3.6.21 \(\int \tan (c+d x) (a+b \tan (c+d x))^{5/2} \, dx\) [521]

Optimal. Leaf size=158 \[ -\frac {(a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d} \]

[Out]

-(a-I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d-(a+I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a
+I*b)^(1/2))/d+2*(a^2-b^2)*(a+b*tan(d*x+c))^(1/2)/d+2/3*a*(a+b*tan(d*x+c))^(3/2)/d+2/5*(a+b*tan(d*x+c))^(5/2)/
d

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Rubi [A]
time = 0.20, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 \left (a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}-\frac {(a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - ((a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*T
an[c + d*x]]/Sqrt[a + I*b]])/d + (2*(a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/d + (2*a*(a + b*Tan[c + d*x])^(3/2))
/(3*d) + (2*(a + b*Tan[c + d*x])^(5/2))/(5*d)

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 3609

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

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 \tan (c+d x) (a+b \tan (c+d x))^{5/2} \, dx &=\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}+\int (-b+a \tan (c+d x)) (a+b \tan (c+d x))^{3/2} \, dx\\ &=\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}+\int \sqrt {a+b \tan (c+d x)} \left (-2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}+\int \frac {-b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}-\frac {1}{2} (i a-b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} (i a+b)^3 \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {(a-i b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac {(a+i b)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}\\ &=\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}+\frac {(i a-b)^3 \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 d}-\frac {(i a+b)^3 \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 d}\\ &=-\frac {(a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {(a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 \left (a^2-b^2\right ) \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 a (a+b \tan (c+d x))^{3/2}}{3 d}+\frac {2 (a+b \tan (c+d x))^{5/2}}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.76, size = 138, normalized size = 0.87 \begin {gather*} \frac {-15 (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-15 (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 \sqrt {a+b \tan (c+d x)} \left (23 a^2-15 b^2+11 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)\right )}{15 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*(a - I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - 15*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*T
an[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(23*a^2 - 15*b^2 + 11*a*b*Tan[c + d*x] + 3*b^2*Tan[c
+ d*x]^2))/(15*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(703\) vs. \(2(132)=264\).
time = 0.14, size = 704, normalized size = 4.46

method	result	size
derivativedivides	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-2 b^{2} \sqrt {a +b \tan \left (d x +c \right )}+\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (-2 \sqrt {a^{2}+b^{2}}\, a^{2}+2 \sqrt {a^{2}+b^{2}}\, b^{2}-\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (2 \sqrt {a^{2}+b^{2}}\, a^{2}-2 \sqrt {a^{2}+b^{2}}\, b^{2}+\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{d}\)	\(704\)
default	\(\frac {\frac {2 \left (a +b \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 a \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {a +b \tan \left (d x +c \right )}-2 b^{2} \sqrt {a +b \tan \left (d x +c \right )}+\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (-2 \sqrt {a^{2}+b^{2}}\, a^{2}+2 \sqrt {a^{2}+b^{2}}\, b^{2}-\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (2 \sqrt {a^{2}+b^{2}}\, a^{2}-2 \sqrt {a^{2}+b^{2}}\, b^{2}+\frac {\left (2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a -3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{2}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{d}\)	\(704\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)*(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/d*(2/5*(a+b*tan(d*x+c))^(5/2)+2/3*a*(a+b*tan(d*x+c))^(3/2)+2*a^2*(a+b*tan(d*x+c))^(1/2)-2*b^2*(a+b*tan(d*x+c
))^(1/2)+1/4*(2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+(2*(a^2+b^
2)^(1/2)+2*a)^(1/2)*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*(a^2+b^2)^(1/2)*a^2+2*(a^2+b^2)^(1/2)*b^2-1/2*(2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-3*(2*(
a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*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*(2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+(2*(a^2+b^2)^(1/2)
+2*a)^(1/2)*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*(
a^2+b^2)^(1/2)*a^2-2*(a^2+b^2)^(1/2)*b^2+1/2*(2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a-3*(2*(a^2+b^2)
^(1/2)+2*a)^(1/2)*a^2+(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*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)))

