3.118 \(\int \frac{(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=224 \[ -\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}-\frac{5 a^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}} \]

[Out]

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

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Rubi [A]  time = 0.424021, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {3872, 2873, 2636, 2640, 2639, 2564, 325, 329, 298, 203, 206, 2570, 2571} \[ -\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left (\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right )}{d e^{3/2}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}-\frac{5 a^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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

Mathematica [C]  time = 10.6163, size = 135, normalized size = 0.6 \[ -\frac{2 a^2 \cos ^4\left (\frac{1}{2} (c+d x)\right ) \cot (c+d x) \sqrt{e \sin (c+d x)} \sec ^4\left (\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right ) \left (\sin ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{3}{2},\frac{7}{4},\sin ^2(c+d x)\right )+6 \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},\sin ^2(c+d x)\right )+6 \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{3}{2},\frac{3}{4},\sin ^2(c+d x)\right )\right )}{3 d e^2 \sqrt{\cos ^2(c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*a^2*Cos[(c + d*x)/2]^4*Cot[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*Sqrt[e*Sin[c + d*x]]*(6*Hypergeometric2F
1[-1/4, 1, 3/4, Sin[c + d*x]^2] + 6*Hypergeometric2F1[-1/4, 3/2, 3/4, Sin[c + d*x]^2] + Hypergeometric2F1[3/4,
 3/2, 7/4, Sin[c + d*x]^2]*Sin[c + d*x]^2))/(3*d*e^2*Sqrt[Cos[c + d*x]^2])

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Maple [A]  time = 2.436, size = 238, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}}{2\,d\cos \left ( dx+c \right ) } \left ( 10\,{e}^{3/2}\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) -5\,{e}^{3/2}\sqrt{-\sin \left ( dx+c \right ) +1}\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{-\sin \left ( dx+c \right ) +1},1/2\,\sqrt{2} \right ) -10\,{e}^{3/2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-8\,{e}^{3/2}\cos \left ( dx+c \right ) +4\,{\it Artanh} \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ) \sqrt{e\sin \left ( dx+c \right ) }e\cos \left ( dx+c \right ) -4\,\sqrt{e\sin \left ( dx+c \right ) }\arctan \left ({\frac{\sqrt{e\sin \left ( dx+c \right ) }}{\sqrt{e}}} \right ) e\cos \left ( dx+c \right ) +2\,{e}^{3/2} \right ){e}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/e^(5/2)/(e*sin(d*x+c))^(1/2)/cos(d*x+c)*a^2*(10*e^(3/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d
*x+c)^(1/2)*EllipticE((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-5*e^(3/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2
)*sin(d*x+c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-10*e^(3/2)*cos(d*x+c)^2-8*e^(3/2)*cos(d*x+c)+4
*arctanh((e*sin(d*x+c))^(1/2)/e^(1/2))*(e*sin(d*x+c))^(1/2)*e*cos(d*x+c)-4*(e*sin(d*x+c))^(1/2)*arctan((e*sin(
d*x+c))^(1/2)/e^(1/2))*e*cos(d*x+c)+2*e^(3/2))/d

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\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 )}}{e^{2} \cos \left (d x + c\right )^{2} - e^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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