3.3.7 \(\int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx\) [207]

Optimal. Leaf size=177 \[ -\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{9/2} f}+\frac {91 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{32 \sqrt {2} a^{9/2} f}+\frac {\tan (e+f x)}{3 a f (a+a \sec (e+f x))^{7/2}}+\frac {11 \tan (e+f x)}{24 a^2 f (a+a \sec (e+f x))^{5/2}}+\frac {27 \tan (e+f x)}{32 a^3 f (a+a \sec (e+f x))^{3/2}} \]

[Out]

-2*arctan(a^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2))/a^(9/2)/f+91/64*arctan(1/2*a^(1/2)*tan(f*x+e)*2^(1/2)/(a+
a*sec(f*x+e))^(1/2))/a^(9/2)/f*2^(1/2)+1/3*tan(f*x+e)/a/f/(a+a*sec(f*x+e))^(7/2)+11/24*tan(f*x+e)/a^2/f/(a+a*s
ec(f*x+e))^(5/2)+27/32*tan(f*x+e)/a^3/f/(a+a*sec(f*x+e))^(3/2)

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Rubi [A]
time = 0.14, antiderivative size = 227, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3972, 482, 541, 536, 209} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{9/2} f}+\frac {91 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{32 \sqrt {2} a^{9/2} f}+\frac {27 \sin (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{64 a^4 f \sqrt {a \sec (e+f x)+a}}+\frac {\sin (e+f x) \cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right )}{24 a^4 f \sqrt {a \sec (e+f x)+a}}+\frac {11 \sin (e+f x) \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right )}{96 a^4 f \sqrt {a \sec (e+f x)+a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[e + f*x]^2/(a + a*Sec[e + f*x])^(9/2),x]

[Out]

(-2*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + a*Sec[e + f*x]]])/(a^(9/2)*f) + (91*ArcTan[(Sqrt[a]*Tan[e + f*x])/(
Sqrt[2]*Sqrt[a + a*Sec[e + f*x]])])/(32*Sqrt[2]*a^(9/2)*f) + (27*Sec[(e + f*x)/2]^2*Sin[e + f*x])/(64*a^4*f*Sq
rt[a + a*Sec[e + f*x]]) + (11*Cos[e + f*x]*Sec[(e + f*x)/2]^4*Sin[e + f*x])/(96*a^4*f*Sqrt[a + a*Sec[e + f*x]]
) + (Cos[e + f*x]^2*Sec[(e + f*x)/2]^6*Sin[e + f*x])/(24*a^4*f*Sqrt[a + a*Sec[e + f*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 482

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

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 3972

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_.), x_Symbol] :> Dist[-2*(a^(m/2 +
n + 1/2)/d), Subst[Int[x^m*((2 + a*x^2)^(m/2 + n - 1/2)/(1 + a*x^2)), x], x, Cot[c + d*x]/Sqrt[a + b*Csc[c +
d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m/2] && IntegerQ[n - 1/2]

Rubi steps

\begin {align*} \int \frac {\tan ^2(e+f x)}{(a+a \sec (e+f x))^{9/2}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )^4} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^3 f}\\ &=\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\text {Subst}\left (\int \frac {1-5 a x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{3 a^4 f}\\ &=\frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\text {Subst}\left (\int \frac {15 a-33 a^2 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{24 a^5 f}\\ &=\frac {27 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{64 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\text {Subst}\left (\int \frac {111 a^2-81 a^3 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{96 a^6 f}\\ &=\frac {27 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{64 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^4 f}-\frac {91 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{32 a^4 f}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{9/2} f}+\frac {91 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{32 \sqrt {2} a^{9/2} f}+\frac {27 \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{64 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {11 \cos (e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{96 a^4 f \sqrt {a+a \sec (e+f x)}}+\frac {\cos ^2(e+f x) \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sin (e+f x)}{24 a^4 f \sqrt {a+a \sec (e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 24.18, size = 5584, normalized size = 31.55 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Tan[e + f*x]^2/(a + a*Sec[e + f*x])^(9/2),x]

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(723\) vs. \(2(148)=296\).
time = 0.95, size = 724, normalized size = 4.09

method	result	size
default	\(-\frac {\left (192 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+273 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+384 \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+546 \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-384 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-314 \left (\cos ^{5}\left (f x +e \right )\right )-546 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-192 \sin \left (f x +e \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+216 \left (\cos ^{4}\left (f x +e \right )\right )-273 \sin \left (f x +e \right ) \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}+348 \left (\cos ^{3}\left (f x +e \right )\right )-88 \left (\cos ^{2}\left (f x +e \right )\right )-162 \cos \left (f x +e \right )\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}}{192 f \sin \left (f x +e \right )^{3} \left (\cos \left (f x +e \right )+1\right )^{2} a^{5}}\)	\(724\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(f*x+e)^2/(a+a*sec(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/192/f*(192*sin(f*x+e)*cos(f*x+e)^4*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2
^(1/2))*2^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+273*sin(f*x+e)*cos(f*x+e)^4*ln((sin(f*x+e)*(-2*cos(f*x+e)
/(cos(f*x+e)+1))^(1/2)-cos(f*x+e)+1)/sin(f*x+e))*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+384*sin(f*x+e)*cos(f*x+e
)^3*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*2^(1/2)*(-2*cos(f*x+e)/(co
s(f*x+e)+1))^(1/2)+546*sin(f*x+e)*cos(f*x+e)^3*ln((sin(f*x+e)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-cos(f*x+e)+
1)/sin(f*x+e))*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-384*sin(f*x+e)*cos(f*x+e)*arctanh(1/2*(-2*cos(f*x+e)/(cos(
f*x+e)+1))^(1/2)*sin(f*x+e)/cos(f*x+e)*2^(1/2))*2^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-314*cos(f*x+e)^5-
546*sin(f*x+e)*cos(f*x+e)*ln((sin(f*x+e)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-cos(f*x+e)+1)/sin(f*x+e))*(-2*co
s(f*x+e)/(cos(f*x+e)+1))^(1/2)-192*sin(f*x+e)*arctanh(1/2*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)/cos(
f*x+e)*2^(1/2))*2^(1/2)*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+216*cos(f*x+e)^4-273*sin(f*x+e)*ln((sin(f*x+e)*(-
2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-cos(f*x+e)+1)/sin(f*x+e))*(-2*cos(f*x+e)/(cos(f*x+e)+1))^(1/2)+348*cos(f*x+
e)^3-88*cos(f*x+e)^2-162*cos(f*x+e))*(a*(cos(f*x+e)+1)/cos(f*x+e))^(1/2)/sin(f*x+e)^3/(cos(f*x+e)+1)^2/a^5

