3.2.38 \(\int \frac {\csc ^4(e+f x)}{(a+b \tan ^2(e+f x))^{3/2}} \, dx\) [138]
Optimal. Leaf size=114 \[ -\frac {(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^3(e+f x)}{3 a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {2 (3 a-4 b) b \tan (e+f x)}{3 a^3 f \sqrt {a+b \tan ^2(e+f x)}} \]
[Out]
-1/3*(3*a-4*b)*cot(f*x+e)/a^2/f/(a+b*tan(f*x+e)^2)^(1/2)-1/3*cot(f*x+e)^3/a/f/(a+b*tan(f*x+e)^2)^(1/2)-2/3*(3*
a-4*b)*b*tan(f*x+e)/a^3/f/(a+b*tan(f*x+e)^2)^(1/2)
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Rubi [A]
time = 0.08, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3744, 464, 277,
197} \begin {gather*} -\frac {2 b (3 a-4 b) \tan (e+f x)}{3 a^3 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^3(e+f x)}{3 a f \sqrt {a+b \tan ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]
[Out]
-1/3*((3*a - 4*b)*Cot[e + f*x])/(a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^3/(3*a*f*Sqrt[a + b*Tan[e +
f*x]^2]) - (2*(3*a - 4*b)*b*Tan[e + f*x])/(3*a^3*f*Sqrt[a + b*Tan[e + f*x]^2])
Rule 197
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]
Rule 277
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Rule 464
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Rule 3744
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff^(m + 1)/f), Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)
^(m/2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2]
Rubi steps
\begin {align*} \int \frac {\csc ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1+x^2}{x^4 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x)}{3 a f \sqrt {a+b \tan ^2(e+f x)}}+\frac {(3 a-4 b) \text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac {(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^3(e+f x)}{3 a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {(2 (3 a-4 b) b) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 a^2 f}\\ &=-\frac {(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot ^3(e+f x)}{3 a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {2 (3 a-4 b) b \tan (e+f x)}{3 a^3 f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.00, size = 119, normalized size = 1.04 \begin {gather*} \frac {\left (-3 a^2-7 a b+12 b^2-2 \left (a^2-6 a b+8 b^2\right ) \cos (2 (e+f x))+\left (a^2-5 a b+4 b^2\right ) \cos (4 (e+f x))\right ) \csc ^3(e+f x) \sec (e+f x)}{6 \sqrt {2} a^3 f \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]
[Out]
((-3*a^2 - 7*a*b + 12*b^2 - 2*(a^2 - 6*a*b + 8*b^2)*Cos[2*(e + f*x)] + (a^2 - 5*a*b + 4*b^2)*Cos[4*(e + f*x)])
*Csc[e + f*x]^3*Sec[e + f*x])/(6*Sqrt[2]*a^3*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x]^2])
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Maple [A]
time = 0.33, size = 170, normalized size = 1.49
| | |
method |
result |
size |
| | |
default |
\(\frac {\left (2 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2}-10 \left (\cos ^{4}\left (f x +e \right )\right ) a b +8 \left (\cos ^{4}\left (f x +e \right )\right ) b^{2}-3 \left (\cos ^{2}\left (f x +e \right )\right ) a^{2}+16 \left (\cos ^{2}\left (f x +e \right )\right ) a b -16 \left (\cos ^{2}\left (f x +e \right )\right ) b^{2}-6 a b +8 b^{2}\right ) \left (\cos ^{3}\left (f x +e \right )\right ) \left (\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}\right )^{\frac {3}{2}}}{3 f \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{2} \sin \left (f x +e \right )^{3} a^{3}}\) |
\(170\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/3/f/(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2*(2*cos(f*x+e)^4*a^2-10*cos(f*x+e)^4*a*b+8*cos(f*x+e)^4*b^2-3*cos(f*x
+e)^2*a^2+16*cos(f*x+e)^2*a*b-16*cos(f*x+e)^2*b^2-6*a*b+8*b^2)*cos(f*x+e)^3*((a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)
