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

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

[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

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Rubi [A]  time = 0.42, 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 = {3872, 2873, 2636, 2642, 2641, 2564, 325, 329, 212, 206, 203, 2570, 2571} \[ \frac {2 a^2 \tan ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {7 a^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}} \]

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 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 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 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 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 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]

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 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 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 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 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]

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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [C]  time = 48.88, size = 169, normalized size = 0.72 \[ -\frac {a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sqrt {e \sin (c+d x)} \sec ^4\left (\frac {1}{2} \sin ^{-1}(\sin (c+d x))\right ) \left (3 \sqrt {\cos ^2(c+d x)} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(c+d x)\right )+4 \sqrt {\cos ^2(c+d x)} \csc ^2(c+d x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\sin ^2(c+d x)\right )+4 \sqrt {\cos ^2(c+d x)} \csc ^2(c+d x) \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};\sin ^2(c+d x)\right )+3\right )}{3 d e^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/3*(a^2*Cos[(c + d*x)/2]^4*(3 + 4*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^2*Hypergeometric2F1[-3/4, 1, 1/4, Sin[c
+ d*x]^2] + 4*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^2*Hypergeometric2F1[-3/4, 3/2, 1/4, Sin[c + d*x]^2] + 3*Sqrt[C
os[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Sin[c + d*x]^2])*Sec[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*Sq
rt[e*Sin[c + d*x]])/(d*e^3)

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fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}\right )} \sqrt {e \sin \left (d x + c\right )}}{{\left (e^{3} \cos \left (d x + c\right )^{2} - e^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]

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]

integral(-(a^2*sec(d*x + c)^2 + 2*a^2*sec(d*x + c) + a^2)*sqrt(e*sin(d*x + c))/((e^3*cos(d*x + c)^2 - e^3)*sin
(d*x + c)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]

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)

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maple [A]  time = 6.25, size = 301, normalized size = 1.29 \[ \frac {a^{2} \left (7 \left (\sin ^{\frac {7}{2}}\left (d x +c \right )\right ) \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) e^{\frac {7}{2}}-14 e^{\frac {7}{2}} \left (\cos ^{4}\left (d x +c \right )\right )-8 e^{\frac {7}{2}} \left (\cos ^{3}\left (d x +c \right )\right )+12 \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} e^{2} \left (\cos ^{3}\left (d x +c \right )\right )+12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) e^{2} \left (\cos ^{3}\left (d x +c \right )\right )+20 e^{\frac {7}{2}} \left (\cos ^{2}\left (d x +c \right )\right )+8 e^{\frac {7}{2}} \cos \left (d x +c \right )-12 \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} e^{2} \cos \left (d x +c \right )-12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) e^{2} \cos \left (d x +c \right )-6 e^{\frac {7}{2}}\right )}{6 e^{\frac {9}{2}} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )-1\right ) d} \]

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)

[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)^(7/2)*(-sin(d*x+c)+1)^(1/2)*(2*
sin(d*x+c)+2)^(1/2)*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*arctan((e*sin(d*x+c))^(1/2)/e^(1/2))*(e*sin(d*x+c))^(3/2)*e^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))*e^2*cos(d*x+c)^3+20*e^(7/2)*cos(d*x+c)^2+8*e^(7/2)*cos(d*x+c)-12*arcta
n((e*sin(d*x+c))^(1/2)/e^(1/2))*(e*sin(d*x+c))^(3/2)*e^2*cos(d*x+c)-12*(e*sin(d*x+c))^(3/2)*arctanh((e*sin(d*x
+c))^(1/2)/e^(1/2))*e^2*cos(d*x+c)-6*e^(7/2))/d

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]

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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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