∫ (a + a sin(e + fx))5∕2 tan2(e + fx) dx [100]

Optimal. Leaf size=118 \[ \frac {124 a^3 \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}+\frac {31 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {9 \sec (e+f x) (a+a \sin (e+f x))^{5/2}}{5 f}-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{7/2}}{5 a f} \]

9/5*sec(f*x+e)*(a+a*sin(f*x+e))^(5/2)/f-2/5*sec(f*x+e)*(a+a*sin(f*x+e))^(7/2)/a/f+124/15*a^3*cos(f*x+e)/f/(a+a

*sin(f*x+e))^(1/2)+31/15*a^2*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2)/f

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Rubi [A]
time = 0.14, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2792, 2934, 2726, 2725} \begin {gather*} \frac {124 a^3 \cos (e+f x)}{15 f \sqrt {a \sin (e+f x)+a}}+\frac {31 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}-\frac {2 \sec (e+f x) (a \sin (e+f x)+a)^{7/2}}{5 a f}+\frac {9 \sec (e+f x) (a \sin (e+f x)+a)^{5/2}}{5 f} \end {gather*}

(124*a^3*Cos[e + f*x])/(15*f*Sqrt[a + a*Sin[e + f*x]]) + (31*a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/(15*f)

+ (9*Sec[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(5*f) - (2*Sec[e + f*x]*(a + a*Sin[e + f*x])^(7/2))/(5*a*f)

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*b*(Cos[c + d*x]/(d*Sqrt[a + b*Sin[c + d*x

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n

- 1)/(d*n)), x] + Dist[a*((2*n - 1)/n), Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Simp[-(a + b*Sin[e + f

*x])^(m + 1)/(b*f*m*Cos[e + f*x]), x] + Dist[1/(b*m), Int[(a + b*Sin[e + f*x])^m*((b*(m + 1) + a*Sin[e + f*x])

/Cos[e + f*x]^2), x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && !LtQ[m, 0]

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)

+ (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c + a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*(p +

1))), x] + Dist[b*((a*d*m + b*c*(m + p + 1))/(a*g^2*(p + 1))), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*

x])^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]

\begin {align*} \int (a+a \sin (e+f x))^{5/2} \tan ^2(e+f x) \, dx &=-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{7/2}}{5 a f}+\frac {2 \int \sec ^2(e+f x) (a+a \sin (e+f x))^{5/2} \left (\frac {7 a}{2}+a \sin (e+f x)\right ) \, dx}{5 a}\\ &=\frac {9 \sec (e+f x) (a+a \sin (e+f x))^{5/2}}{5 f}-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{7/2}}{5 a f}-\frac {1}{10} (31 a) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=\frac {31 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {9 \sec (e+f x) (a+a \sin (e+f x))^{5/2}}{5 f}-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{7/2}}{5 a f}-\frac {1}{15} \left (62 a^2\right ) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=\frac {124 a^3 \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}+\frac {31 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {9 \sec (e+f x) (a+a \sin (e+f x))^{5/2}}{5 f}-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{7/2}}{5 a f}\\ \end {align*}

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Mathematica [A]
time = 5.32, size = 60, normalized size = 0.51 \begin {gather*} \frac {a^2 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} (330+22 \cos (2 (e+f x))-185 \sin (e+f x)+3 \sin (3 (e+f x)))}{30 f} \end {gather*}

(a^2*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(330 + 22*Cos[2*(e + f*x)] - 185*Sin[e + f*x] + 3*Sin[3*(e + f*x)

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method	result	size

default	\(-\frac {2 a^{3} \left (1+\sin \left (f x +e \right )\right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )+11 \left (\sin ^{2}\left (f x +e \right )\right )+44 \sin \left (f x +e \right )-88\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\)	\(67\)

int((a+a*sin(f*x+e))^(5/2)*tan(f*x+e)^2,x,method=_RETURNVERBOSE)

-2/15*a^3*(1+sin(f*x+e))*(3*sin(f*x+e)^3+11*sin(f*x+e)^2+44*sin(f*x+e)-88)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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Maxima [A]
time = 0.51, size = 207, normalized size = 1.75 \begin {gather*} -\frac {8 \, {\left (22 \, a^{\frac {5}{2}} - \frac {22 \, a^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {55 \, a^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {50 \, a^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {55 \, a^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {22 \, a^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {22 \, a^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )}}{15 \, f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \end {gather*}

integrate((a+a*sin(f*x+e))^(5/2)*tan(f*x+e)^2,x, algorithm="maxima")

-8/15*(22*a^(5/2) - 22*a^(5/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 55*a^(5/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^

2 - 50*a^(5/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 55*a^(5/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 22*a^(5/

2)*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 22*a^(5/2)*sin(f*x + e)^6/(cos(f*x + e) + 1)^6)/(f*(sin(f*x + e)/(cos

(f*x + e) + 1) - 1)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(5/2))

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Fricas [A]
time = 0.36, size = 75, normalized size = 0.64 \begin {gather*} \frac {2 \, {\left (11 \, a^{2} \cos \left (f x + e\right )^{2} + 77 \, a^{2} + {\left (3 \, a^{2} \cos \left (f x + e\right )^{2} - 47 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{15 \, f \cos \left (f x + e\right )} \end {gather*}

integrate((a+a*sin(f*x+e))^(5/2)*tan(f*x+e)^2,x, algorithm="fricas")

