\(\int \csc ^5(e+f x) (a+b \tan ^2(e+f x))^2 \, dx\) [48]

Optimal result

Integrand size = 23, antiderivative size = 123 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=-\frac {\left (3 a^2+24 a b+8 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}+\frac {b^2 \sec ^3(e+f x)}{3 f} \]

[Out]

-1/8*(3*a^2+24*a*b+8*b^2)*arctanh(cos(f*x+e))/f-1/8*a*(a+8*b)*cot(f*x+e)*csc(f*x+e)/f+1/4*(a^2+8*a*b+4*b^2)*se
c(f*x+e)/f-1/4*a^2*csc(f*x+e)^4*sec(f*x+e)/f+1/3*b^2*sec(f*x+e)^3/f

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3745, 474, 466, 1167, 213} \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=-\frac {\left (3 a^2+24 a b+8 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}+\frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {b^2 \sec ^3(e+f x)}{3 f} \]

[In]

Int[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]

[Out]

-1/8*((3*a^2 + 24*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/f - (a*(a + 8*b)*Cot[e + f*x]*Csc[e + f*x])/(8*f) + ((a^
2 + 8*a*b + 4*b^2)*Sec[e + f*x])/(4*f) - (a^2*Csc[e + f*x]^4*Sec[e + f*x])/(4*f) + (b^2*Sec[e + f*x]^3)/(3*f)

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 474

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^2, x_Symbol] :> Simp[(-(b*c - a*
d)^2)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b^2*e*n*(p + 1))), x] + Dist[1/(a*b^2*n*(p + 1)), Int[(e*x)^m*(a +
b*x^n)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + b^2*c^2*n*(p + 1) + a*b*d^2*n*(p + 1)*x^n, x], x], x] /; FreeQ[{a
, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 3745

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sec[e + f*x], x]}, Dist[1/(f*ff^m), Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m
+ 1)), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4 \left (a-b+b x^2\right )^2}{\left (-1+x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f} \\ & = -\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}+\frac {\text {Subst}\left (\int \frac {x^4 \left (a^2+8 a b-4 b^2+4 b^2 x^2\right )}{\left (-1+x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{4 f} \\ & = -\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac {\text {Subst}\left (\int \frac {-a (a+8 b)-2 a (a+8 b) x^2-8 b^2 x^4}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 f} \\ & = -\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac {\text {Subst}\left (\int \left (-2 \left (a^2+8 a b+4 b^2\right )-8 b^2 x^2+\frac {-3 a^2-24 a b-8 b^2}{-1+x^2}\right ) \, dx,x,\sec (e+f x)\right )}{8 f} \\ & = -\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}+\frac {b^2 \sec ^3(e+f x)}{3 f}+\frac {\left (3 a^2+24 a b+8 b^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 f} \\ & = -\frac {\left (3 a^2+24 a b+8 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac {\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac {a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}+\frac {b^2 \sec ^3(e+f x)}{3 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(447\) vs. \(2(123)=246\).

Time = 7.51 (sec) , antiderivative size = 447, normalized size of antiderivative = 3.63 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\frac {\left (-3 a^2-8 a b\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}-\frac {a^2 \csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {\left (-3 a^2-24 a b-8 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {\left (3 a^2+24 a b+8 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {\left (3 a^2+8 a b\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {a^2 \sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {b^2}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {b^2}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {-12 a b \sin \left (\frac {1}{2} (e+f x)\right )-7 b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {12 a b \sin \left (\frac {1}{2} (e+f x)\right )+7 b^2 \sin \left (\frac {1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

Integrate[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]

[Out]

