3.211 \(\int \frac {\tan ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx\)

Optimal. Leaf size=71 \[ -\frac {a^2 \log \left (a+b \tan ^2(e+f x)\right )}{2 b^2 f (a-b)}-\frac {\log (\cos (e+f x))}{f (a-b)}+\frac {\tan ^2(e+f x)}{2 b f} \]

[Out]

-ln(cos(f*x+e))/(a-b)/f-1/2*a^2*ln(a+b*tan(f*x+e)^2)/(a-b)/b^2/f+1/2*tan(f*x+e)^2/b/f

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Rubi [A]  time = 0.10, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3670, 446, 72} \[ -\frac {a^2 \log \left (a+b \tan ^2(e+f x)\right )}{2 b^2 f (a-b)}-\frac {\log (\cos (e+f x))}{f (a-b)}+\frac {\tan ^2(e+f x)}{2 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Log[Cos[e + f*x]]/((a - b)*f)) - (a^2*Log[a + b*Tan[e + f*x]^2])/(2*(a - b)*b^2*f) + Tan[e + f*x]^2/(2*b*f)

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 446

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

Rule 3670

Int[((d_.)*tan[(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)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\tan ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(1+x) (a+b x)} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b}+\frac {1}{(a-b) (1+x)}-\frac {a^2}{(a-b) b (a+b x)}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {\log (\cos (e+f x))}{(a-b) f}-\frac {a^2 \log \left (a+b \tan ^2(e+f x)\right )}{2 (a-b) b^2 f}+\frac {\tan ^2(e+f x)}{2 b f}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 64, normalized size = 0.90 \[ \frac {-\frac {a^2 \log \left (a+b \tan ^2(e+f x)\right )}{b^2 (a-b)}-\frac {2 \log (\cos (e+f x))}{a-b}+\frac {\tan ^2(e+f x)}{b}}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Log[Cos[e + f*x]])/(a - b) - (a^2*Log[a + b*Tan[e + f*x]^2])/((a - b)*b^2) + Tan[e + f*x]^2/b)/(2*f)

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fricas [A]  time = 0.48, size = 92, normalized size = 1.30 \[ -\frac {a^{2} \log \left (\frac {b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - {\left (a^{2} - b^{2}\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, {\left (a b^{2} - b^{3}\right )} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^2*log((b*tan(f*x + e)^2 + a)/(tan(f*x + e)^2 + 1)) - (a*b - b^2)*tan(f*x + e)^2 - (a^2 - b^2)*log(1/(t
an(f*x + e)^2 + 1)))/((a*b^2 - b^3)*f)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (2*pi/x/2)>(-2*pi/
x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check si
gn: (2*pi/x/2)>(-2*pi/x/2)2/f*(-1/(-4*a+4*b)*ln(abs((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))+1/(1-cos(f*x+exp(1
)))*(1+cos(f*x+exp(1)))+2))-a^3/(4*a^2*b^2-4*a*b^3)*ln(abs(((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))+1/(1-cos(f
*x+exp(1)))*(1+cos(f*x+exp(1))))*a-2*a+4*b))+(a+b)*1/4/b^2*ln(abs((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))+1/(1
-cos(f*x+exp(1)))*(1+cos(f*x+exp(1)))-2))+(-((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))+1/(1-cos(f*x+exp(1)))*(1+
cos(f*x+exp(1))))*a-((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))+1/(1-cos(f*x+exp(1)))*(1+cos(f*x+exp(1))))*b+2*a+
6*b)*1/4/b^2/((1-cos(f*x+exp(1)))/(1+cos(f*x+exp(1)))+1/(1-cos(f*x+exp(1)))*(1+cos(f*x+exp(1)))-2))

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maple [A]  time = 0.18, size = 72, normalized size = 1.01 \[ \frac {\tan ^{2}\left (f x +e \right )}{2 b f}-\frac {a^{2} \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 \left (a -b \right ) b^{2} f}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a -b \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*tan(f*x+e)^2/b/f-1/2*a^2*ln(a+b*tan(f*x+e)^2)/(a-b)/b^2/f+1/2/f/(a-b)*ln(1+tan(f*x+e)^2)

