Integrand size = 27, antiderivative size = 355 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx=-\frac {(c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b)^3 f}+\frac {(c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b)^3 f}+\frac {\sqrt {b c-a d} \left (8 a^3 b c d-56 a b^3 c d+a^4 d^2+b^4 \left (8 c^2-15 d^2\right )-6 a^2 b^2 \left (4 c^2-3 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{4 b^{3/2} \left (a^2+b^2\right )^3 f}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {(b c-a d) \left (8 a b c+a^2 d+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{4 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \]
-(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(I*a+b)^3/f+( c+I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(I*a-b)^3/f+1/4 *(8*a^3*b*c*d-56*a*b^3*c*d+a^4*d^2+b^4*(8*c^2-15*d^2)-6*a^2*b^2*(4*c^2-3*d ^2))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))*(-a*d+b*c)^( 1/2)/b^(3/2)/(a^2+b^2)^3/f-1/2*(-a*d+b*c)^2*(c+d*tan(f*x+e))^(1/2)/b/(a^2+ b^2)/f/(a+b*tan(f*x+e))^2-1/4*(-a*d+b*c)*(a^2*d+8*a*b*c+9*b^2*d)*(c+d*tan( f*x+e))^(1/2)/b/(a^2+b^2)^2/f/(a+b*tan(f*x+e))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2946\) vs. \(2(355)=710\).
Time = 6.76 (sec) , antiderivative size = 2946, normalized size of antiderivative = 8.30 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Result too large to show} \]
-1/2*(b^2*(c + d*Tan[e + f*x])^(7/2))/((a^2 + b^2)*(b*c - a*d)*f*(a + b*Ta n[e + f*x])^2) - (-((b*d*(c + d*Tan[e + f*x])^(5/2))/(f*(a + b*Tan[e + f*x ]))) + (2*((-3*b*d*(b*c - a*d)*(c + d*Tan[e + f*x])^(3/2))/(2*f*(a + b*Tan [e + f*x])) + (2*((-3*b*d*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(4*f*(a + b*Tan[e + f*x])) - (2*(-((((I*Sqrt[c - I*d]*(b*(b*c - a*d)*((3*a*b^3*(b* c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d) *(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b ^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16) + a*((3*b^2*(b*c - a*d)*( (b^2*d)/2 - a*(b*c - a*d))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)) )/16 + (-(b*c) + (a*d)/2)*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c* d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16) - (d*((3*b^4*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c ^3 - 6*c*d^2)))/16 - a*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*a*b^2*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16)))/2) - I*(a*(b*c - a*d)*((3*a*b^3*(b*c - a*d)*(b*c^3 - 3*a*c^ 2*d - 3*b*c*d^2 + a*d^3))/4 - (3*b^3*d*(b*c - a*d)*(6*a*b*c*d + a^2*d^2 - b^2*(3*c^2 - 4*d^2)))/16 + (3*b^3*(b*c - a*d)*(9*b^2*c^2*d + a^2*d^3 + a*b *(4*c^3 - 6*c*d^2)))/16) - b*((3*b^2*(b*c - a*d)*((b^2*d)/2 - a*(b*c - a*d ))*(9*b^2*c^2*d + a^2*d^3 + a*b*(4*c^3 - 6*c*d^2)))/16 + (-(b*c) + (a*d)/2 )*((-3*b^4*(b*c - a*d)*(b*c^3 - 3*a*c^2*d - 3*b*c*d^2 + a*d^3))/4 - (3*...