((-3*a^2 - 8*a*b)*Csc[(e + f*x)/2]^2)/(32*f) - (a^2*Csc[(e + f*x)/2]^4)/(64*f) + ((-3*a^2 - 24*a*b - 8*b^2)*Lo
g[Cos[(e + f*x)/2]])/(8*f) + ((3*a^2 + 24*a*b + 8*b^2)*Log[Sin[(e + f*x)/2]])/(8*f) + ((3*a^2 + 8*a*b)*Sec[(e
+ f*x)/2]^2)/(32*f) + (a^2*Sec[(e + f*x)/2]^4)/(64*f) + b^2/(12*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (
b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*
x)/2] + Sin[(e + f*x)/2])^3) + b^2/(12*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) + (-12*a*b*Sin[(e + f*x)/2]
- 7*b^2*Sin[(e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) + (12*a*b*Sin[(e + f*x)/2] + 7*b^2*Sin[(
e + f*x)/2])/(6*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {a^{2} \left (\left (-\frac {\csc \left (f x +e \right )^{3}}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\frac {1}{2 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {3}{2 \cos \left (f x +e \right )}+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+b^{2} \left (\frac {1}{3 \cos \left (f x +e \right )^{3}}+\frac {1}{\cos \left (f x +e \right )}+\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )\right )}{f}\)	\(145\)
default	\(\frac {a^{2} \left (\left (-\frac {\csc \left (f x +e \right )^{3}}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{8}\right )+2 a b \left (-\frac {1}{2 \sin \left (f x +e \right )^{2} \cos \left (f x +e \right )}+\frac {3}{2 \cos \left (f x +e \right )}+\frac {3 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+b^{2} \left (\frac {1}{3 \cos \left (f x +e \right )^{3}}+\frac {1}{\cos \left (f x +e \right )}+\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )\right )}{f}\)	\(145\)
risch	\(\frac {{\mathrm e}^{i \left (f x +e \right )} \left (9 a^{2} {\mathrm e}^{12 i \left (f x +e \right )}+72 a b \,{\mathrm e}^{12 i \left (f x +e \right )}+24 b^{2} {\mathrm e}^{12 i \left (f x +e \right )}-6 a^{2} {\mathrm e}^{10 i \left (f x +e \right )}-48 a b \,{\mathrm e}^{10 i \left (f x +e \right )}-16 b^{2} {\mathrm e}^{10 i \left (f x +e \right )}-105 a^{2} {\mathrm e}^{8 i \left (f x +e \right )}-72 a b \,{\mathrm e}^{8 i \left (f x +e \right )}-152 b^{2} {\mathrm e}^{8 i \left (f x +e \right )}-180 a^{2} {\mathrm e}^{6 i \left (f x +e \right )}+96 a b \,{\mathrm e}^{6 i \left (f x +e \right )}+288 b^{2} {\mathrm e}^{6 i \left (f x +e \right )}-105 a^{2} {\mathrm e}^{4 i \left (f x +e \right )}-72 a b \,{\mathrm e}^{4 i \left (f x +e \right )}-152 b^{2} {\mathrm e}^{4 i \left (f x +e \right )}-6 a^{2} {\mathrm e}^{2 i \left (f x +e \right )}-48 a b \,{\mathrm e}^{2 i \left (f x +e \right )}-16 b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 a^{2}+72 a b +24 b^{2}\right )}{12 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b}{f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{f}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b}{f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{f}\)	\(460\)

[In]

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

[Out]

1/f*(a^2*((-1/4*csc(f*x+e)^3-3/8*csc(f*x+e))*cot(f*x+e)+3/8*ln(csc(f*x+e)-cot(f*x+e)))+2*a*b*(-1/2/sin(f*x+e)^
2/cos(f*x+e)+3/2/cos(f*x+e)+3/2*ln(csc(f*x+e)-cot(f*x+e)))+b^2*(1/3/cos(f*x+e)^3+1/cos(f*x+e)+ln(csc(f*x+e)-co
t(f*x+e))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (113) = 226\).

Time = 0.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.31 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\frac {6 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - 10 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 16 \, {\left (6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 16 \, b^{2} - 3 \, {\left ({\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{7} - 2 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{5} + {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right )}{48 \, {\left (f \cos \left (f x + e\right )^{7} - 2 \, f \cos \left (f x + e\right )^{5} + f \cos \left (f x + e\right )^{3}\right )}} \]

[In]

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

[Out]

1/48*(6*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^6 - 10*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^4 + 16*(6*a*b + b^2
)*cos(f*x + e)^2 + 16*b^2 - 3*((3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^7 - 2*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x +
e)^5 + (3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^3)*log(1/2*cos(f*x + e) + 1/2) + 3*((3*a^2 + 24*a*b + 8*b^2)*cos(
f*x + e)^7 - 2*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^5 + (3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^3)*log(-1/2*cos
(f*x + e) + 1/2))/(f*cos(f*x + e)^7 - 2*f*cos(f*x + e)^5 + f*cos(f*x + e)^3)