/cos(f*x+e)^2)^(3/2)/sin(f*x+e)^3/a^3
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Maxima [A]
time = 0.30, size = 151, normalized size = 1.32 \begin {gather*} -\frac {\frac {6 \, b \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a^{2}} - \frac {8 \, b^{2} \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a^{3}} + \frac {3}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a \tan \left (f x + e\right )} - \frac {4 \, b}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a^{2} \tan \left (f x + e\right )} + \frac {1}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a \tan \left (f x + e\right )^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
-1/3*(6*b*tan(f*x + e)/(sqrt(b*tan(f*x + e)^2 + a)*a^2) - 8*b^2*tan(f*x + e)/(sqrt(b*tan(f*x + e)^2 + a)*a^3)
+ 3/(sqrt(b*tan(f*x + e)^2 + a)*a*tan(f*x + e)) - 4*b/(sqrt(b*tan(f*x + e)^2 + a)*a^2*tan(f*x + e)) + 1/(sqrt(
b*tan(f*x + e)^2 + a)*a*tan(f*x + e)^3))/f
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Fricas [A]
time = 4.04, size = 163, normalized size = 1.43 \begin {gather*} -\frac {{\left (2 \, {\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{5} - {\left (3 \, a^{2} - 16 \, a b + 16 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (3 \, a b - 4 \, b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{3 \, {\left ({\left (a^{4} - a^{3} b\right )} f \cos \left (f x + e\right )^{4} - a^{3} b f - {\left (a^{4} - 2 \, a^{3} b\right )} f \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
-1/3*(2*(a^2 - 5*a*b + 4*b^2)*cos(f*x + e)^5 - (3*a^2 - 16*a*b + 16*b^2)*cos(f*x + e)^3 - 2*(3*a*b - 4*b^2)*co
s(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/(((a^4 - a^3*b)*f*cos(f*x + e)^4 - a^3*b*f - (a^
4 - 2*a^3*b)*f*cos(f*x + e)^2)*sin(f*x + e))
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)**4/(a+b*tan(f*x+e)**2)**(3/2),x)
[Out]
Integral(csc(e + f*x)**4/(a + b*tan(e + f*x)**2)**(3/2), x)
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^4/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
integrate(csc(f*x + e)^4/(b*tan(f*x + e)^2 + a)^(3/2), x)
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Mupad [B]
time = 35.94, size = 2500, normalized size = 21.93 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sin(e + f*x)^4*(a + b*tan(e + f*x)^2)^(3/2)),x)
[Out]
((a + (b*(exp(e*2i + f*x*2i)*1i - 1i)^2)/(exp(e*2i + f*x*2i) + 1)^2)^(1/2)*(2*exp(e*2i + f*x*2i) + exp(e*4i +
f*x*4i) + 1)*(((a + 3*b)*(((a + 3*b)*(((a + 3*b)*(((a + 3*b)*(((a + 3*b)*((((((a - b)*(a - 2*b) - (a + 2*b)^2)
*(a - b)^2*(a + 2*b))/(a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))
*(a - b)^5)/(3072*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(a + 3*b))/(3072*a^4*f
*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) - ((a - b)^7*(a + 2*b)*(3*a + b))/(1024*a^4*f*(a*b^2 - a^2*b)*(a*b
- a^2)*(a*1i - b*1i))))/(a - b) + ((a + 3*b)*(((a + 3*b)*((((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*(a +
2*b))/(a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)^5)/(30
72*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(a + 3*b))/(3072*a^4*f*(a*b^2 - a^2*b
)*(a*b - a^2)*(a*1i - b*1i)) - ((a - b)^7*(a + 2*b)*(3*a + b))/(1024*a^4*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i -
b*1i))))/(a - b) + (((a + 2*b)^3 + (((a - b)*(a - 2*b) - (a + 2*b)^2)*((a - b)*(a + 2*b) - (a - b)*(a + 3*b))
*(a + 2*b))/(a*b - a^2))*(a - b)^5)/(3072*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^6*(a + 2*b
)*(9*a + 4*b))/(768*a^3*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) - (((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a