2/15*(11*a^2*cos(f*x + e)^2 + 77*a^2 + (3*a^2*cos(f*x + e)^2 - 47*a^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)/

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

Exception raised: SystemError >> excessive stack use: stack is 6189 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1502 vs. \(2 (110) = 220\).
time = 44.08, size = 1502, normalized size = 12.73 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((a+a*sin(f*x+e))^(5/2)*tan(f*x+e)^2,x, algorithm="giac")

-1/3840*sqrt(2)*(1080*pi*a^2*floor(-1/8*(pi - 2*f*x - 2*e)/pi + 1/2)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-

1/8*pi + 1/4*f*x + 1/4*e)^11 + 1080*pi*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(tan(-1/8*pi + 1/4*f*x + 1/4

*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^11 + 135*(pi - 2*f*x - 2*e)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1

/8*pi + 1/4*f*x + 1/4*e)^11 - 1080*a^2*arctan(1/tan(-1/8*pi + 1/4*f*x + 1/4*e))*sgn(cos(-1/4*pi + 1/2*f*x + 1/

2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^11 + 3840*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x +

1/4*e)^12 + 5400*pi*a^2*floor(-1/8*(pi - 2*f*x - 2*e)/pi + 1/2)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*

pi + 1/4*f*x + 1/4*e)^9 + 5400*pi*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(tan(-1/8*pi + 1/4*f*x + 1/4*e))*

tan(-1/8*pi + 1/4*f*x + 1/4*e)^9 + 675*(pi - 2*f*x - 2*e)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi

+ 1/4*f*x + 1/4*e)^9 - 5400*a^2*arctan(1/tan(-1/8*pi + 1/4*f*x + 1/4*e))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*t

an(-1/8*pi + 1/4*f*x + 1/4*e)^9 + 99840*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)

^10 + 10800*pi*a^2*floor(-1/8*(pi - 2*f*x - 2*e)/pi + 1/2)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1

/4*f*x + 1/4*e)^7 + 10800*pi*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(tan(-1/8*pi + 1/4*f*x + 1/4*e))*tan(-

1/8*pi + 1/4*f*x + 1/4*e)^7 + 1350*(pi - 2*f*x - 2*e)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/

4*f*x + 1/4*e)^7 - 10800*a^2*arctan(1/tan(-1/8*pi + 1/4*f*x + 1/4*e))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(

-1/8*pi + 1/4*f*x + 1/4*e)^7 + 200960*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^8

+ 10800*pi*a^2*floor(-1/8*(pi - 2*f*x - 2*e)/pi + 1/2)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*

f*x + 1/4*e)^5 + 10800*pi*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(tan(-1/8*pi + 1/4*f*x + 1/4*e))*tan(-1/8

*pi + 1/4*f*x + 1/4*e)^5 + 1350*(pi - 2*f*x - 2*e)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f

*x + 1/4*e)^5 - 10800*a^2*arctan(1/tan(-1/8*pi + 1/4*f*x + 1/4*e))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/

8*pi + 1/4*f*x + 1/4*e)^5 + 406528*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^6 +

5400*pi*a^2*floor(-1/8*(pi - 2*f*x - 2*e)/pi + 1/2)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x

+ 1/4*e)^3 + 5400*pi*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(tan(-1/8*pi + 1/4*f*x + 1/4*e))*tan(-1/8*pi +

1/4*f*x + 1/4*e)^3 + 675*(pi - 2*f*x - 2*e)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1

/4*e)^3 - 5400*a^2*arctan(1/tan(-1/8*pi + 1/4*f*x + 1/4*e))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi +

1/4*f*x + 1/4*e)^3 + 200960*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^4 + 1080*pi

*a^2*floor(-1/8*(pi - 2*f*x - 2*e)/pi + 1/2)*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e

) + 1080*pi*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sgn(tan(-1/8*pi + 1/4*f*x + 1/4*e))*tan(-1/8*pi + 1/4*f*x

+ 1/4*e) + 135*(pi - 2*f*x - 2*e)*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e) - 108

0*a^2*arctan(1/tan(-1/8*pi + 1/4*f*x + 1/4*e))*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4

*e) + 99840*a^2*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*tan(-1/8*pi + 1/4*f*x + 1/4*e)^2 + 3840*a^2*sgn(cos(-1/4*p

i + 1/2*f*x + 1/2*e)))*sqrt(a)/((tan(-1/8*pi + 1/4*f*x + 1/4*e)^11 + 5*tan(-1/8*pi + 1/4*f*x + 1/4*e)^9 + 10*t

an(-1/8*pi + 1/4*f*x + 1/4*e)^7 + 10*tan(-1/8*pi + 1/4*f*x + 1/4*e)^5 + 5*tan(-1/8*pi + 1/4*f*x + 1/4*e)^3 + t

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}

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3.1.100 \(\int (a+a \sin (e+f x))^{5/2} \tan ^2(e+f x) \, dx\) [100]