Sympy [F]

\[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{2} \csc ^{5}{\left (e + f x \right )}\, dx \]

[In]

integrate(csc(f*x+e)**5*(a+b*tan(f*x+e)**2)**2,x)

[Out]

Integral((a + b*tan(e + f*x)**2)**2*csc(e + f*x)**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.33 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=-\frac {3 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) + 1\right ) - 3 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \log \left (\cos \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - 5 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 8 \, b^{2}\right )}}{\cos \left (f x + e\right )^{7} - 2 \, \cos \left (f x + e\right )^{5} + \cos \left (f x + e\right )^{3}}}{48 \, f} \]

[In]

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

[Out]

-1/48*(3*(3*a^2 + 24*a*b + 8*b^2)*log(cos(f*x + e) + 1) - 3*(3*a^2 + 24*a*b + 8*b^2)*log(cos(f*x + e) - 1) - 2
*(3*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^6 - 5*(3*a^2 + 24*a*b + 8*b^2)*cos(f*x + e)^4 + 8*(6*a*b + b^2)*cos(
f*x + e)^2 + 8*b^2)/(cos(f*x + e)^7 - 2*cos(f*x + e)^5 + cos(f*x + e)^3))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (113) = 226\).

Time = 0.69 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.18 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=-\frac {\frac {24 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {48 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {3 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 12 \, {\left (3 \, a^{2} + 24 \, a b + 8 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right ) + \frac {3 \, {\left (a^{2} - \frac {8 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {16 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {144 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {48 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) - 1\right )}^{2}} - \frac {256 \, {\left (3 \, a b + 2 \, b^{2} + \frac {6 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {3 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 1\right )}^{3}}}{192 \, f} \]

[In]

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

[Out]

-1/192*(24*a^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 48*a*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 3*a^2*(c
os(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 - 12*(3*a^2 + 24*a*b + 8*b^2)*log(abs(-cos(f*x + e) + 1)/abs(cos(f*x +
e) + 1)) + 3*(a^2 - 8*a^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) - 16*a*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1
) + 18*a^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 144*a*b*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2 + 48*
b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)*(cos(f*x + e) + 1)^2/(cos(f*x + e) - 1)^2 - 256*(3*a*b + 2*b^2
+ 6*a*b*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 3*b^2*(cos(f*x + e) - 1)/(cos(f*x + e) + 1) + 3*a*b*(cos(f*x +
e) - 1)^2/(cos(f*x + e) + 1)^2 + 3*b^2*(cos(f*x + e) - 1)^2/(cos(f*x + e) + 1)^2)/((cos(f*x + e) - 1)/(cos(f*
x + e) + 1) + 1)^3)/f

Mupad [B] (verification not implemented)

Time = 10.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.98 \[ \int \csc ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{64\,f}+\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{8}+3\,a\,b+b^2\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {5\,a^2}{4}+4\,b\,a\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (2\,a^2+68\,a\,b+64\,b^2\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {21\,a^2}{4}+76\,a\,b+\frac {128\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {23\,a^2}{4}+140\,a\,b+64\,b^2\right )+\frac {a^2}{4}}{f\,\left (-16\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-48\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^2}{8}+\frac {b\,a}{4}\right )}{f} \]

[In]

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

[Out]

(a^2*tan(e/2 + (f*x)/2)^4)/(64*f) + (log(tan(e/2 + (f*x)/2))*(3*a*b + (3*a^2)/8 + b^2))/f - (tan(e/2 + (f*x)/2
)^2*(4*a*b + (5*a^2)/4) - tan(e/2 + (f*x)/2)^8*(68*a*b + 2*a^2 + 64*b^2) - tan(e/2 + (f*x)/2)^4*(76*a*b + (21*
a^2)/4 + (128*b^2)/3) + tan(e/2 + (f*x)/2)^6*(140*a*b + (23*a^2)/4 + 64*b^2) + a^2/4)/(f*(16*tan(e/2 + (f*x)/2
)^4 - 48*tan(e/2 + (f*x)/2)^6 + 48*tan(e/2 + (f*x)/2)^8 - 16*tan(e/2 + (f*x)/2)^10)) + (tan(e/2 + (f*x)/2)^2*(
(a*b)/4 + a^2/8))/f