3.97 \(\int \csc ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx\)

Optimal. Leaf size=187 \[ -\frac{\left (3 a^2+6 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 f}-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a f} \]

[Out]

-((3*a^2 + 6*a*b - b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(3/2)*f) + (Sqrt[
b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - ((3*a + b)*Cot[e + f*x]*Csc[e + f*x]*Sq
rt[a - b + b*Sec[e + f*x]^2])/(8*a*f) - (Cot[e + f*x]*Csc[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(4*f)

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Rubi [A]  time = 0.232347, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3664, 467, 578, 523, 217, 206, 377, 207} \[ -\frac{\left (3 a^2+6 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right )}{f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 f}-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*a^2 + 6*a*b - b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(3/2)*f) + (Sqrt[
b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - ((3*a + b)*Cot[e + f*x]*Csc[e + f*x]*Sq
rt[a - b + b*Sec[e + f*x]^2])/(8*a*f) - (Cot[e + f*x]*Csc[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(4*f)

Rule 3664

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]

Rule 467

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)/(b*n*(p + 1)), x] - Dist[e^n/(b*n*(p + 1)), Int[(e*x)^
(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m - n + 1) + d*(m + n*(q - 1) + 1)*x^n, x], x], x] /;
FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 0] && GtQ[m - n + 1, 0] &
& IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 578

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c -
a*d)*(p + 1)), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rule 523

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 207

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

Rubi steps

\begin{align*} \int \csc ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4 \sqrt{a-b+b x^2}}{\left (-1+x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{4 f}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 (a-b)+4 b x^2\right )}{\left (-1+x^2\right )^2 \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{4 f}\\ &=-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{8 a f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{4 f}+\frac{\operatorname{Subst}\left (\int \frac{(a-b) (3 a+b)+8 a b x^2}{\left (-1+x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 a f}\\ &=-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{8 a f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{4 f}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}+\frac{\left (3 a^2+6 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-1+x^2\right ) \sqrt{a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{8 a f}\\ &=-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{8 a f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{4 f}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{f}+\frac{\left (3 a^2+6 a b-b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+a x^2} \, dx,x,\frac{\sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{8 a f}\\ &=-\frac{\left (3 a^2+6 a b-b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b+b \sec ^2(e+f x)}}\right )}{f}-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{8 a f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a-b+b \sec ^2(e+f x)}}{4 f}\\ \end{align*}

