3.4.40 \(\int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx\) [340]

Optimal. Leaf size=74 \[ \frac {\sqrt {2} a \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}} \]

[Out]

a*arctanh(1/2*(d^(1/2)+d^(1/2)*tan(f*x+e))*2^(1/2)/(d*tan(f*x+e))^(1/2))*2^(1/2)/d^(3/2)/f-2*a/d/f/(d*tan(f*x+
e))^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3610, 3613, 214} \begin {gather*} \frac {\sqrt {2} a \tanh ^{-1}\left (\frac {\sqrt {d} \tan (e+f x)+\sqrt {d}}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2),x]

[Out]

(Sqrt[2]*a*ArcTanh[(Sqrt[d] + Sqrt[d]*Tan[e + f*x])/(Sqrt[2]*Sqrt[d*Tan[e + f*x]])])/(d^(3/2)*f) - (2*a)/(d*f*
Sqrt[d*Tan[e + f*x]])

Rule 214

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

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3613

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-2*(d^2/f),
Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x
] && EqQ[c^2 - d^2, 0]

Rubi steps

\begin {align*} \int \frac {a+a \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx &=-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}+\frac {\int \frac {a d-a d \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{d^2}\\ &=-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{-2 a^2 d^2+d x^2} \, dx,x,\frac {a d+a d \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{f}\\ &=\frac {\sqrt {2} a \tanh ^{-1}\left (\frac {\sqrt {d}+\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{d^{3/2} f}-\frac {2 a}{d f \sqrt {d \tan (e+f x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.14, size = 64, normalized size = 0.86 \begin {gather*} -\frac {(1+i) a \left (\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-i \tan (e+f x)\right )-i \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};i \tan (e+f x)\right )\right )}{d f \sqrt {d \tan (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Tan[e + f*x])/(d*Tan[e + f*x])^(3/2),x]

[Out]

((-1 - I)*a*(Hypergeometric2F1[-1/2, 1, 1/2, (-I)*Tan[e + f*x]] - I*Hypergeometric2F1[-1/2, 1, 1/2, I*Tan[e +
f*x]]))/(d*f*Sqrt[d*Tan[e + f*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(292\) vs. \(2(61)=122\).
time = 0.20, size = 293, normalized size = 3.96

method	result	size
derivativedivides	\(\frac {a \left (-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}+\frac {\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d}\right )}{f}\)	\(293\)
default	\(\frac {a \left (-\frac {2}{d \sqrt {d \tan \left (f x +e \right )}}+\frac {\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (d^{2}\right )^{\frac {1}{4}}}}{d}\right )}{f}\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*tan(f*x+e))/(d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/f*a*(-2/d/(d*tan(f*x+e))^(1/2)+2/d*(1/8/d*(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(
1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2
)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))-1/8/(d^2)^(1/4)*2
^(1/2)*(ln((d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)+(d^2)^(1/4)*(d*ta
n(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(
d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))))

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Maxima [A]
time = 0.50, size = 102, normalized size = 1.38 \begin {gather*} \frac {a {\left (\frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {4 \, a}{\sqrt {d \tan \left (f x + e\right )}}}{2 \, d f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

1/2*(a*(sqrt(2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d) - sqrt(2)*log(d*tan(f*x
+ e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d)/sqrt(d)) - 4*a/sqrt(d*tan(f*x + e)))/(d*f)

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Fricas [A]
time = 1.36, size = 205, normalized size = 2.77 \begin {gather*} \left [\frac {\sqrt {2} a \sqrt {d} \log \left (\frac {\tan \left (f x + e\right )^{2} + \frac {2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt {d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right ) - 4 \, \sqrt {d \tan \left (f x + e\right )} a}{2 \, d^{2} f \tan \left (f x + e\right )}, -\frac {\sqrt {2} a d \sqrt {-\frac {1}{d}} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-\frac {1}{d}} {\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right ) + 2 \, \sqrt {d \tan \left (f x + e\right )} a}{d^{2} f \tan \left (f x + e\right )}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

[1/2*(sqrt(2)*a*sqrt(d)*log((tan(f*x + e)^2 + 2*sqrt(2)*sqrt(d*tan(f*x + e))*(tan(f*x + e) + 1)/sqrt(d) + 4*ta
n(f*x + e) + 1)/(tan(f*x + e)^2 + 1))*tan(f*x + e) - 4*sqrt(d*tan(f*x + e))*a)/(d^2*f*tan(f*x + e)), -(sqrt(2)
*a*d*sqrt(-1/d)*arctan(1/2*sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(-1/d)*(tan(f*x + e) + 1)/tan(f*x + e))*tan(f*x +
e) + 2*sqrt(d*tan(f*x + e))*a)/(d^2*f*tan(f*x + e))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a \left (\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\tan {\left (e + f x \right )}}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))**(3/2),x)

[Out]

a*(Integral((d*tan(e + f*x))**(-3/2), x) + Integral(tan(e + f*x)/(d*tan(e + f*x))**(3/2), x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (64) = 128\).
time = 0.66, size = 253, normalized size = 3.42 \begin {gather*} -\frac {\frac {8 \, a}{\sqrt {d \tan \left (f x + e\right )} f} - \frac {2 \, \sqrt {2} {\left (a d \sqrt {{\left | d \right |}} - a {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d^{2} f} - \frac {2 \, \sqrt {2} {\left (a d \sqrt {{\left | d \right |}} - a {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d^{2} f} - \frac {\sqrt {2} {\left (a d \sqrt {{\left | d \right |}} + a {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d^{2} f} + \frac {\sqrt {2} {\left (a d \sqrt {{\left | d \right |}} + a {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d^{2} f}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*tan(f*x+e))/(d*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

-1/4*(8*a/(sqrt(d*tan(f*x + e))*f) - 2*sqrt(2)*(a*d*sqrt(abs(d)) - a*abs(d)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)
*sqrt(abs(d)) + 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/(d^2*f) - 2*sqrt(2)*(a*d*sqrt(abs(d)) - a*abs(d)^(3/2))*
arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/(d^2*f) - sqrt(2)*(a*d*sqrt(
abs(d)) + a*abs(d)^(3/2))*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/(d^2*f) + s
qrt(2)*(a*d*sqrt(abs(d)) + a*abs(d)^(3/2))*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + ab
s(d))/(d^2*f))/d

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Mupad [B]
time = 4.58, size = 84, normalized size = 1.14 \begin {gather*} -\frac {2\,a}{d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (-1-\mathrm {i}\right )}{d^{3/2}\,f}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,\left (1-\mathrm {i}\right )}{d^{3/2}\,f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(e + f*x))/(d*tan(e + f*x))^(3/2),x)

[Out]

((-1)^(1/4)*a*atanh(((-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*(1 - 1i))/(d^(3/2)*f) - ((-1)^(1/4)*a*atan(((
-1)^(1/4)*(d*tan(e + f*x))^(1/2))/d^(1/2))*(1 + 1i))/(d^(3/2)*f) - (2*a)/(d*f*(d*tan(e + f*x))^(1/2))

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