3.6.41 \(\int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx\) [541]

Optimal. Leaf size=116 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {2 a}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \]

[Out]

-arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(3/2)/d-arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(
a+I*b)^(3/2)/d+2*a/(a^2+b^2)/d/(a+b*tan(d*x+c))^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3610, 3620, 3618, 65, 214} \begin {gather*} \frac {2 a}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(3/2)*d)) - ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt
[a + I*b]]/((a + I*b)^(3/2)*d) + (2*a)/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, 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 3618

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

Rule 3620

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

Rubi steps

\begin {align*} \int \frac {\tan (c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx &=\frac {2 a}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\int \frac {b+a \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{a^2+b^2}\\ &=\frac {2 a}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}-\frac {\int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a-b)}+\frac {\int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (i a+b)}\\ &=\frac {2 a}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b) d}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b) d}\\ &=\frac {2 a}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(i a-b) b d}-\frac {\text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b (i a+b) d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{3/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{3/2} d}+\frac {2 a}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d 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 = 0.17, size = 100, normalized size = 0.86 \begin {gather*} \frac {(a+i b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan (c+d x)}{a-i b}\right )+(a-i b) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan (c+d x)}{a+i b}\right )}{\left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + I*b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Tan[c + d*x])/(a - I*b)] + (a - I*b)*Hypergeometric2F1[-1/2,
1, 1/2, (a + b*Tan[c + d*x])/(a + I*b)])/((a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(798\) vs. \(2(98)=196\).
time = 0.13, size = 799, normalized size = 6.89

method	result	size
derivativedivides	\(\frac {\frac {\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 a^{4}+2 b^{4}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 a^{4}-2 b^{4}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{a^{2}+b^{2}}+\frac {2 a}{\left (a^{2}+b^{2}\right ) \sqrt {a +b \tan \left (d x +c \right )}}}{d}\)	\(799\)
default	\(\frac {\frac {\frac {\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 a^{4}+2 b^{4}-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 a^{4}-2 b^{4}+\frac {\left (\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, b^{2}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a \,b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{a^{2}+b^{2}}+\frac {2 a}{\left (a^{2}+b^{2}\right ) \sqrt {a +b \tan \left (d x +c \right )}}}{d}\)	\(799\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(2/(a^2+b^2)*(1/4/(a^2+b^2)^(3/2)*(1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2+(2*(a^2+b^2)^(1/
2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^2-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^2)*l
n(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))+2*(-2*a^4+2*b^4-1/2*((2
*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2+(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^2-2*(2*(a^2+b^
2)^(1/2)+2*a)^(1/2)*a^3-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/
2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)))+
1/4/(a^2+b^2)^(3/2)*(-1/2*((2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2+(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^
2+b^2)^(1/2)*b^2-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^2)*ln(-b*tan(d*x+c)-a
+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-(a^2+b^2)^(1/2))+2*(2*a^4-2*b^4+1/2*((2*(a^2+b^2)^(1/2)+
2*a)^(1/2)*(a^2+b^2)^(1/2)*a^2+(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*b^2-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/
2)*a^3-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a*b^2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arc
tan((-2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))))+2*a/(a^2+b^2)/(
a+b*tan(d*x+c))^(1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5638 vs. \(2 (94) = 188\).
time = 1.52, size = 5638, normalized size = 48.60 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*sqrt(2)*((a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*d^5*cos(d*x + c)^2 + 2*(a^9*b +
4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 +
4*a^2*b^8 + b^10)*d^5)*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*
sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((9*a^4*b^2 - 6*a^2*b^4
+ b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*(1/((a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arctan(-((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*b^8 - 2*a^
2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b
^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^9 + 8*a^7*b^2 + 6*a^5*b
^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8
+ 6*a^2*b^10 + b^12)*d^4)) + sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^4*b^10 - 5
*a^2*b^12 - b^14)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^
4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (a^11 + 5*a^9*b^2 + 10*a^
7*b^4 + 10*a^5*b^6 + 5*a^3*b^8 + a*b^10)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b
^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^
5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b
^6))*sqrt(((9*a^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*
d^4))*cos(d*x + c) + sqrt(2)*((9*a^9 + 12*a^7*b^2 - 2*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d^3*sqrt(1/((a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6)*d^4))*cos(d*x + c) + (9*a^6 - 15*a^4*b^2 + 7*a^2*b^4 - b^6)*d*cos(d*x + c))*sqrt((a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3
*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(1/4) + (9*a^5 - 6*a^3*b^2 + a*b^4)*cos(d*x + c) + (9*a^4*b - 6*a^2*b^3 + b^
5)*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4) + sqrt(2)*((3*a^16 + 14*a^1
4*b^2 + 22*a^12*b^4 + 6*a^10*b^6 - 20*a^8*b^8 - 22*a^6*b^10 - 6*a^4*b^12 + 2*a^2*b^14 + b^16)*d^7*sqrt((9*a^4*
b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*s
qrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^13 + 14*a^11*b^2 + 25*a^9*b^4 + 20*a^7*b^6 + 5*a^5*b^8
- 2*a^3*b^10 - a*b^12)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 +
15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6
- 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(
d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4))/(9*a^4*b^2 - 6*a
^2*b^4 + b^6)) + 4*sqrt(2)*((a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*d^5*cos(d*x + c)^2 +
2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*d^5*cos(d*x + c)*sin(d*x + c) + (a^8*b^2 + 4*a^6*b^4 +
6*a^4*b^6 + 4*a^2*b^8 + b^10)*d^5)*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*
a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((9*a^4*b^2 -
6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*(1/((a
^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4))^(3/4)*arctan(((3*a^12 + 14*a^10*b^2 + 25*a^8*b^4 + 20*a^6*b^6 + 5*a^4*
b^8 - 2*a^2*b^10 - b^12)*d^4*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6
+ 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (3*a^9 + 8*a^7*b^2
+ 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 1
5*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)) - sqrt(2)*((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a^6*b^8 - 9*a^
4*b^10 - 5*a^2*b^12 - b^14)*d^7*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b
^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4))*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)) + (a^11 + 5*a^9*b
^2 + 10*a^7*b^4 + 10*a^5*b^6 + 5*a^3*b^8 + a*b^10)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - (
a^9 - 6*a^5*b^4 - 8*a^3*b^6 - 3*a*b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d^4)))/(9*a^4*b^2 - 6*a
^2*b^4 + b^6))*sqrt(((9*a^8 + 12*a^6*b^2 - 2*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2*sqrt(1/((a^6 + 3*a^4*b^2 + 3*a^2*b
^4 + b^6)*d^4))*cos(d*x + c) - sqrt(2)*((9*a^9 ...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)/(a+b*tan(d*x+c))**(3/2),x)