- b)^2*(a + 2*b))/(a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a
- b)^4*(3*a + b))/(1024*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(a + 3*b))/(307
2*a^4*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/(a - b) - ((a - b)^8*(a + 2*b))/(3072*a^4*f*(a*b^2 - a^2*
b)*(a*b - a^2)*(a*1i - b*1i)) + ((a - b)^4*(a + 2*b)*(56*a^3*b - 84*a^4 - 8*b^4 + 36*a^2*b^2))/(3072*a^4*f*(a*
b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) - (((a + 2*b)^3 + (((a - b)*(a - 2*b) - (a + 2*b)^2)*((a - b)*(a + 2*b
) - (a - b)*(a + 3*b))*(a + 2*b))/(a*b - a^2))*(a - b)^4*(3*a + b))/(1024*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1
i - b*1i)) + (((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*(a + 2*b))/(a*b - a^2) + (((a - b)*(a + 2*b) - (a
- b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)^3*(9*a + 4*b))/(768*a^3*f*(a*b^2 - a^2*b)*(a + 2*b)*
(a*1i - b*1i))))/(a - b) + ((a + 3*b)*(((a + 3*b)*((((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*(a + 2*b))/(
a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)^5)/(3072*a^4*f
*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(a + 3*b))/(3072*a^4*f*(a*b^2 - a^2*b)*(a*b -
a^2)*(a*1i - b*1i)) - ((a - b)^7*(a + 2*b)*(3*a + b))/(1024*a^4*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)))
)/(a - b) + (((a + 2*b)^3 + (((a - b)*(a - 2*b) - (a + 2*b)^2)*((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a + 2*
b))/(a*b - a^2))*(a - b)^5)/(3072*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^6*(a + 2*b)*(9*a +
4*b))/(768*a^3*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) - (((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*
(a + 2*b))/(a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)^4*
(3*a + b))/(1024*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(a + 3*b))/(3072*a^4*f*
(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/(a - b) - (((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*(a + 2*b
))/(a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)^5)/(3072*a
^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + (((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*(a + 2*b))/(a*b
- a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)*(56*a^3*b - 84*a^
4 - 8*b^4 + 36*a^2*b^2))/(3072*a^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) - ((a - b)^7*(a + 2*b)*(a + 3*b)
)/(3072*a^4*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(3*a + b))/(1024*a^4*f*(a*b^2
- a^2*b)*(a*b - a^2)*(a*1i - b*1i)) - ((a - b)^4*(a + 2*b)*(64*a^3*b - 72*a*b^3 - 182*a^4 + 30*b^4 + 96*a^2*b^
2))/(3072*a^4*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) + (((a + 2*b)^3 + (((a - b)*(a - 2*b) - (a + 2*b)^2
)*((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a + 2*b))/(a*b - a^2))*(a - b)^3*(9*a + 4*b))/(768*a^3*f*(a*b^2 - a
^2*b)*(a + 2*b)*(a*1i - b*1i))))/(a - b) - ((a + 3*b)*((((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*(a + 2*b
))/(a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)^5)/(3072*a
^4*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(a + 3*b))/(3072*a^4*f*(a*b^2 - a^2*b)*(a
*b - a^2)*(a*1i - b*1i)) - ((a - b)^7*(a + 2*b)*(3*a + b))/(1024*a^4*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1
i))))/(a - b) + ((a + 3*b)*(((a + 3*b)*(((a + 3*b)*((((((a - b)*(a - 2*b) - (a + 2*b)^2)*(a - b)^2*(a + 2*b))/
(a*b - a^2) + (((a - b)*(a + 2*b) - (a - b)*(a + 3*b))*(a - b)*(a + 2*b)^2)/(a*b - a^2))*(a - b)^5)/(3072*a^4*
f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^7*(a + 2*b)*(a + 3*b))/(3072*a^4*f*(a*b^2 - a^2*b)*(a*b
- a^2)*(a*1i - b*1i)) - ((a - b)^7*(a + 2*b)*(3...
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