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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))^(5/2),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. 6683 vs. \(2 (128) = 256\).
time = 5.71, size = 6683, normalized size = 42.30 \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))^(5/2),x, algorithm="fricas")

[Out]

-1/60*(60*sqrt(2)*d^5*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2
+ 5*a*b^4)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^
6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4
)^(3/4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4)*arctan(((5*a^18 + 25*a^16*b^2 +
36*a^14*b^4 - 28*a^12*b^6 - 154*a^10*b^8 - 210*a^8*b^10 - 140*a^6*b^12 - 44*a^4*b^14 - 3*a^2*b^16 + b^18)*d^4
*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 11
0*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^23 + 35*a^21*b^2 + 91*a^19*b^4 + 69*a^17*b^6 - 174*a^15*b^8 - 546*a
^13*b^10 - 714*a^11*b^12 - 534*a^9*b^14 - 231*a^7*b^16 - 49*a^5*b^18 - a^3*b^20 + a*b^22)*d^2*sqrt((25*a^8*b^2
- 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + sqrt(2)*((5*a^10 - 5*a^8*b^2 - 14*a^6*b^4 + 6*a^4*b^6
+ 9*a^2*b^8 - b^10)*d^7*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^
8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^15 - 5*a^13*b^2 - 39*a^11*b^4 - 9*a^9*b^6 +
79*a^7*b^8 + 81*a^5*b^10 + 19*a^3*b^12 - 3*a*b^14)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*
b^8 + b^10)/d^4))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 + 5*
a*b^4)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*b^
4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^10 + 5*a^8*b^2
+ 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)^(3/4) - sqrt(2)*((a^2 - b^2)*d^7*sqrt((a^10 + 5*a^8*b^2 + 1
0*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^
10)/d^4) + (a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8
+ b^10)/d^4))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 + 5*a*b
^4)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^14 - 25*a^12*b^2 - 115*a^10*b^4 + 35*a^8*b^6 + 171*a^6*b^8 + 53
*a^4*b^10 - 17*a^2*b^12 + b^14)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*
cos(d*x + c) + sqrt(2)*((25*a^11 - 175*a^9*b^2 + 410*a^7*b^4 - 350*a^5*b^6 + 61*a^3*b^8 - 3*a*b^10)*d^3*sqrt((
a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*cos(d*x + c) + (25*a^16 - 50*a^14*b^2 - 90
*a^12*b^4 + 150*a^10*b^6 + 136*a^8*b^8 - 118*a^6*b^10 - 70*a^4*b^12 + 18*a^2*b^14 - b^16)*d*cos(d*x + c))*sqrt
((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b^2 + 5*a*b^4)*d^2*sqrt((a^10
+ 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a
^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*
b^6 + 5*a^2*b^8 + b^10)/d^4)^(1/4) + (25*a^19 + 25*a^17*b^2 - 140*a^15*b^4 - 220*a^13*b^6 + 126*a^11*b^8 + 430
*a^9*b^10 + 260*a^7*b^12 + 20*a^5*b^14 - 15*a^3*b^16 + a*b^18)*cos(d*x + c) + (25*a^18*b + 25*a^16*b^3 - 140*a
^14*b^5 - 220*a^12*b^7 + 126*a^10*b^9 + 430*a^8*b^11 + 260*a^6*b^13 + 20*a^4*b^15 - 15*a^2*b^17 + b^19)*sin(d*
x + c))/((a^2 + b^2)*cos(d*x + c)))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)^(3/4
))/(25*a^26*b^2 + 125*a^24*b^4 + 110*a^22*b^6 - 530*a^20*b^8 - 1469*a^18*b^10 - 921*a^16*b^12 + 1716*a^14*b^14
+ 3924*a^12*b^16 + 3471*a^10*b^18 + 1531*a^8*b^20 + 254*a^6*b^22 - 34*a^4*b^24 - 11*a^2*b^26 + b^28))*cos(d*x
+ c)^2 + 60*sqrt(2)*d^5*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^5 - 10*a^3*b
^2 + 5*a*b^4)*d^2*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4))/(25*a^8*b^2 - 100
*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/
d^4)^(3/4)*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4)*arctan(-((5*a^18 + 25*a^16*b
^2 + 36*a^14*b^4 - 28*a^12*b^6 - 154*a^10*b^8 - 210*a^8*b^10 - 140*a^6*b^12 - 44*a^4*b^14 - 3*a^2*b^16 + b^18)
*d^4*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((25*a^8*b^2 - 100*a^6*b^4
+ 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^23 + 35*a^21*b^2 + 91*a^19*b^4 + 69*a^17*b^6 - 174*a^15*b^8 - 5
46*a^13*b^10 - 714*a^11*b^12 - 534*a^9*b^14 - 231*a^7*b^16 - 49*a^5*b^18 - a^3*b^20 + a*b^22)*d^2*sqrt((25*a^8
*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) - sqrt(2)*((5*a^10 - 5*a^8*b^2 - 14*a^6*b^4 + 6*a^4
*b^6 + 9*a^2*b^8 - b^10)*d^7*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)/d^4)*sqrt((2
5*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/d^4) + (5*a^15 - 5*a^13*b^2 - 39*a^11*b^4 - 9*a^9*b
^6 + 79*a^7*b^8 + 81*a^5*b^10 + 19*a^3*b^12 - 3...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}} \tan {\left (c + d x \right )}\, 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))**(5/2),x)

