3.65 \(\int \frac{\csc ^4(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=164 \[ -\frac{5 \sqrt{b} (3 a-4 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 f (a+b)^{9/2}}-\frac{b (7 a-4 b) \tan (e+f x)}{8 f (a+b)^4 \left (a+b \tan ^2(e+f x)+b\right )}-\frac{a b \tan (e+f x)}{4 f (a+b)^3 \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac{\cot ^3(e+f x)}{3 f (a+b)^3}-\frac{(a-2 b) \cot (e+f x)}{f (a+b)^4} \]

[Out]

(-5*(3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) - ((a - 2*b)*Cot[e + f
*x])/((a + b)^4*f) - Cot[e + f*x]^3/(3*(a + b)^3*f) - (a*b*Tan[e + f*x])/(4*(a + b)^3*f*(a + b + b*Tan[e + f*x
]^2)^2) - ((7*a - 4*b)*b*Tan[e + f*x])/(8*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2))

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Rubi [A]  time = 0.25045, antiderivative size = 164, 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 = {4132, 456, 1259, 1261, 205} \[ -\frac{5 \sqrt{b} (3 a-4 b) \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 f (a+b)^{9/2}}-\frac{b (7 a-4 b) \tan (e+f x)}{8 f (a+b)^4 \left (a+b \tan ^2(e+f x)+b\right )}-\frac{a b \tan (e+f x)}{4 f (a+b)^3 \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac{\cot ^3(e+f x)}{3 f (a+b)^3}-\frac{(a-2 b) \cot (e+f x)}{f (a+b)^4} \]

Antiderivative was successfully verified.

[In]

Int[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

(-5*(3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*(a + b)^(9/2)*f) - ((a - 2*b)*Cot[e + f
*x])/((a + b)^4*f) - Cot[e + f*x]^3/(3*(a + b)^3*f) - (a*b*Tan[e + f*x])/(4*(a + b)^3*f*(a + b + b*Tan[e + f*x
]^2)^2) - ((7*a - 4*b)*b*Tan[e + f*x])/(8*(a + b)^4*f*(a + b + b*Tan[e + f*x]^2))

Rule 4132

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = Fr
eeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p)/(
1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Integer
Q[n/2]

Rule 456

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[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1259

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[((-d)^(
m/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*(d + e*x^2)^(q + 1))/(2*e^(2*p + m/2)*(q + 1)), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1*(2*(-d)^(-(m/2) + 1)*e^(2*p)*(q + 1)*
(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2)))/(d + e*x^2)], x], x]
, x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1261

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\csc ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{x^4 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{a b \tan (e+f x)}{4 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{b \operatorname{Subst}\left (\int \frac{-\frac{4}{b (a+b)}-\frac{4 a x^2}{b (a+b)^2}+\frac{3 a x^4}{(a+b)^3}}{x^4 \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac{a b \tan (e+f x)}{4 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{(7 a-4 b) b \tan (e+f x)}{8 (a+b)^4 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-8 b (a+b)-8 (a-b) b x^2+\frac{(7 a-4 b) b^2 x^4}{a+b}}{x^4 \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{8 b (a+b)^3 f}\\ &=-\frac{a b \tan (e+f x)}{4 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{(7 a-4 b) b \tan (e+f x)}{8 (a+b)^4 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{8 b}{x^4}+\frac{8 b (-a+2 b)}{(a+b) x^2}+\frac{5 (3 a-4 b) b^2}{(a+b) \left (a+b+b x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{8 b (a+b)^3 f}\\ &=-\frac{(a-2 b) \cot (e+f x)}{(a+b)^4 f}-\frac{\cot ^3(e+f x)}{3 (a+b)^3 f}-\frac{a b \tan (e+f x)}{4 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{(7 a-4 b) b \tan (e+f x)}{8 (a+b)^4 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac{(5 (3 a-4 b) b) \operatorname{Subst}\left (\int \frac{1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 (a+b)^4 f}\\ &=-\frac{5 (3 a-4 b) \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b}}\right )}{8 (a+b)^{9/2} f}-\frac{(a-2 b) \cot (e+f x)}{(a+b)^4 f}-\frac{\cot ^3(e+f x)}{3 (a+b)^3 f}-\frac{a b \tan (e+f x)}{4 (a+b)^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac{(7 a-4 b) b \tan (e+f x)}{8 (a+b)^4 f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end{align*}

