\sin (c+d x)}}{d}-\frac{5 a^2 e \sqrt{e \csc (c+d x)} E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{d}+\frac{3 a^2 e \sqrt{e \csc (c+d x)} \sin (c+d x) \tan (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 4.72268, size = 195, normalized size = 0.81 \[ \frac{2 a^2 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) (e \csc (c+d x))^{3/2} \sec ^4\left (\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right ) \left (5 \sqrt{-\cot ^2(c+d x)} \sqrt{\csc (c+d x)} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{3}{2},\frac{7}{4},\csc ^2(c+d x)\right )-6 \sqrt{\csc (c+d x)}-6 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)}+3 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left (\sqrt{\csc (c+d x)}\right )+3 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left (\sqrt{\csc (c+d x)}\right )\right )}{3 d \csc ^{\frac{3}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]
[Out]
(2*a^2*Cos[(c + d*x)/2]^4*(e*Csc[c + d*x])^(3/2)*(3*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + 3*ArcTan
h[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2] - 6*Sqrt[Csc[c + d*x]] - 6*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]]
+ 5*Sqrt[-Cot[c + d*x]^2]*Sqrt[Csc[c + d*x]]*Hypergeometric2F1[3/4, 3/2, 7/4, Csc[c + d*x]^2])*Sec[c + d*x]*Se
c[ArcCsc[Csc[c + d*x]]/2]^4)/(3*d*Csc[c + d*x]^(3/2))
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Maple [C] time = 0.197, size = 1559, normalized size = 6.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x)
[Out]
-1/2*a^2/d*2^(1/2)*(2*I*cos(d*x+c)^2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*
x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*EllipticPi(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c)
)^(1/2),1/2-1/2*I,1/2*2^(1/2))-2*I*cos(d*x+c)^2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((I*cos(d*x+c)+sin(d*x+c
)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*EllipticPi(((I*cos(d*x+c)+sin(d*x+c)-I)
/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))-2*cos(d*x+c)^2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((I*cos(d*x+c)+
sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*EllipticPi(((I*cos(d*x+c)+sin(
d*x+c)-I)/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))-2*cos(d*x+c)^2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((I*co
s(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*EllipticPi(((I*cos(d*
x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))-10*cos(d*x+c)^2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1
/2)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*EllipticE((
(I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+5*cos(d*x+c)^2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2
)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*EllipticF(((I
*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+2*I*cos(d*x+c)*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(1/2)*
((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*EllipticPi(((I*
cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))-2*I*cos(d*x+c)*(-I*(-1+cos(d*x+c))/sin(d*x+c
))^(1/2)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)*Ellipt
icPi(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))-2*(-I*(-1+cos(d*x+c))/sin(d*x+c))^(
1/2)*cos(d*x+c)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c))^(1/2)
*EllipticPi(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))-2*(-I*(-1+cos(d*x+c))/sin(d*
x+c))^(1/2)*cos(d*x+c)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*x+c)
)^(1/2)*EllipticPi(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))-10*(-I*(-1+cos(d*x+c)
)/sin(d*x+c))^(1/2)*cos(d*x+c)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/si
n(d*x+c))^(1/2)*EllipticE(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+5*(-I*(-1+cos(d*x+c))/si
n(d*x+c))^(1/2)*cos(d*x+c)*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*((-I*cos(d*x+c)+sin(d*x+c)+I)/sin(d*
x+c))^(1/2)*EllipticF(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+9*cos(d*x+c)*2^(1/2)-2^(1/2)
)*(e/sin(d*x+c))^(3/2)*sin(d*x+c)/cos(d*x+c)
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Maxima [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((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x, algorithm="maxima")
[Out]
Timed out
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} e \csc \left (d x + c\right ) \sec \left (d x + c\right )^{2} + 2 \, a^{2} e \csc \left (d x + c\right ) \sec \left (d x + c\right ) + a^{2} e \csc \left (d x + c\right )\right )} \sqrt{e \csc \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x, algorithm="fricas")
[Out]
integral((a^2*e*csc(d*x + c)*sec(d*x + c)^2 + 2*a^2*e*csc(d*x + c)*sec(d*x + c) + a^2*e*csc(d*x + c))*sqrt(e*c
sc(d*x + c)), x)
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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((e*csc(d*x+c))**(3/2)*(a+a*sec(d*x+c))**2,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \csc \left (d x + c\right )\right )^{\frac{3}{2}}{\left (a \sec \left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*csc(d*x+c))^(3/2)*(a+a*sec(d*x+c))^2,x, algorithm="giac")
[Out]
integrate((e*csc(d*x + c))^(3/2)*(a*sec(d*x + c) + a)^2, x)