[Out]

Integral((a + b*tan(c + d*x))**(5/2)*tan(c + d*x), 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))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 11.83, size = 2191, normalized size = 13.87 \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))^(5/2),x)

[Out]

atan(((((8*(4*b^6*d^2 - 4*a^4*b^2*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((5*a*b^4 - a^4*b*5i + a^5 -
b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3
*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*((5*a*b^
4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i - (((8*(4*b^6*d^2 - 4*a^4*b^2*d^2))/
d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d
^2))^(1/2))*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c
+ d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10
i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i)/((((8*(4*b^6*d^2 - 4*a^4*b^2*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2
)*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*((5*a*b^4 - a^4*b*5i + a^5 -
b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4
*b^4 - a^6*b^2))/d^2)*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (((8*(4
*b^6*d^2 - 4*a^4*b^2*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2
*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2)
)^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*((5*a*b^4 - a^4*b*5i
+ a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(3*a*b^10 + 8*a^3*b^8 + 6*a^5*b^6 - a^9*b^2))/
d^3))*((5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*2i + atan(((((8*(4*b^6*d^
2 - 4*a^4*b^2*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10
i - 10*a^3*b^2)/(4*d^2))^(1/2))*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)
- (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*((5*a*b^4 + a^4*b*5i + a^5 +
b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*1i - (((8*(4*b^6*d^2 - 4*a^4*b^2*d^2))/d^3 + 64*a*b^2*(a +
b*tan(c + d*x))^(1/2)*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*((5*a*b^
4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b*tan(c + d*x))^(1/2)*(b^8 -
15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d
^2))^(1/2)*1i)/((((8*(4*b^6*d^2 - 4*a^4*b^2*d^2))/d^3 - 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((5*a*b^4 + a^4*b*
5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2))*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10
i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(a + b*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2
)*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (((8*(4*b^6*d^2 - 4*a^4*b^2
*d^2))/d^3 + 64*a*b^2*(a + b*tan(c + d*x))^(1/2)*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^
2)/(4*d^2))^(1/2))*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) + (16*(a + b
*tan(c + d*x))^(1/2)*(b^8 - 15*a^2*b^6 + 15*a^4*b^4 - a^6*b^2))/d^2)*((5*a*b^4 + a^4*b*5i + a^5 + b^5*1i - a^2
*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2) - (16*(3*a*b^10 + 8*a^3*b^8 + 6*a^5*b^6 - a^9*b^2))/d^3))*((5*a*b^4 + a^
4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)/(4*d^2))^(1/2)*2i + (2*(a + b*tan(c + d*x))^(5/2))/(5*d) - (
(2*(a^2 + b^2))/d - (4*a^2)/d)*(a + b*tan(c + d*x))^(1/2) + (2*a*(a + b*tan(c + d*x))^(3/2))/(3*d)

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