\(\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx\) [460]
Optimal result
Integrand size = 35, antiderivative size = 175 \[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {(i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{3/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{3/2} d}+\frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}
\]
[Out]
(I*A-B)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))/(I*a-b)^(3/2)/d+(I*A+B)*arctanh((I*a+b)^
(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))/(I*a+b)^(3/2)/d+2*b*(A*b-B*a)*tan(d*x+c)^(1/2)/a/(a^2+b^2)/d/(a
+b*tan(d*x+c))^(1/2)
Rubi [A] (verified)
Time = 0.94 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of
steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3690, 3697, 3696, 95, 209,
212} \[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {(-B+i A) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{3/2}}+\frac {(B+i A) \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{3/2}}
\]
[In]
Int[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]
[Out]
((I*A - B)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a - b)^(3/2)*d) + ((I*A +
B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/((I*a + b)^(3/2)*d) + (2*b*(A*b - a*B
)*Sqrt[Tan[c + d*x]])/(a*(a^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]])
Rule 95
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 3690
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n
+ 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e +
f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(
m + n + 2)) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ
[a, 0])))
Rule 3696
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[A^2/f, Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e
+ f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 +
B^2, 0]
Rule 3697
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A + I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 -
I*Tan[e + f*x]), x], x] + Dist[(A - I*B)/2, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(1 + I*Tan[e +
f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[A^2
+ B^2, 0]
Rubi steps \begin{align*}
\text {integral}& = \frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {2 \int \frac {\frac {1}{2} a (a A+b B)-\frac {1}{2} a (A b-a B) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )} \\ & = \frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {(A-i B) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)}+\frac {(A+i B) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)} \\ & = \frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {(A-i B) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a-i b) d}+\frac {(A+i B) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a+i b) d} \\ & = \frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}}+\frac {(A-i B) \text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a-i b) d}+\frac {(A+i B) \text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b) d} \\ & = \frac {(i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{3/2} d}+\frac {(i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{3/2} d}+\frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a \left (a^2+b^2\right ) d \sqrt {a+b \tan (c+d x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.89 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.15
\[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {\frac {\sqrt [4]{-1} (-i a+b) (A-i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}+\frac {\sqrt [4]{-1} (a-i b) (-i A+B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}+\frac {2 b (A b-a B) \sqrt {\tan (c+d x)}}{a \sqrt {a+b \tan (c+d x)}}}{\left (a^2+b^2\right ) d}
\]
[In]
Integrate[(A + B*Tan[c + d*x])/(Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2)),x]
[Out]
(((-1)^(1/4)*((-I)*a + b)*(A - I*B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d
*x]]])/Sqrt[-a + I*b] + ((-1)^(1/4)*(a - I*b)*((-I)*A + B)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]]
)/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[a + I*b] + (2*b*(A*b - a*B)*Sqrt[Tan[c + d*x]])/(a*Sqrt[a + b*Tan[c + d*x]])
)/((a^2 + b^2)*d)
Maple [B] (warning: unable to verify)
Leaf count of result is larger than twice the leaf count of optimal. \(4926\) vs. \(2(149)=298\).
Time = 29.70 (sec) , antiderivative size = 4927, normalized size of antiderivative =
28.15
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| method | result | size |
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| default |
\(\text {Expression too large to display}\) |
\(4927\) |
| parts |
\(\text {Expression too large to display}\) |
\(1559557\) |
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[In]
int((A+B*tan(d*x+c))/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/4/d*2^(1/2)/a/(b+(a^2+b^2)^(1/2))^(1/2)/(a^2+b^2)^(3/2)*(A*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c
)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+
c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)))*a^2*(-b+(a^2+
b^2)^(1/2))^(1/2)*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^
(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-A*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(
d*x+c))-2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-
b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)))*b^2*(-b+(a^2+b^2)^(1/2))^(1/2)*((1-cos(d
*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(
1/2)-A*(a^2+b^2)^(1/2)*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*(
(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)
^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)))*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*
b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*b-A*ln(1/(
1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*((1-cos(d*x+c))*(a*(1-cos(d*x+
c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*
(1-cos(d*x+c))-a*sin(d*x+c)))*a^2*(-b+(a^2+b^2)^(1/2))^(1/2)*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-
2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+A*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x
+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc
