Integrand size = 35, antiderivative size = 457 \[ \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {(i a-b)^{5/2} (i A-B) \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\left (40 a^3 A b-320 a A b^3-5 a^4 B-240 a^2 b^2 B+128 b^4 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{64 b^{3/2} d}-\frac {(i a+b)^{5/2} (i A+B) \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}+\frac {\left (40 a^2 A b-64 A b^3-5 a^3 B-112 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{64 b d \sqrt {\cot (c+d x)}}+\frac {\left (40 a A b-5 a^2 B-48 b^2 B\right ) (a+b \tan (c+d x))^{3/2}}{96 b d \sqrt {\cot (c+d x)}}+\frac {(8 A b-a B) (a+b \tan (c+d x))^{5/2}}{24 b d \sqrt {\cot (c+d x)}}+\frac {B (a+b \tan (c+d x))^{7/2}}{4 b d \sqrt {\cot (c+d x)}} \]
-(I*a-b)^(5/2)*(I*A-B)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+ c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d+1/64*(40*A*a^3*b-320*A*a*b^ 3-5*B*a^4-240*B*a^2*b^2+128*B*b^4)*arctanh(b^(1/2)*tan(d*x+c)^(1/2)/(a+b*t an(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/b^(3/2)/d-(I*a+b)^(5/2 )*(I*A+B)*arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*c ot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d+1/64*(40*A*a^2*b-64*A*b^3-5*B*a^3-112*B *a*b^2)*(a+b*tan(d*x+c))^(1/2)/b/d/cot(d*x+c)^(1/2)+1/96*(40*A*a*b-5*B*a^2 -48*B*b^2)*(a+b*tan(d*x+c))^(3/2)/b/d/cot(d*x+c)^(1/2)+1/24*(8*A*b-B*a)*(a +b*tan(d*x+c))^(5/2)/b/d/cot(d*x+c)^(1/2)+1/4*B*(a+b*tan(d*x+c))^(7/2)/b/d /cot(d*x+c)^(1/2)
Time = 6.25 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} \left (-192 \sqrt [4]{-1} (-a+i b)^{5/2} b (i A+B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+192 (-1)^{3/4} (a+i b)^{5/2} b (A+i B) \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-3 \left (-40 a^2 A b+64 A b^3+5 a^3 B+112 a b^2 B\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}-2 \left (-40 a A b+5 a^2 B+48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}+8 (8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}+48 B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}-\frac {3 \sqrt {a} \left (-40 a^3 A b+320 a A b^3+5 a^4 B+240 a^2 b^2 B-128 b^4 B\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{\sqrt {b} \sqrt {a+b \tan (c+d x)}}\right )}{192 b d} \]
(Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(-192*(-1)^(1/4)*(-a + I*b)^(5/2)*b *(I*A + B)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] + 192*(-1)^(3/4)*(a + I*b)^(5/2)*b*(A + I*B)*ArcTan[((-1) ^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - 3*(-4 0*a^2*A*b + 64*A*b^3 + 5*a^3*B + 112*a*b^2*B)*Sqrt[Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]] - 2*(-40*a*A*b + 5*a^2*B + 48*b^2*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2) + 8*(8*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(5/2) + 48*B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(7/2) - (3*S qrt[a]*(-40*a^3*A*b + 320*a*A*b^3 + 5*a^4*B + 240*a^2*b^2*B - 128*b^4*B)*A rcSinh[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]]*Sqrt[1 + (b*Tan[c + d*x])/a]) /(Sqrt[b]*Sqrt[a + b*Tan[c + d*x]])))/(192*b*d)
Time = 2.76 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.92, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 4729, 3042, 4090, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4130, 27, 3042, 4138, 2035, 2257, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot (c+d x)^{3/2}}dx\) |
\(\Big \downarrow \) 4729 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \tan (c+d x)^{3/2} (a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))dx\) |
\(\Big \downarrow \) 4090 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\int -\frac {(a+b \tan (c+d x))^{5/2} \left (-\left ((8 A b-a B) \tan ^2(c+d x)\right )+8 b B \tan (c+d x)+a B\right )}{2 \sqrt {\tan (c+d x)}}dx}{4 b}+\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\int \frac {(a+b \tan (c+d x))^{5/2} \left (-\left ((8 A b-a B) \tan ^2(c+d x)\right )+8 b B \tan (c+d x)+a B\right )}{\sqrt {\tan (c+d x)}}dx}{8 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\int \frac {(a+b \tan (c+d x))^{5/2} \left (-\left ((8 A b-a B) \tan (c+d x)^2\right )+8 b B \tan (c+d x)+a B\right )}{\sqrt {\tan (c+d x)}}dx}{8 b}\right )\) |
