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

Optimal. Leaf size=123 \[ \frac{\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac{\left (3 a^2+24 a b+8 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac{a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac{b^2 \sec ^3(e+f x)}{3 f} \]

[Out]

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

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Rubi [A]  time = 0.129721, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3664, 463, 455, 1153, 207} \[ \frac{\left (a^2+8 a b+4 b^2\right ) \sec (e+f x)}{4 f}-\frac{\left (3 a^2+24 a b+8 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac{a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac{b^2 \sec ^3(e+f x)}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((3*a^2 + 24*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/(8*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 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 463

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 455

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 1153

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 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) \left (a+b \tan ^2(e+f x)\right )^2 \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 ) \operatorname{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 ) \tanh ^{-1}(\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]  time = 6.18752, size = 447, normalized size = 3.63 \[ \frac{\left (3 a^2+24 a b+8 b^2\right ) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )\right )}{8 f}+\frac{\left (-3 a^2-24 a b-8 b^2\right ) \log \left (\cos \left (\frac{1}{2} (e+f x)\right )\right )}{8 f}+\frac{\left (-3 a^2-8 a b\right ) \csc ^2\left (\frac{1}{2} (e+f x)\right )}{32 f}+\frac{\left (3 a^2+8 a b\right ) \sec ^2\left (\frac{1}{2} (e+f x)\right )}{32 f}-\frac{a^2 \csc ^4\left (\frac{1}{2} (e+f x)\right )}{64 f}+\frac{a^2 \sec ^4\left (\frac{1}{2} (e+f x)\right )}{64 f}+\frac{-12 a b \sin \left (\frac{1}{2} (e+f x)\right )-7 b^2 \sin \left (\frac{1}{2} (e+f x)\right )}{6 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}+\frac{12 a b \sin \left (\frac{1}{2} (e+f x)\right )+7 b^2 \sin \left (\frac{1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}+\frac{b^2 \sin \left (\frac{1}{2} (e+f x)\right )}{6 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{b^2}{12 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}+\frac{b^2}{12 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2}-\frac{b^2 \sin \left (\frac{1}{2} (e+f x)\right )}{6 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]

Warning: Unable to verify antiderivative.

[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]))

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Maple [A]  time = 0.058, size = 183, normalized size = 1.5 \begin{align*}{\frac{{b}^{2}}{3\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}}}+{\frac{{b}^{2}}{f\cos \left ( fx+e \right ) }}+{\frac{{b}^{2}\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{f}}-{\frac{ab}{f \left ( \sin \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) }}+3\,{\frac{ab}{f\cos \left ( fx+e \right ) }}+3\,{\frac{ab\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{f}}-{\frac{{a}^{2}\cot \left ( fx+e \right ) \left ( \csc \left ( fx+e \right ) \right ) ^{3}}{4\,f}}-{\frac{3\,{a}^{2}\csc \left ( fx+e \right ) \cot \left ( fx+e \right ) }{8\,f}}+{\frac{3\,{a}^{2}\ln \left ( \csc \left ( fx+e \right ) -\cot \left ( fx+e \right ) \right ) }{8\,f}} \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)^2,x)

[Out]

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

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Maxima [A]  time = 1.01137, size = 220, normalized size = 1.79 \begin{align*} -\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} \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)^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

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Fricas [B]  time = 2.15241, size = 699, normalized size = 5.68 \begin{align*} \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 )}} \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)^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)

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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)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.68271, size = 564, normalized size = 4.59 \begin{align*} -\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{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\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} \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)^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(-(cos(f*x + e) - 1)/(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*(c
os(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