3.2.37 \(\int \frac {\csc ^2(e+f x)}{(a+b \tan ^2(e+f x))^{3/2}} \, dx\) [137]
Optimal. Leaf size=62 \[ -\frac {\cot (e+f x)}{a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {2 b \tan (e+f x)}{a^2 f \sqrt {a+b \tan ^2(e+f x)}} \]
[Out]
-cot(f*x+e)/a/f/(a+b*tan(f*x+e)^2)^(1/2)-2*b*tan(f*x+e)/a^2/f/(a+b*tan(f*x+e)^2)^(1/2)
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Rubi [A]
time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {3744, 277, 197}
\begin {gather*} -\frac {2 b \tan (e+f x)}{a^2 f \sqrt {a+b \tan ^2(e+f x)}}-\frac {\cot (e+f x)}{a f \sqrt {a+b \tan ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]
[Out]
-(Cot[e + f*x]/(a*f*Sqrt[a + b*Tan[e + f*x]^2])) - (2*b*Tan[e + f*x])/(a^2*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 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 ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x)}{a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=-\frac {\cot (e+f x)}{a f \sqrt {a+b \tan ^2(e+f x)}}-\frac {2 b \tan (e+f x)}{a^2 f \sqrt {a+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 74, normalized size = 1.19 \begin {gather*} -\frac {(a+2 b+(a-2 b) \cos (2 (e+f x))) \csc (e+f x) \sec (e+f x)}{\sqrt {2} a^2 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]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]
[Out]
-(((a + 2*b + (a - 2*b)*Cos[2*(e + f*x)])*Csc[e + f*x]*Sec[e + f*x])/(Sqrt[2]*a^2*f*Sqrt[(a + b + (a - b)*Cos[
2*(e + f*x)])*Sec[e + f*x]^2]))
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Maple [A]
time = 0.40, size = 109, normalized size = 1.76
| | |
| method |
result |
size |
| | |
| default |
\(-\frac {\left (a \left (\cos ^{2}\left (f x +e \right )\right )-2 \left (\cos ^{2}\left (f x +e \right )\right ) b +2 b \right ) \left (\cos ^{3}\left (f x +e \right )\right ) \left (\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}\right )^{\frac {3}{2}}}{f \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{2} \sin \left (f x +e \right ) a^{2}}\) |
\(109\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(f*x+e)^2/(a+b*tan(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
[Out]
-1/f/(a*cos(f*x+e)^2-cos(f*x+e)^2*b+b)^2*(a*cos(f*x+e)^2-2*cos(f*x+e)^2*b+2*b)*cos(f*x+e)^3*((a*cos(f*x+e)^2-c
os(f*x+e)^2*b+b)/cos(f*x+e)^2)^(3/2)/sin(f*x+e)/a^2
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Maxima [A]
time = 0.28, size = 62, normalized size = 1.00 \begin {gather*} -\frac {\frac {2 \, b \tan \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a^{2}} + \frac {1}{\sqrt {b \tan \left (f x + e\right )^{2} + a} a \tan \left (f x + e\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(f*x+e)^2/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="maxima")
[Out]
-(2*b*tan(f*x + e)/(sqrt(b*tan(f*x + e)^2 + a)*a^2) + 1/(sqrt(b*tan(f*x + e)^2 + a)*a*tan(f*x + e)))/f
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Fricas [A]
time = 1.12, size = 96, normalized size = 1.55 \begin {gather*} -\frac {{\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \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}}}}{{\left (a^{2} b f + {\left (a^{3} - a^{2} 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)^2/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="fricas")
[Out]
-((a - 2*b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a^2*b*f + (
a^3 - a^2*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 ^{2}{\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)**2/(a+b*tan(f*x+e)**2)**(3/2),x)
[Out]
Integral(csc(e + f*x)**2/(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)^2/(a+b*tan(f*x+e)^2)^(3/2),x, algorithm="giac")
[Out]
integrate(csc(f*x + e)^2/(b*tan(f*x + e)^2 + a)^(3/2), x)
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Mupad [B]
time = 18.30, size = 2978, normalized size = 48.03 \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)^2*(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)*(exp(e*2i + f*x*2i)*(((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))/(8*f*(
a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) - (3*(a - b)^4*(a + 2*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1
i)) + ((a - b)^3*(a + 2*b)*(a + 3*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/(a - b) + (3*(a - b)^4
*(a + 2*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) - (3*((((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))/
(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) +
((a - b)^3*(a + 2*b)*(a + 3*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/(a - b) + (3*(a - b)^4*(a +
2*b))/(8*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*f*(a*
b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + (3*((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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) - ((a -
b)^3*(a + 2*b)*(a + 3*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))) - exp(e*4i + f*x*4i)*(((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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) - (3*(a - b)^4*(a +
2*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) + ((a - b)^3*(a + 2*b)*(a + 3*b))/(8*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))/(
8*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) - (3*(a - b)^4*(a + 2*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i
- b*1i)) + ((a - b)^3*(a + 2*b)*(a + 3*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/(a - b) + (3*(a -
b)^4*(a + 2*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) - (3*((((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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i
)) + ((a - b)^3*(a + 2*b)*(a + 3*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/(a - b) - ((a - b)^4*(a
+ 2*b))/(4*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) + (3*((((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))/(8*
f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) - (3*((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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i))) +
((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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) - (3*(
a - b)^4*(a + 2*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i)) + ((a - b)^3*(a + 2*b)*(a + 3*b))/(8*f*(a*
b^2 - a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/(a - b) + (3*(a - b)^4*(a + 2*b))/(8*f*(a*b^2 - a^2*b)*(a*b - a^2)*(
a*1i - b*1i)) - (3*((((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))/(8*f*(a*b^2 - a^2*b)*(a + 2*b)*(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*f*(a*b^2 - a^2*b)*(a + 2*b)*(a*1i - b*1i)) + ((a - b)^3*(a + 2*b)*(a + 3*b))/(8*f*(a*b^2 -
a^2*b)*(a*b - a^2)*(a*1i - b*1i))))/((exp(e*2i + f*x*2i) + 1)*(b - a - exp(e*2i + f*x*2i)*(a + 3*b) + exp(e*4i
+ f*x*4i)*(a + 3*b) + exp(e*6i + f*x*6i)*(a - b)))
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