\(\Big \downarrow \) 4130 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{3} \int \frac {(a+b \tan (c+d x))^{3/2} \left (-\left (\left (-5 B a^2+40 A b a-48 b^2 B\right ) \tan ^2(c+d x)\right )+48 b (A b+a B) \tan (c+d x)+a (8 A b+5 a B)\right )}{2 \sqrt {\tan (c+d x)}}dx-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \int \frac {(a+b \tan (c+d x))^{3/2} \left (-\left (\left (-5 B a^2+40 A b a-48 b^2 B\right ) \tan ^2(c+d x)\right )+48 b (A b+a B) \tan (c+d x)+a (8 A b+5 a B)\right )}{\sqrt {\tan (c+d x)}}dx-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \int \frac {(a+b \tan (c+d x))^{3/2} \left (-\left (\left (-5 B a^2+40 A b a-48 b^2 B\right ) \tan (c+d x)^2\right )+48 b (A b+a B) \tan (c+d x)+a (8 A b+5 a B)\right )}{\sqrt {\tan (c+d x)}}dx-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 4130 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {1}{2} \int \frac {3 \sqrt {a+b \tan (c+d x)} \left (-\left (\left (-5 B a^3+40 A b a^2-112 b^2 B a-64 A b^3\right ) \tan ^2(c+d x)\right )+64 b \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (5 B a^2+24 A b a-16 b^2 B\right )\right )}{2 \sqrt {\tan (c+d x)}}dx-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \int \frac {\sqrt {a+b \tan (c+d x)} \left (-\left (\left (-5 B a^3+40 A b a^2-112 b^2 B a-64 A b^3\right ) \tan ^2(c+d x)\right )+64 b \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (5 B a^2+24 A b a-16 b^2 B\right )\right )}{\sqrt {\tan (c+d x)}}dx-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \int \frac {\sqrt {a+b \tan (c+d x)} \left (-\left (\left (-5 B a^3+40 A b a^2-112 b^2 B a-64 A b^3\right ) \tan (c+d x)^2\right )+64 b \left (B a^2+2 A b a-b^2 B\right ) \tan (c+d x)+a \left (5 B a^2+24 A b a-16 b^2 B\right )\right )}{\sqrt {\tan (c+d x)}}dx-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 4130 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\int \frac {-\left (\left (-5 B a^4+40 A b a^3-240 b^2 B a^2-320 A b^3 a+128 b^4 B\right ) \tan ^2(c+d x)\right )+128 b \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 B a^3+88 A b a^2-144 b^2 B a-64 A b^3\right )}{2 \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {-\left (\left (-5 B a^4+40 A b a^3-240 b^2 B a^2-320 A b^3 a+128 b^4 B\right ) \tan ^2(c+d x)\right )+128 b \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 B a^3+88 A b a^2-144 b^2 B a-64 A b^3\right )}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {-\left (\left (-5 B a^4+40 A b a^3-240 b^2 B a^2-320 A b^3 a+128 b^4 B\right ) \tan (c+d x)^2\right )+128 b \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 B a^3+88 A b a^2-144 b^2 B a-64 A b^3\right )}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx-\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 4138 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int \frac {-\left (\left (-5 B a^4+40 A b a^3-240 b^2 B a^2-320 A b^3 a+128 b^4 B\right ) \tan ^2(c+d x)\right )+128 b \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 B a^3+88 A b a^2-144 b^2 B a-64 A b^3\right )}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{2 d}-\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int \frac {-\left (\left (-5 B a^4+40 A b a^3-240 b^2 B a^2-320 A b^3 a+128 b^4 B\right ) \tan ^2(c+d x)\right )+128 b \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)+a \left (5 B a^3+88 A b a^2-144 b^2 B a-64 A b^3\right )}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\sqrt {\tan (c+d x)}}{d}-\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 2257 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int \left (\frac {5 B a^4-40 A b a^3+240 b^2 B a^2+320 A b^3 a-128 b^4 B}{\sqrt {a+b \tan (c+d x)}}+\frac {128 \left (b \left (A a^3-3 b B a^2-3 A b^2 a+b^3 B\right )+b \left (B a^3+3 A b a^2-3 b^2 B a-A b^3\right ) \tan (c+d x)\right )}{\sqrt {a+b \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}\right )d\sqrt {\tan (c+d x)}}{d}-\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}\right )-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}\right )-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}}{8 