Mathematica [C]  time = 4.42902, size = 994, normalized size = 6.06 \[ \frac{(\cos (2 (e+f x)) a+a+2 b) \sec ^6(e+f x) \left (\frac{480 (3 a-4 b) b \tan ^{-1}\left (\frac{\sec (f x) (\cos (2 e)-i \sin (2 e)) (a \sin (2 e+f x)-(a+2 b) \sin (f x))}{2 \sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 (e+f x)) a+a+2 b)^2 (\cos (2 e)-i \sin (2 e))}{\sqrt{a+b} \sqrt{b (\cos (e)-i \sin (e))^4}}-\frac{\csc (e) \csc ^3(e+f x) \sec (2 e) \left (224 \sin (2 e-f x) a^4-224 \sin (2 e+f x) a^4+176 \sin (4 e+f x) a^4+48 \sin (2 e+3 f x) a^4-96 \sin (4 e+3 f x) a^4+48 \sin (6 e+3 f x) a^4+16 \sin (2 e+5 f x) a^4+16 \sin (6 e+5 f x) a^4+16 \sin (4 e+7 f x) a^4+16 \sin (8 e+7 f x) a^4+576 b \sin (2 e-f x) a^3-657 b \sin (2 e+f x) a^3+569 b \sin (4 e+f x) a^3+111 b \sin (2 e+3 f x) a^3-152 b \sin (4 e+3 f x) a^3+192 b \sin (6 e+3 f x) a^3+72 b \sin (4 e+5 f x) a^3+27 b \sin (6 e+5 f x) a^3+45 b \sin (8 e+5 f x) a^3-83 b \sin (4 e+7 f x) a^3+27 b \sin (6 e+7 f x) a^3-56 b \sin (8 e+7 f x) a^3+124 b^2 \sin (2 e-f x) a^2-538 b^2 \sin (2 e+f x) a^2+666 b^2 \sin (4 e+f x) a^2+360 b^2 \sin (2 e+3 f x) a^2+146 b^2 \sin (4 e+3 f x) a^2+558 b^2 \sin (6 e+3 f x) a^2-598 b^2 \sin (2 e+5 f x) a^2+150 b^2 \sin (4 e+5 f x) a^2-388 b^2 \sin (6 e+5 f x) a^2-60 b^2 \sin (8 e+5 f x) a^2+6 b^2 \sin (4 e+7 f x) a^2-6 b^2 \sin (6 e+7 f x) a^2-2184 b^3 \sin (2 e-f x) a+984 b^3 \sin (2 e+f x) a+1704 b^3 \sin (4 e+f x) a+312 b^3 \sin (2 e+3 f x) a-728 b^3 \sin (4 e+3 f x) a-168 b^3 \sin (6 e+3 f x) a+48 b^3 \sin (2 e+5 f x) a-48 b^3 \sin (4 e+5 f x) a+4 \left (44 a^4+122 b a^3+63 b^2 a^2+126 b^3 a+36 b^4\right ) \sin (f x)+\left (-96 a^4-71 b a^3+344 b^2 a^2-1208 b^3 a+48 b^4\right ) \sin (3 f x)+144 b^4 \sin (2 e-f x)+144 b^4 \sin (2 e+f x)-144 b^4 \sin (4 e+f x)-48 b^4 \sin (2 e+3 f x)-48 b^4 \sin (4 e+3 f x)+48 b^4 \sin (6 e+3 f x)\right )}{a}\right )}{6144 (a+b)^4 f \left (b \sec ^2(e+f x)+a\right )^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[e + f*x]^4/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