[Out]

Integral(tan(c + d*x)/(a + b*tan(c + d*x))**(3/2), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [B]
time = 6.30, size = 2844, normalized size = 24.52 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)/(a + b*tan(c + d*x))^(3/2),x)

[Out]

(2*a)/(d*(a^2 + b^2)*(a + b*tan(c + d*x))^(1/2)) - atan(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i
))^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(32*a^6*b^6*d^4 - 48*a^2*b^10*d^4 - 3
2*a^4*b^8*d^4 - 16*b^12*d^4 + 48*a^8*b^4*d^4 + 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2
*b*d^2*3i))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6
*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5))/4))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 -
32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2)*1i + (1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(((1/
(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(16*b^12*d^4 + 48*a^2*b^10*d^4 + 32*a^4*b^8*d^4 - 3
2*a^6*b^6*d^4 - 48*a^8*b^4*d^4 - 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1
/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9
*b^4*d^5 + 64*a^11*b^2*d^5))/4))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^
3 - 16*a^8*b^2*d^3))/2)*1i)/((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(((1/(a^3*d^2 + b^3
*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(16*b^12*d^4 + 48*a^2*b^10*d^4 + 32*a^4*b^8*d^4 - 32*a^6*b^6*d^4
- 48*a^8*b^4*d^4 - 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(a + b*tan
(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*
a^11*b^2*d^5))/4))/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2
*d^3))/2) - (1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2
*d^2 - a^2*b*d^2*3i))^(1/2)*(32*a^6*b^6*d^4 - 48*a^2*b^10*d^4 - 32*a^4*b^8*d^4 - 16*b^12*d^4 + 48*a^8*b^4*d^4
+ 16*a^10*b^2*d^4 + ((1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^2*3i))^(1/2)*(a + b*tan(c + d*x))^(1/2)*
(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5))/4)
)/2 + ((a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))/2) + 16*a*
b^8*d^2 + 48*a^3*b^6*d^2 + 48*a^5*b^4*d^2 + 16*a^7*b^2*d^2))*(1/(a^3*d^2 + b^3*d^2*1i - 3*a*b^2*d^2 - a^2*b*d^
2*3i))^(1/2)*1i - atan((((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*((1i/(4*(a^3*d^2*1
i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5
+ 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) - 32*b^12*d^4 - 96*a^2*b^10*d^4 - 64
*a^4*b^8*d^4 + 64*a^6*b^6*d^4 + 96*a^8*b^4*d^4 + 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 +
32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2))
)^(1/2)*1i + ((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(32*b^12*d^4 + (1i/(4*(a^3*d^
2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*
d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) + 96*a^2*b^10*d^4 + 64*a^4*b^8*d^
4 - 64*a^6*b^6*d^4 - 96*a^8*b^4*d^4 - 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*
d^3 - 32*a^6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*1i)
/(((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(32*b^12*d^4 + (1i/(4*(a^3*d^2*1i + b^3*
d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a
^5*b^8*d^5 + 640*a^7*b^6*d^5 + 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) + 96*a^2*b^10*d^4 + 64*a^4*b^8*d^4 - 64*a^6*
b^6*d^4 - 96*a^8*b^4*d^4 - 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^
6*b^4*d^3 - 16*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2) - ((1i/(4*(a^3
*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*((1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b
*d^2)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^12*d^5 + 320*a^3*b^10*d^5 + 640*a^5*b^8*d^5 + 640*a^7*b^6*d^5
+ 320*a^9*b^4*d^5 + 64*a^11*b^2*d^5) - 32*b^12*d^4 - 96*a^2*b^10*d^4 - 64*a^4*b^8*d^4 + 64*a^6*b^6*d^4 + 96*a
^8*b^4*d^4 + 32*a^10*b^2*d^4) + (a + b*tan(c + d*x))^(1/2)*(16*b^10*d^3 + 32*a^2*b^8*d^3 - 32*a^6*b^4*d^3 - 16
*a^8*b^2*d^3))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2) + 16*a*b^8*d^2 + 48*a^3*b^6*
d^2 + 48*a^5*b^4*d^2 + 16*a^7*b^2*d^2))*(1i/(4*(a^3*d^2*1i + b^3*d^2 - a*b^2*d^2*3i - 3*a^2*b*d^2)))^(1/2)*2i

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