(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)
))*b^2*(-b+(a^2+b^2)^(1/2))^(1/2)*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))
-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)+A*(a^2+b^2)^(1/2)*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(
d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot
(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)))*((1-cos(
d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))
^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*b+2*B*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1
-cos(d*x+c))-2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/
2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)))*a*b*(-b+(a^2+b^2)^(1/2))^(1/2)*((1-
cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/
2))^(1/2)+B*(a^2+b^2)^(1/2)*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c)
)-2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2
+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)))*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)
^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*a-2*B
*ln(1/(1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*((1-cos(d*x+c))*(a*(1-c
os(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+
c)-2*b*(1-cos(d*x+c))-a*sin(d*x+c)))*a*b*(-b+(a^2+b^2)^(1/2))^(1/2)*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*
x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)-B*(a^2+b^2)^(1/2)*ln(1/(1-co
s(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^
2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2*b*(1-c
os(d*x+c))-a*sin(d*x+c)))*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(
d*x+c))^(1/2)*(-b+(a^2+b^2)^(1/2))^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*a+4*A*arctan(((-b+(a^2+b^2)^(1/2))^(1/2)*(c
sc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+
c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))*a^2*b*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(
d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)+2*A*(a^2+b^2)^(1/2)*arctan(((-b+(a^2+b^2)^(1/2))^(1/
2)*(csc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc
(d*x+c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d
*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*a^2+4*A*arctan((-(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+
c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/
2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))*a^2*b*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^
2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)+2*A*(a^2+b^2)^(1/2)*arctan((-(-b+(a^2+b^2)^(1/2))^(1/2)*(cs
c(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c
))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^
2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*a^2-8*A*(a^2+b^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*b^2*(csc(
d*x+c)-cot(d*x+c))-2*B*arctan(((-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*
x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^
(1/2))^(1/2))*a^3*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^
(1/2)+2*B*arctan(((-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d
*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))
*a*b^2*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)+2*B*(
a^2+b^2)^(1/2)*arctan(((-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*
csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(
1/2))*((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*a*b-2*
B*arctan((-(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2
-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))*a^3*((
1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)+2*B*arctan((-(
-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*
x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))*a*b^2*((1-cos(d*x+
c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)+2*B*(a^2+b^2)^(1/2)*arct
an((-(-b+(a^2+b^2)^(1/2))^(1/2)*(csc(d*x+c)-cot(d*x+c))+((1-cos(d*x+c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(
csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2))/(1-cos(d*x+c))*sin(d*x+c)/(b+(a^2+b^2)^(1/2))^(1/2))*((1-cos(d*x+
c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c))-a)*csc(d*x+c))^(1/2)*a*b+8*B*(a^2+b^2)^(1/2)*
(b+(a^2+b^2)^(1/2))^(1/2)*a*b*(csc(d*x+c)-cot(d*x+c)))*((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d
*x+c))-a)/((1-cos(d*x+c))^2*csc(d*x+c)^2-1))^(1/2)/(a*(1-cos(d*x+c))^2*csc(d*x+c)^2-2*b*(csc(d*x+c)-cot(d*x+c)
)-a)/(-1/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*(csc(d*x+c)-cot(d*x+c)))^(1/2)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 18563 vs. \(2 (143) = 286\).
Time = 6.95 (sec) , antiderivative size = 18563, normalized size of antiderivative = 106.07
\[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display}
\]
[In]
integrate((A+B*tan(d*x+c))/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
Too large to include
Sympy [F]
\[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {A + B \tan {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}} \sqrt {\tan {\left (c + d x \right )}}}\, dx
\]
[In]
integrate((A+B*tan(d*x+c))/tan(d*x+c)**(1/2)/(a+b*tan(d*x+c))**(3/2),x)
[Out]
Integral((A + B*tan(c + d*x))/((a + b*tan(c + d*x))**(3/2)*sqrt(tan(c + d*x))), x)
Maxima [F]
\[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\int { \frac {B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\tan \left (d x + c\right )}} \,d x }
\]
[In]
integrate((A+B*tan(d*x+c))/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
integrate((B*tan(d*x + c) + A)/((b*tan(d*x + c) + a)^(3/2)*sqrt(tan(d*x + c))), x)
Giac [F(-1)]
Timed out. \[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\text {Timed out}
\]
[In]
integrate((A+B*tan(d*x+c))/tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(3/2),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \frac {A+B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\int \frac {A+B\,\mathrm {tan}\left (c+d\,x\right )}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x
\]
[In]
int((A + B*tan(c + d*x))/(tan(c + d*x)^(1/2)*(a + b*tan(c + d*x))^(3/2)),x)
[Out]
int((A + B*tan(c + d*x))/(tan(c + d*x)^(1/2)*(a + b*tan(c + d*x))^(3/2)), x)