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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(tan(f*x+e)^2/(a+a*sec(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

integrate(tan(f*x + e)^2/(a*sec(f*x + e) + a)^(9/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (158) = 316\).
time = 3.46, size = 733, normalized size = 4.14 \begin {gather*} \left [-\frac {273 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) + 384 \, {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right ) - 4 \, {\left (157 \, \cos \left (f x + e\right )^{3} + 206 \, \cos \left (f x + e\right )^{2} + 81 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{384 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{5} f \cos \left (f x + e\right )^{3} + 6 \, a^{5} f \cos \left (f x + e\right )^{2} + 4 \, a^{5} f \cos \left (f x + e\right ) + a^{5} f\right )}}, -\frac {273 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 384 \, {\left (\cos \left (f x + e\right )^{4} + 4 \, \cos \left (f x + e\right )^{3} + 6 \, \cos \left (f x + e\right )^{2} + 4 \, \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) - 2 \, {\left (157 \, \cos \left (f x + e\right )^{3} + 206 \, \cos \left (f x + e\right )^{2} + 81 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{192 \, {\left (a^{5} f \cos \left (f x + e\right )^{4} + 4 \, a^{5} f \cos \left (f x + e\right )^{3} + 6 \, a^{5} f \cos \left (f x + e\right )^{2} + 4 \, a^{5} f \cos \left (f x + e\right ) + a^{5} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^2/(a+a*sec(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

[-1/384*(273*sqrt(2)*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(-a)*log(
(2*sqrt(2)*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + 3*a*cos(f*x + e)^2 + 2
*a*cos(f*x + e) - a)/(cos(f*x + e)^2 + 2*cos(f*x + e) + 1)) + 384*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f
*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(-a)*log((2*a*cos(f*x + e)^2 - 2*sqrt(-a)*sqrt((a*cos(f*x + e) + a)/cos(f*
x + e))*cos(f*x + e)*sin(f*x + e) + a*cos(f*x + e) - a)/(cos(f*x + e) + 1)) - 4*(157*cos(f*x + e)^3 + 206*cos(
f*x + e)^2 + 81*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 4*
a^5*f*cos(f*x + e)^3 + 6*a^5*f*cos(f*x + e)^2 + 4*a^5*f*cos(f*x + e) + a^5*f), -1/192*(273*sqrt(2)*(cos(f*x +
e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x + e)^2 + 4*cos(f*x + e) + 1)*sqrt(a)*arctan(sqrt(2)*sqrt((a*cos(f*x + e) +
a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*x + e))) - 384*(cos(f*x + e)^4 + 4*cos(f*x + e)^3 + 6*cos(f*x +
e)^2 + 4*cos(f*x + e) + 1)*sqrt(a)*arctan(sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*cos(f*x + e)/(sqrt(a)*sin(f*
x + e))) - 2*(157*cos(f*x + e)^3 + 206*cos(f*x + e)^2 + 81*cos(f*x + e))*sqrt((a*cos(f*x + e) + a)/cos(f*x + e
))*sin(f*x + e))/(a^5*f*cos(f*x + e)^4 + 4*a^5*f*cos(f*x + e)^3 + 6*a^5*f*cos(f*x + e)^2 + 4*a^5*f*cos(f*x + e
) + a^5*f)]

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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(tan(f*x+e)**2/(a+a*sec(f*x+e))**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3878 deep

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Giac [A]
time = 1.38, size = 109, normalized size = 0.62 \begin {gather*} \frac {\sqrt {-a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a} {\left (2 \, {\left (\frac {4 \, \sqrt {2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} - \frac {19 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \frac {111 \, \sqrt {2}}{a^{5} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}\right )} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{192 \, f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(f*x+e)^2/(a+a*sec(f*x+e))^(9/2),x, algorithm="giac")

[Out]

1/192*sqrt(-a*tan(1/2*f*x + 1/2*e)^2 + a)*(2*(4*sqrt(2)*tan(1/2*f*x + 1/2*e)^2/(a^5*sgn(cos(f*x + e))) - 19*sq
rt(2)/(a^5*sgn(cos(f*x + e))))*tan(1/2*f*x + 1/2*e)^2 + 111*sqrt(2)/(a^5*sgn(cos(f*x + e))))*tan(1/2*f*x + 1/2
*e)/f

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e + f*x)^2/(a + a/cos(e + f*x))^(9/2),x)

[Out]

int(tan(e + f*x)^2/(a + a/cos(e + f*x))^(9/2), x)

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