b}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {B \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{7/2}}{4 b d}-\frac {-\frac {(8 A b-a B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{5/2}}{3 d}+\frac {1}{6} \left (-\frac {\left (-5 a^2 B+40 a A b-48 b^2 B\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}}{2 d}+\frac {3}{4} \left (-\frac {\left (-5 a^3 B+40 a^2 A b-112 a b^2 B-64 A b^3\right ) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}{d}+\frac {-\frac {\left (-5 a^4 B+40 a^3 A b-240 a^2 b^2 B-320 a A b^3+128 b^4 B\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {b}}+64 b (-b+i a)^{5/2} (-B+i A) \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+64 b (b+i a)^{5/2} (B+i A) \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\right )\right )}{8 b}\right )\) |
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*((B*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(7/2))/(4*b*d) - (-1/3*((8*A*b - a*B)*Sqrt[Tan[c + d*x]]*(a + b*Ta n[c + d*x])^(5/2))/d + (-1/2*((40*a*A*b - 5*a^2*B - 48*b^2*B)*Sqrt[Tan[c + d*x]]*(a + b*Tan[c + d*x])^(3/2))/d + (3*((64*(I*a - b)^(5/2)*b*(I*A - B) *ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]] - ((4 0*a^3*A*b - 320*a*A*b^3 - 5*a^4*B - 240*a^2*b^2*B + 128*b^4*B)*ArcTanh[(Sq rt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/Sqrt[b] + 64*b*(I*a + b)^(5/2)*(I*A + B)*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b* Tan[c + d*x]]])/d - ((40*a^2*A*b - 64*A*b^3 - 5*a^3*B - 112*a*b^2*B)*Sqrt[ Tan[c + d*x]]*Sqrt[a + b*Tan[c + d*x]])/d))/4)/6)/(8*b))
3.7.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m + n)) Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Ta n[e + f*x])^n*Simp[a^2*A*d*(m + n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m - 1) - b *(A*b + a*B)*d*(m + n))*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] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(3239\) vs. \(2(389)=778\).
Time = 1.05 (sec) , antiderivative size = 3240, normalized size of antiderivative = 7.09
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3240\) |
default | \(\text {Expression too large to display}\) | \(3240\) |
1/192/d*((b+a*cot(d*x+c))/cot(d*x+c))^(1/2)*(384*B*arctanh((b+a*cot(d*x+c) )^(1/2)/b^(1/2))*a*b^4*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*cot(d*x+c)^4-48*A*ln( a*cot(d*x+c)+b+(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)+(a^2+b ^2)^(1/2))*b^(9/2)*(2*(a^2+b^2)^(1/2)+2*b)^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^( 1/2)*cot(d*x+c)^4-384*A*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d *x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*a^2*b^(5/2)*(a^2+b^2)^(1/2)*c ot(d*x+c)^4+384*A*arctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b) ^(1/2))/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*a^2*b^(5/2)*(a^2+b^2)^(1/2)*cot(d*x +c)^4-192*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2) )/(2*(a^2+b^2)^(1/2)-2*b)^(1/2))*a^3*b^(3/2)*(a^2+b^2)^(1/2)*cot(d*x+c)^4+ 192*B*arctan(((2*(a^2+b^2)^(1/2)+2*b)^(1/2)-2*(b+a*cot(d*x+c))^(1/2))/(2*( a^2+b^2)^(1/2)-2*b)^(1/2))*a*b^(7/2)*(a^2+b^2)^(1/2)*cot(d*x+c)^4+192*B*ar ctan((2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2) ^(1/2)-2*b)^(1/2))*a^3*b^(3/2)*(a^2+b^2)^(1/2)*cot(d*x+c)^4-192*B*arctan(( 2*(b+a*cot(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*b)^(1/2))/(2*(a^2+b^2)^(1/2) -2*b)^(1/2))*a*b^(7/2)*(a^2+b^2)^(1/2)*cot(d*x+c)^4+264*A*a^3*b^(3/2)*(b+a *cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*cot(d*x+c)^3-192*A*a*b^(7 /2)*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*cot(d*x+c)^3-432* B*a^2*b^(5/2)*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^(1/2)*cot(d*x +c)^3+208*A*a^2*b^(5/2)*(b+a*cot(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)-2*b)^...
Leaf count of result is larger than twice the leaf count of optimal. 17849 vs. \(2 (382) = 764\).
Time = 9.42 (sec) , antiderivative size = 35731, normalized size of antiderivative = 78.19 \[ \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \tan (c+d x))^{5/2} (A+B \tan (c+d x))}{\cot ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]