3.3.91 \(\int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx\) [291]

Optimal. Leaf size=222 \[ -\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]

[Out]

-4*a^2/d/e/(e*csc(d*x+c))^(1/2)-2/3*a^2*cos(d*x+c)/d/e/(e*csc(d*x+c))^(1/2)+a^2*sec(d*x+c)/d/e/(e*csc(d*x+c))^
(1/2)+2*a^2*arctan(sin(d*x+c)^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+2*a^2*arctanh(sin(d*x+c)^(1/2))
/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)+1/3*a^2*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*
x)*EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))/d/e/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3963, 3957, 2952, 2715, 2720, 2644, 327, 335, 218, 212, 209, 2646} \begin {gather*} \frac {2 a^2 \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}-\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(3/2),x]

[Out]

(-4*a^2)/(d*e*Sqrt[e*Csc[c + d*x]]) - (2*a^2*Cos[c + d*x])/(3*d*e*Sqrt[e*Csc[c + d*x]]) + (a^2*Sec[c + d*x])/(
d*e*Sqrt[e*Csc[c + d*x]]) + (2*a^2*ArcTan[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) +
(2*a^2*ArcTanh[Sqrt[Sin[c + d*x]]])/(d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]]) - (a^2*EllipticF[(c - Pi/2
+ d*x)/2, 2])/(3*d*e*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 335

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 2644

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 2646

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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]

Rule 2952

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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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 3963

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]

Rubi steps

\begin {align*} \int \frac {(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx &=\frac {\int (a+a \sec (c+d x))^2 \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int (-a-a \cos (c+d x))^2 \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {\int \left (a^2 \sin ^{\frac {3}{2}}(c+d x)+2 a^2 \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x)+a^2 \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x)\right ) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=\frac {a^2 \int \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a^2 \int \sec ^2(c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \int \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {a^2 \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x^{3/2}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ &=-\frac {4 a^2}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a^2 \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a^2 \sec (c+d x)}{d e \sqrt {e \csc (c+d x)}}+\frac {2 a^2 \tan ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a^2 \tanh ^{-1}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}\\ \end {align*}

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Mathematica [A]
time = 30.01, size = 302, normalized size = 1.36 \begin {gather*} \frac {\left (1+\cos \left (2 \left (\frac {c}{2}+\frac {d x}{2}\right )\right )\right )^2 \cos (c+d x) \left (-1+\csc ^2(c+d x)\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^2 \left (-\frac {6 \text {ArcTan}\left (\sqrt {\csc (c+d x)}\right )+3 \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \log \left (1+\sqrt {\csc (c+d x)}\right )+\frac {\csc (c+d x) F\left (\left .\text {ArcSin}\left (\sqrt {\csc (c+d x)}\right )\right |-1\right ) \sqrt {1-\sin ^2(c+d x)}}{\sqrt {1-\csc ^2(c+d x)}}}{3 d}-\frac {-2+\csc ^2(c+d x) \left (-1+12 \sqrt {1-\sin ^2(c+d x)}\right )}{3 d \csc ^{\frac {5}{2}}(c+d x) \sqrt {1-\sin ^2(c+d x)}}\right )}{4 \left (1+\cos \left (2 \left (\frac {c}{2}+\frac {1}{2} \left (-c+\csc ^{-1}(\csc (c+d x))\right )\right )\right )\right )^2 \sqrt {\csc (c+d x)} (e \csc (c+d x))^{3/2} \sqrt {1-\sin ^2(c+d x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(3/2),x]

[Out]

((1 + Cos[2*(c/2 + (d*x)/2)])^2*Cos[c + d*x]*(-1 + Csc[c + d*x]^2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2
*(-1/3*(6*ArcTan[Sqrt[Csc[c + d*x]]] + 3*Log[1 - Sqrt[Csc[c + d*x]]] - 3*Log[1 + Sqrt[Csc[c + d*x]]] + (Csc[c
+ d*x]*EllipticF[ArcSin[Sqrt[Csc[c + d*x]]], -1]*Sqrt[1 - Sin[c + d*x]^2])/Sqrt[1 - Csc[c + d*x]^2])/d - (-2 +
Csc[c + d*x]^2*(-1 + 12*Sqrt[1 - Sin[c + d*x]^2]))/(3*d*Csc[c + d*x]^(5/2)*Sqrt[1 - Sin[c + d*x]^2])))/(4*(1
+ Cos[2*(c/2 + (-c + ArcCsc[Csc[c + d*x]])/2)])^2*Sqrt[Csc[c + d*x]]*(e*Csc[c + d*x])^(3/2)*Sqrt[1 - Sin[c + d
*x]^2])

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Maple [C] Result contains complex when optimal does not.
time = 0.19, size = 763, normalized size = 3.44

method	result	size
default	\(-\frac {a^{2} \left (6 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}+6 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}-13 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}+6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}-6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-\sin \left (d x +c \right )-i}{\sin \left (d x +c \right )}}+2 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}+10 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-15 \sqrt {2}\, \cos \left (d x +c \right )+3 \sqrt {2}\right ) \sqrt {2}}{6 d \left (-1+\cos \left (d x +c \right )\right ) \cos \left (d x +c \right ) \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )}\)	\(763\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*a^2/d*(6*I*cos(d*x+c)*sin(d*x+c)*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))*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*(-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)+6*I*cos(d*x+c)*sin(d*x+c)*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))*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*(-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)-13*I*cos(d*x+c)*sin(d*x+c)*((I*cos(d*x+c)+sin(d*x+c)-I)/si
n(d*x+c))^(1/2)*(-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)*Elliptic
F(((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2),1/2*2^(1/2))+6*cos(d*x+c)*sin(d*x+c)*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))*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*x+c))^(1/2)*(-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)-6*cos(d*x+c)*sin(d*x+c)*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))*((I*cos(d*x+c)+sin(d*x+c)-I)/sin(d*
x+c))^(1/2)*(-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)+2*cos(d*x+c)
^3*2^(1/2)+10*cos(d*x+c)^2*2^(1/2)-15*2^(1/2)*cos(d*x+c)+3*2^(1/2))/(-1+cos(d*x+c))/cos(d*x+c)/(e/sin(d*x+c))^
(3/2)/sin(d*x+c)*2^(1/2)

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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((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

e^(-3/2)*integrate((a*sec(d*x + c) + a)^2/csc(d*x + c)^(3/2), x)

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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((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**2/(e*csc(d*x+c))**(3/2),x)

[Out]

a**2*(Integral((e*csc(c + d*x))**(-3/2), x) + Integral(2*sec(c + d*x)/(e*csc(c + d*x))**(3/2), x) + Integral(s
ec(c + d*x)**2/(e*csc(c + d*x))**(3/2), x))

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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((a+a*sec(d*x+c))^2/(e*csc(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^2/(e*csc(d*x + c))^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(3/2),x)

[Out]

int((a + a/cos(c + d*x))^2/(e/sin(c + d*x))^(3/2), x)

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