Mathematica [B]  time = 6.53037, size = 1059, normalized size = 5.66 \[ \frac{\sqrt{\frac{\cos (2 (e+f x)) a+a+b-b \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\frac{(-3 a \cos (e+f x)-b \cos (e+f x)) \csc ^2(e+f x)}{8 a}-\frac{1}{4} \cot (e+f x) \csc ^3(e+f x)\right )}{f}+\frac{\frac{\left (3 a^2-2 b a-b^2\right ) \left (2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )}{\sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}\right )+\sqrt{b} \left (\tanh ^{-1}\left (\frac{-a \tan ^2\left (\frac{1}{2} (e+f x)\right )+2 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a}{\sqrt{a} \sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}\right )+\tanh ^{-1}\left (\frac{2 b+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )}{\sqrt{a} \sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}\right )\right )\right ) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right ) \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right ) \sqrt{\frac{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}{\left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2} \sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}-\frac{\left (3 a^2+14 b a-b^2\right ) \left (2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{b} \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )}{\sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}\right )-\sqrt{b} \left (\tanh ^{-1}\left (\frac{-a \tan ^2\left (\frac{1}{2} (e+f x)\right )+2 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a}{\sqrt{a} \sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}\right )+\tanh ^{-1}\left (\frac{2 b+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )}{\sqrt{a} \sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}\right )\right )\right ) (\cos (e+f x)+1) \sqrt{\frac{\cos (2 (e+f x))+1}{(\cos (e+f x)+1)^2}} \sqrt{\frac{a+b+(a-b) \cos (2 (e+f x))}{\cos (2 (e+f x))+1}} \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right ) \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right ) \sqrt{\frac{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}{\left (\tan ^2\left (\frac{1}{2} (e+f x)\right )+1\right )^2}}}{4 \sqrt{a} \sqrt{b} \sqrt{a+b+(a-b) \cos (2 (e+f x))} \sqrt{\left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2} \sqrt{4 b \tan ^2\left (\frac{1}{2} (e+f x)\right )+a \left (\tan ^2\left (\frac{1}{2} (e+f x)\right )-1\right )^2}}}{8 a f} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[(a + b + a*Cos[2*(e + f*x)] - b*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(((-3*a*Cos[e + f*x] - b*Cos[e
+ f*x])*Csc[e + f*x]^2)/(8*a) - (Cot[e + f*x]*Csc[e + f*x]^3)/4))/f + (-((3*a^2 + 14*a*b - b^2)*(2*Sqrt[a]*Ar
cTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] - Sqrt[
b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 +
Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 +
a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e + f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[
(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*
Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(4*Sqrt[a]*Sqrt[b]*
Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1 + Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 +
Tan[(e + f*x)/2]^2)^2]) + ((3*a^2 - 2*a*b - b^2)*(2*Sqrt[a]*ArcTanh[(Sqrt[b]*(1 + Tan[(e + f*x)/2]^2))/Sqrt[4
*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]] + Sqrt[b]*(ArcTanh[(a - a*Tan[(e + f*x)/2]^2 + 2*b*Tan
[(e + f*x)/2]^2)/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])] + ArcTanh[(2*b + a*(-
1 + Tan[(e + f*x)/2]^2))/(Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2])]))*(1 + Cos[e
+ f*x])*Sqrt[(1 + Cos[2*(e + f*x)])/(1 + Cos[e + f*x])^2]*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])/(1 + Cos[2*(
e + f*x)])]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e +
f*x)/2]^2)^2)/(1 + Tan[(e + f*x)/2]^2)^2])/(4*Sqrt[a]*Sqrt[b]*Sqrt[a + b + (a - b)*Cos[2*(e + f*x)]]*Sqrt[(-1
+ Tan[(e + f*x)/2]^2)^2]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]))/(8*a*f)

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Maple [B]  time = 0.296, size = 5378, normalized size = 28.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tan \left (f x + e\right )^{2} + a} \csc \left (f x + e\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 8.87839, size = 3058, normalized size = 16.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/16*(((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*
sqrt(a)*log(-2*((a - b)*cos(f*x + e)^2 + 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x +
e) + a + b)/(cos(f*x + e)^2 - 1)) - 8*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(b)*log(-((a - b)
*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos(f*x + e)
^2) - 2*((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x
+ e)^2))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), 1/8*(((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 -
2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt(((a - b)*cos(f*x
+ e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) + 4*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f*x + e)^2 + a^2)*sqrt(b)*log(
-((a - b)*cos(f*x + e)^2 + 2*sqrt(b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + 2*b)/cos
(f*x + e)^2) + ((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/c
os(f*x + e)^2))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), -1/16*(16*(a^2*cos(f*x + e)^4 - 2*a^2
*cos(f*x + e)^2 + a^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)
/b) + ((3*a^2 + 6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*sq
rt(a)*log(-2*((a - b)*cos(f*x + e)^2 + 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e
) + a + b)/(cos(f*x + e)^2 - 1)) - 2*((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)
*cos(f*x + e)^2 + b)/cos(f*x + e)^2))/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f), 1/8*(((3*a^2 +
6*a*b - b^2)*cos(f*x + e)^4 - 2*(3*a^2 + 6*a*b - b^2)*cos(f*x + e)^2 + 3*a^2 + 6*a*b - b^2)*sqrt(-a)*arctan(sq
rt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/a) - 8*(a^2*cos(f*x + e)^4 - 2*a^2*cos(f
*x + e)^2 + a^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*cos(f*x + e)/b) +
((3*a^2 + a*b)*cos(f*x + e)^3 - (5*a^2 + a*b)*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2))
/(a^2*f*cos(f*x + e)^4 - 2*a^2*f*cos(f*x + e)^2 + a^2*f)]

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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(csc(f*x+e)**5*(a+b*tan(f*x+e)**2)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \tan \left (f x + e\right )^{2} + a} \csc \left (f x + e\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^5*(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

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