3.2.19 \(\int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx\) [119]

Optimal. Leaf size=234 \[ \frac {2 a^2 \text {ArcTan}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {7 a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3} \]

[Out]

2*a^2*arctan((e*sin(d*x+c))^(1/2)/e^(1/2))/d/e^(5/2)+2*a^2*arctanh((e*sin(d*x+c))^(1/2)/e^(1/2))/d/e^(5/2)-4/3
*a^2/d/e/(e*sin(d*x+c))^(3/2)-2/3*a^2*cos(d*x+c)/d/e/(e*sin(d*x+c))^(3/2)-2/3*a^2*sec(d*x+c)/d/e/(e*sin(d*x+c)
)^(3/2)-7/3*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))*sin(d*x+c)^(1/2)/d/e^2/(e*sin(d*x+c))^(1/2)+5/3*a^2*sec(d*x+c)*(e*sin(d*x+c))^(1/2)/d/e^3

________________________________________________________________________________________

Rubi [A]
time = 0.29, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3957, 2952, 2716, 2721, 2720, 2644, 331, 335, 218, 212, 209, 2650, 2651} \begin {gather*} \frac {2 a^2 \text {ArcTan}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {7 a^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(5/2),x]

[Out]

(2*a^2*ArcTan[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^(5/2)) + (2*a^2*ArcTanh[Sqrt[e*Sin[c + d*x]]/Sqrt[e]])/(d*e^
(5/2)) - (4*a^2)/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a^2*Cos[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a^2
*Sec[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (7*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3
*d*e^2*Sqrt[e*Sin[c + d*x]]) + (5*a^2*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d*e^3)

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 331

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 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 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 2651

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 2716

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 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 2721

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

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]

Rubi steps

\begin {align*} \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx &=\int \frac {(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx\\ &=\int \left (\frac {a^2}{(e \sin (c+d x))^{5/2}}+\frac {2 a^2 \sec (c+d x)}{(e \sin (c+d x))^{5/2}}+\frac {a^2 \sec ^2(c+d x)}{(e \sin (c+d x))^{5/2}}\right ) \, dx\\ &=a^2 \int \frac {1}{(e \sin (c+d x))^{5/2}} \, dx+a^2 \int \frac {\sec ^2(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx+\left (2 a^2\right ) \int \frac {\sec (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx\\ &=-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {a^2 \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 e^2}+\frac {\left (5 a^2\right ) \int \frac {\sec ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{3 e^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x^{5/2} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e}\\ &=-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e^3}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{6 e^2}+\frac {\left (a^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {2 a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e^3}+\frac {\left (5 a^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{6 e^2 \sqrt {e \sin (c+d x)}}\\ &=-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {7 a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e^2}\\ &=\frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {7 a^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 40.43, size = 241, normalized size = 1.03 \begin {gather*} \frac {a^2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \cot \left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \text {ArcSin}(\sin (c+d x))\right ) \left (-1-8 \sqrt {\cos ^2(c+d x)}-7 \cos (2 (c+d x))+12 \text {ArcTan}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sin ^{\frac {3}{2}}(c+d x)+14 \sqrt {\cos ^2(c+d x)} F\left (\left .\text {ArcSin}\left (\sqrt {\sin (c+d x)}\right )\right |-1\right ) \sin ^{\frac {3}{2}}(c+d x)-6 \sqrt {\cos ^2(c+d x)} \log \left (1-\sqrt {\sin (c+d x)}\right ) \sin ^{\frac {3}{2}}(c+d x)+6 \sqrt {\cos ^2(c+d x)} \log \left (1+\sqrt {\sin (c+d x)}\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{12 d e^2 \sqrt {e \sin (c+d x)}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(5/2),x]

[Out]

(a^2*Cos[(c + d*x)/2]^2*Cot[(c + d*x)/2]*Sec[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*(-1 - 8*Sqrt[Cos[c + d*x]^
2] - 7*Cos[2*(c + d*x)] + 12*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2]*Sin[c + d*x]^(3/2) + 14*Sqrt[Cos[
c + d*x]^2]*EllipticF[ArcSin[Sqrt[Sin[c + d*x]]], -1]*Sin[c + d*x]^(3/2) - 6*Sqrt[Cos[c + d*x]^2]*Log[1 - Sqrt
[Sin[c + d*x]]]*Sin[c + d*x]^(3/2) + 6*Sqrt[Cos[c + d*x]^2]*Log[1 + Sqrt[Sin[c + d*x]]]*Sin[c + d*x]^(3/2)))/(
12*d*e^2*Sqrt[e*Sin[c + d*x]])