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maxima [A]  time = 0.76, size = 76, normalized size = 1.07 \[ -\frac {\frac {a^{2} \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a b^{2} - b^{3}} - \frac {{\left (a + b\right )} \log \left (\sin \left (f x + e\right )^{2} - 1\right )}{b^{2}} + \frac {1}{b \sin \left (f x + e\right )^{2} - b}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a^2*log(-(a - b)*sin(f*x + e)^2 + a)/(a*b^2 - b^3) - (a + b)*log(sin(f*x + e)^2 - 1)/b^2 + 1/(b*sin(f*x
+ e)^2 - b))/f

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mupad [B]  time = 11.86, size = 74, normalized size = 1.04 \[ \frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f\,\left (a-b\right )}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2}{2\,b\,f}-\frac {a^2\,\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}{2\,f\,\left (a\,b^2-b^3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tan(e + f*x)^2 + 1)/(2*f*(a - b)) + tan(e + f*x)^2/(2*b*f) - (a^2*log(a + b*tan(e + f*x)^2))/(2*f*(a*b^2 -
b^3))

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sympy [A]  time = 16.36, size = 348, normalized size = 4.90 \[ \begin {cases} \tilde {\infty } x \tan ^{3}{\relax (e )} & \text {for}\: a = 0 \wedge b = 0 \wedge f = 0 \\- \frac {2 \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \tan ^{2}{\left (e + f x \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} - \frac {2 \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} + \frac {\tan ^{4}{\left (e + f x \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} - \frac {2}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} & \text {for}\: a = b \\\frac {\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {\tan ^{4}{\left (e + f x \right )}}{4 f} - \frac {\tan ^{2}{\left (e + f x \right )}}{2 f}}{a} & \text {for}\: b = 0 \\\frac {x \tan ^{5}{\relax (e )}}{a + b \tan ^{2}{\relax (e )}} & \text {for}\: f = 0 \\- \frac {a^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a b^{2} f - 2 b^{3} f} - \frac {a^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a b^{2} f - 2 b^{3} f} + \frac {a b \tan ^{2}{\left (e + f x \right )}}{2 a b^{2} f - 2 b^{3} f} + \frac {b^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 a b^{2} f - 2 b^{3} f} - \frac {b^{2} \tan ^{2}{\left (e + f x \right )}}{2 a b^{2} f - 2 b^{3} f} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*tan(e)**3, Eq(a, 0) & Eq(b, 0) & Eq(f, 0)), (-2*log(tan(e + f*x)**2 + 1)*tan(e + f*x)**2/(2*b
*f*tan(e + f*x)**2 + 2*b*f) - 2*log(tan(e + f*x)**2 + 1)/(2*b*f*tan(e + f*x)**2 + 2*b*f) + tan(e + f*x)**4/(2*
b*f*tan(e + f*x)**2 + 2*b*f) - 2/(2*b*f*tan(e + f*x)**2 + 2*b*f), Eq(a, b)), ((log(tan(e + f*x)**2 + 1)/(2*f)
+ tan(e + f*x)**4/(4*f) - tan(e + f*x)**2/(2*f))/a, Eq(b, 0)), (x*tan(e)**5/(a + b*tan(e)**2), Eq(f, 0)), (-a*
*2*log(-I*sqrt(a)*sqrt(1/b) + tan(e + f*x))/(2*a*b**2*f - 2*b**3*f) - a**2*log(I*sqrt(a)*sqrt(1/b) + tan(e + f
*x))/(2*a*b**2*f - 2*b**3*f) + a*b*tan(e + f*x)**2/(2*a*b**2*f - 2*b**3*f) + b**2*log(tan(e + f*x)**2 + 1)/(2*
a*b**2*f - 2*b**3*f) - b**2*tan(e + f*x)**2/(2*a*b**2*f - 2*b**3*f), True))

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