((a + 2*b + a*Cos[2*(e + f*x)])*Sec[e + f*x]^6*((480*(3*a - 4*b)*b*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-
((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(a + 2*b + a*Cos[2*(e
+ f*x)])^2*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) - (Csc[e]*Csc[e + f*x]^3*Sec[
2*e]*(4*(44*a^4 + 122*a^3*b + 63*a^2*b^2 + 126*a*b^3 + 36*b^4)*Sin[f*x] + (-96*a^4 - 71*a^3*b + 344*a^2*b^2 -
1208*a*b^3 + 48*b^4)*Sin[3*f*x] + 224*a^4*Sin[2*e - f*x] + 576*a^3*b*Sin[2*e - f*x] + 124*a^2*b^2*Sin[2*e - f*
x] - 2184*a*b^3*Sin[2*e - f*x] + 144*b^4*Sin[2*e - f*x] - 224*a^4*Sin[2*e + f*x] - 657*a^3*b*Sin[2*e + f*x] -
538*a^2*b^2*Sin[2*e + f*x] + 984*a*b^3*Sin[2*e + f*x] + 144*b^4*Sin[2*e + f*x] + 176*a^4*Sin[4*e + f*x] + 569*
a^3*b*Sin[4*e + f*x] + 666*a^2*b^2*Sin[4*e + f*x] + 1704*a*b^3*Sin[4*e + f*x] - 144*b^4*Sin[4*e + f*x] + 48*a^
4*Sin[2*e + 3*f*x] + 111*a^3*b*Sin[2*e + 3*f*x] + 360*a^2*b^2*Sin[2*e + 3*f*x] + 312*a*b^3*Sin[2*e + 3*f*x] -
48*b^4*Sin[2*e + 3*f*x] - 96*a^4*Sin[4*e + 3*f*x] - 152*a^3*b*Sin[4*e + 3*f*x] + 146*a^2*b^2*Sin[4*e + 3*f*x]
- 728*a*b^3*Sin[4*e + 3*f*x] - 48*b^4*Sin[4*e + 3*f*x] + 48*a^4*Sin[6*e + 3*f*x] + 192*a^3*b*Sin[6*e + 3*f*x]
+ 558*a^2*b^2*Sin[6*e + 3*f*x] - 168*a*b^3*Sin[6*e + 3*f*x] + 48*b^4*Sin[6*e + 3*f*x] + 16*a^4*Sin[2*e + 5*f*x
] - 598*a^2*b^2*Sin[2*e + 5*f*x] + 48*a*b^3*Sin[2*e + 5*f*x] + 72*a^3*b*Sin[4*e + 5*f*x] + 150*a^2*b^2*Sin[4*e
+ 5*f*x] - 48*a*b^3*Sin[4*e + 5*f*x] + 16*a^4*Sin[6*e + 5*f*x] + 27*a^3*b*Sin[6*e + 5*f*x] - 388*a^2*b^2*Sin[
6*e + 5*f*x] + 45*a^3*b*Sin[8*e + 5*f*x] - 60*a^2*b^2*Sin[8*e + 5*f*x] + 16*a^4*Sin[4*e + 7*f*x] - 83*a^3*b*Si
n[4*e + 7*f*x] + 6*a^2*b^2*Sin[4*e + 7*f*x] + 27*a^3*b*Sin[6*e + 7*f*x] - 6*a^2*b^2*Sin[6*e + 7*f*x] + 16*a^4*
Sin[8*e + 7*f*x] - 56*a^3*b*Sin[8*e + 7*f*x]))/a))/(6144*(a + b)^4*f*(a + b*Sec[e + f*x]^2)^3)

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Maple [B]  time = 0.122, size = 306, normalized size = 1.9 \begin{align*} -{\frac{1}{3\,f \left ( a+b \right ) ^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}}-{\frac{a}{f \left ( a+b \right ) ^{4}\tan \left ( fx+e \right ) }}+2\,{\frac{b}{f \left ( a+b \right ) ^{4}\tan \left ( fx+e \right ) }}-{\frac{7\,{b}^{2} \left ( \tan \left ( fx+e \right ) \right ) ^{3}a}{8\,f \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{{b}^{3} \left ( \tan \left ( fx+e \right ) \right ) ^{3}}{2\,f \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{9\,b\tan \left ( fx+e \right ){a}^{2}}{8\,f \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{5\,{b}^{2}\tan \left ( fx+e \right ) a}{8\,f \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}+{\frac{{b}^{3}\tan \left ( fx+e \right ) }{2\,f \left ( a+b \right ) ^{4} \left ( a+b+b \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{15\,ab}{8\,f \left ( a+b \right ) ^{4}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}}+{\frac{5\,{b}^{2}}{2\,f \left ( a+b \right ) ^{4}}\arctan \left ({b\tan \left ( fx+e \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x)

[Out]