________________________________________________________________________________________

Maple [A]
time = 0.28, size = 301, normalized size = 1.29

method	result	size
default	\(\frac {a^{2} \left (7 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) e^{\frac {7}{2}}-14 e^{\frac {7}{2}} \left (\cos ^{4}\left (d x +c \right )\right )-8 e^{\frac {7}{2}} \left (\cos ^{3}\left (d x +c \right )\right )+12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \left (\cos ^{3}\left (d x +c \right )\right ) e^{2}+12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \left (\cos ^{3}\left (d x +c \right )\right ) e^{2}+20 e^{\frac {7}{2}} \left (\cos ^{2}\left (d x +c \right )\right )+8 e^{\frac {7}{2}} \cos \left (d x +c \right )-12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right ) e^{2}-12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right ) e^{2}-6 e^{\frac {7}{2}}\right )}{6 e^{\frac {9}{2}} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )-1\right ) d}\)	\(301\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^2/(e*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6/e^(9/2)/(e*sin(d*x+c))^(3/2)/cos(d*x+c)/(cos(d*x+c)^2-1)*a^2*(7*(-sin(d*x+c)+1)^(1/2)*(2*sin(d*x+c)+2)^(1/
2)*sin(d*x+c)^(7/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))*e^(7/2)-14*e^(7/2)*cos(d*x+c)^4-8*e^(7/2)*cos
(d*x+c)^3+12*(e*sin(d*x+c))^(3/2)*arctanh((e*sin(d*x+c))^(1/2)/e^(1/2))*cos(d*x+c)^3*e^2+12*(e*sin(d*x+c))^(3/
2)*arctan((e*sin(d*x+c))^(1/2)/e^(1/2))*cos(d*x+c)^3*e^2+20*e^(7/2)*cos(d*x+c)^2+8*e^(7/2)*cos(d*x+c)-12*(e*si
n(d*x+c))^(3/2)*arctanh((e*sin(d*x+c))^(1/2)/e^(1/2))*cos(d*x+c)*e^2-12*(e*sin(d*x+c))^(3/2)*arctan((e*sin(d*x
+c))^(1/2)/e^(1/2))*cos(d*x+c)*e^2-6*e^(7/2))/d

________________________________________________________________________________________

Maxima [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*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.65, size = 291, normalized size = 1.24 \begin {gather*} \frac {7 \, \sqrt {-i} {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - \sqrt {2} a^{2} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 7 \, \sqrt {i} {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{2} - \sqrt {2} a^{2} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 6 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \arctan \left (\frac {\sin \left (d x + c\right ) - 1}{2 \, \sqrt {\sin \left (d x + c\right )}}\right ) + 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (\frac {\cos \left (d x + c\right )^{2} - 4 \, {\left (\sin \left (d x + c\right ) + 1\right )} \sqrt {\sin \left (d x + c\right )} - 6 \, \sin \left (d x + c\right ) - 2}{\cos \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 2}\right ) + 2 \, {\left (7 \, a^{2} \cos \left (d x + c\right ) - 3 \, a^{2}\right )} \sqrt {\sin \left (d x + c\right )}}{6 \, {\left (d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} - d \cos \left (d x + c\right ) e^{\frac {5}{2}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2/(e*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/6*(7*sqrt(-I)*(sqrt(2)*a^2*cos(d*x + c)^2 - sqrt(2)*a^2*cos(d*x + c))*weierstrassPInverse(4, 0, cos(d*x + c)
+ I*sin(d*x + c)) + 7*sqrt(I)*(sqrt(2)*a^2*cos(d*x + c)^2 - sqrt(2)*a^2*cos(d*x + c))*weierstrassPInverse(4,
0, cos(d*x + c) - I*sin(d*x + c)) + 6*(a^2*cos(d*x + c)^2 - a^2*cos(d*x + c))*arctan(1/2*(sin(d*x + c) - 1)/sq
rt(sin(d*x + c))) + 3*(a^2*cos(d*x + c)^2 - a^2*cos(d*x + c))*log((cos(d*x + c)^2 - 4*(sin(d*x + c) + 1)*sqrt(
sin(d*x + c)) - 6*sin(d*x + c) - 2)/(cos(d*x + c)^2 + 2*sin(d*x + c) - 2)) + 2*(7*a^2*cos(d*x + c) - 3*a^2)*sq
rt(sin(d*x + c)))/(d*cos(d*x + c)^2*e^(5/2) - d*cos(d*x + c)*e^(5/2))

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**2/(e*sin(d*x+c))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3007 deep

________________________________________________________________________________________

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*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^2/(e*sin(d*x + c))^(5/2), x)

________________________________________________________________________________________

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 (e\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,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))^(5/2),x)

[Out]

int((a + a/cos(c + d*x))^2/(e*sin(c + d*x))^(5/2), x)

________________________________________________________________________________________