-1/3/f/(a+b)^3/tan(f*x+e)^3-1/f/(a+b)^4/tan(f*x+e)*a+2/f/(a+b)^4/tan(f*x+e)*b-7/8/f/(a+b)^4*b^2/(a+b+b*tan(f*x
+e)^2)^2*tan(f*x+e)^3*a+1/2/f/(a+b)^4*b^3/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)^3-9/8/f/(a+b)^4*b/(a+b+b*tan(f*x+e
)^2)^2*tan(f*x+e)*a^2-5/8/f/(a+b)^4*b^2/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)*a+1/2/f/(a+b)^4*b^3/(a+b+b*tan(f*x+e
)^2)^2*tan(f*x+e)-15/8/f/(a+b)^4*b/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(1/2))*a+5/2/f/(a+b)^4*b^2/((
a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.787752, size = 2288, normalized size = 13.95 \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)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[-1/96*(4*(16*a^3 - 83*a^2*b + 6*a*b^2)*cos(f*x + e)^7 - 4*(24*a^3 - 134*a^2*b + 145*a*b^2 - 12*b^3)*cos(f*x +
e)^5 - 20*(15*a^2*b - 32*a*b^2 + 16*b^3)*cos(f*x + e)^3 + 15*((3*a^3 - 4*a^2*b)*cos(f*x + e)^6 - (3*a^3 - 10*
a^2*b + 8*a*b^2)*cos(f*x + e)^4 - 3*a*b^2 + 4*b^3 - (6*a^2*b - 11*a*b^2 + 4*b^3)*cos(f*x + e)^2)*sqrt(-b/(a +
b))*log(((a^2 + 8*a*b + 8*b^2)*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 - 4*((a^2 + 3*a*b + 2*b^2)*co
s(f*x + e)^3 - (a*b + b^2)*cos(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(
f*x + e)^2 + b^2))*sin(f*x + e) - 60*(3*a*b^2 - 4*b^3)*cos(f*x + e))/(((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3
+ a^2*b^4)*f*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*cos(f*x + e)^4 -
(2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a^2
*b^4 + 4*a*b^5 + b^6)*f)*sin(f*x + e)), -1/48*(2*(16*a^3 - 83*a^2*b + 6*a*b^2)*cos(f*x + e)^7 - 2*(24*a^3 - 13
4*a^2*b + 145*a*b^2 - 12*b^3)*cos(f*x + e)^5 - 10*(15*a^2*b - 32*a*b^2 + 16*b^3)*cos(f*x + e)^3 - 15*((3*a^3 -
4*a^2*b)*cos(f*x + e)^6 - (3*a^3 - 10*a^2*b + 8*a*b^2)*cos(f*x + e)^4 - 3*a*b^2 + 4*b^3 - (6*a^2*b - 11*a*b^2
+ 4*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x
+ e)*sin(f*x + e)))*sin(f*x + e) - 30*(3*a*b^2 - 4*b^3)*cos(f*x + e))/(((a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^
3 + a^2*b^4)*f*cos(f*x + e)^6 - (a^6 + 2*a^5*b - 2*a^4*b^2 - 8*a^3*b^3 - 7*a^2*b^4 - 2*a*b^5)*f*cos(f*x + e)^4
- (2*a^5*b + 7*a^4*b^2 + 8*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 - b^6)*f*cos(f*x + e)^2 - (a^4*b^2 + 4*a^3*b^3 + 6*a
^2*b^4 + 4*a*b^5 + b^6)*f)*sin(f*x + e))]

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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)**4/(a+b*sec(f*x+e)**2)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.31419, size = 371, normalized size = 2.26 \begin{align*} -\frac{\frac{15 \,{\left (\pi \left \lfloor \frac{f x + e}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (f x + e\right )}{\sqrt{a b + b^{2}}}\right )\right )}{\left (3 \, a b - 4 \, b^{2}\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sqrt{a b + b^{2}}} + \frac{3 \,{\left (7 \, a b^{2} \tan \left (f x + e\right )^{3} - 4 \, b^{3} \tan \left (f x + e\right )^{3} + 9 \, a^{2} b \tan \left (f x + e\right ) + 5 \, a b^{2} \tan \left (f x + e\right ) - 4 \, b^{3} \tan \left (f x + e\right )\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} + \frac{8 \,{\left (3 \, a \tan \left (f x + e\right )^{2} - 6 \, b \tan \left (f x + e\right )^{2} + a + b\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \tan \left (f x + e\right )^{3}}}{24 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(f*x+e)^4/(a+b*sec(f*x+e)^2)^3,x, algorithm="giac")

[Out]

-1/24*(15*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b^2)))*(3*a*b - 4*b^2)/((a^4
+ 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sqrt(a*b + b^2)) + 3*(7*a*b^2*tan(f*x + e)^3 - 4*b^3*tan(f*x + e)^3 +
9*a^2*b*tan(f*x + e) + 5*a*b^2*tan(f*x + e) - 4*b^3*tan(f*x + e))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)
*(b*tan(f*x + e)^2 + a + b)^2) + 8*(3*a*tan(f*x + e)^2 - 6*b*tan(f*x + e)^2 + a + b)/((a^4 + 4*a^3*b + 6*a^2*b
^2 + 4*a*b^3 + b^4)*tan(f*x + e)^3))/f