Time = 3.12 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.13, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {3042, 4048, 27, 3042, 4132, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}
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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {\int \frac {a^2 d^3+9 b^2 c^2 d+\left (\left (a^2+4 b^2\right ) d^2-3 b c (b c-2 a d)\right ) \tan ^2(e+f x) d+a b \left (4 c^3-6 c d^2\right )+4 b \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right ) \tan (e+f x)}{2 (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}dx}{2 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^2 d^3+9 b^2 c^2 d+\left (\left (a^2+4 b^2\right ) d^2-3 b c (b c-2 a d)\right ) \tan ^2(e+f x) d+a b \left (4 c^3-6 c d^2\right )-4 b \left (b c^3-3 a d c^2-3 b d^2 c+a d^3\right ) \tan (e+f x)}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}dx}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {a^2 d^3+9 b^2 c^2 d+\left (\left (a^2+4 b^2\right ) d^2-3 b c (b c-2 a d)\right ) \tan (e+f x)^2 d+a b \left (4 c^3-6 c d^2\right )-4 b \left (b c^3-3 a d c^2-3 b d^2 c+a d^3\right ) \tan (e+f x)}{(a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}dx}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int -\frac {-d (b c-a d)^2 \left (d a^2+8 b c a+9 b^2 d\right ) \tan ^2(e+f x)-8 b (b c-a d) \left (-\left (\left (3 c^2 d-d^3\right ) a^2\right )+2 b c \left (c^2-3 d^2\right ) a+b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x)+(b c-a d) \left (-c \left (8 c^2-15 d^2\right ) b^3+a d \left (40 c^2-7 d^2\right ) b^2+a^2 \left (8 c^3-17 c d^2\right ) b+a^3 d^3\right )}{2 (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{\left (a^2+b^2\right ) (b c-a d)}-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-d (b c-a d)^2 \left (d a^2+8 b c a+9 b^2 d\right ) \tan ^2(e+f x)-8 b (b c-a d) \left (-\left (\left (3 c^2 d-d^3\right ) a^2\right )+2 b c \left (c^2-3 d^2\right ) a+b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x)+(b c-a d) \left (-c \left (8 c^2-15 d^2\right ) b^3+a d \left (40 c^2-7 d^2\right ) b^2+a^2 c \left (8 c^2-17 d^2\right ) b+a^3 d^3\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-d (b c-a d)^2 \left (d a^2+8 b c a+9 b^2 d\right ) \tan (e+f x)^2-8 b (b c-a d) \left (-\left (\left (3 c^2 d-d^3\right ) a^2\right )+2 b c \left (c^2-3 d^2\right ) a+b^2 d \left (3 c^2-d^2\right )\right ) \tan (e+f x)+(b c-a d) \left (-c \left (8 c^2-15 d^2\right ) b^3+a d \left (40 c^2-7 d^2\right ) b^2+a^2 c \left (8 c^2-17 d^2\right ) b+a^3 d^3\right )}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4136 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {8 \left (b (b c-a d) (a c+b d) \left (\left (c^2-3 d^2\right ) a^2+8 b c d a-b^2 \left (3 c^2-d^2\right )\right )-b (b c-a d)^2 \left (\left (3 c^2-d^2\right ) a^2+8 b c d a-b^2 \left (c^2-3 d^2\right )\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {8 \int \frac {b (b c-a d) (a c+b d) \left (\left (c^2-3 d^2\right ) a^2+8 b c d a-b^2 \left (3 c^2-d^2\right )\right )-b (b c-a d)^2 \left (\left (3 c^2-d^2\right ) a^2+8 b c d a-b^2 \left (c^2-3 d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {8 \int \frac {b (b c-a d) (a c+b d) \left (\left (c^2-3 d^2\right ) a^2+8 b c d a-b^2 \left (3 c^2-d^2\right )\right )-b (b c-a d)^2 \left (\left (3 c^2-d^2\right ) a^2+8 b c d a-b^2 \left (c^2-3 d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}}{4 b \left (a^2+b^2\right )}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {1}{2} b (a+i b)^3 (c-i d)^3 (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a-i b)^3 (c+i d)^3 (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {1}{2} b (a+i b)^3 (c-i d)^3 (b c-a d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} b (a-i b)^3 (c+i d)^3 (b c-a d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {i b (a+i b)^3 (c-i d)^3 (b c-a d) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i b (a-i b)^3 (c+i d)^3 (b c-a d) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {i b (a-i b)^3 (c+i d)^3 (b c-a d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i b (a+i b)^3 (c-i d)^3 (b c-a d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {b (a+i b)^3 (c-i d)^3 (b c-a d) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {b (a-i b)^3 (c+i d)^3 (b c-a d) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {8 \left (\frac {b (a-i b)^3 (c+i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 (c-i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {(b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {8 \left (\frac {b (a-i b)^3 (c+i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 (c-i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {2 (b c-a d)^2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {8 \left (\frac {b (a-i b)^3 (c+i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 (c-i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}+\frac {-\frac {(b c-a d) \left (a^2 d+8 a b c+9 b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {2 \left (a^4 d^2+8 a^3 b c d-6 a^2 b^2 \left (4 c^2-3 d^2\right )-56 a b^3 c d+b^4 \left (8 c^2-15 d^2\right )\right ) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )}+\frac {8 \left (\frac {b (a-i b)^3 (c+i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {b (a+i b)^3 (c-i d)^{5/2} (b c-a d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right ) (b c-a d)}}{4 b \left (a^2+b^2\right )}\) |
-1/2*((b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*(a + b*Tan[ e + f*x])^2) + (((8*(((a + I*b)^3*b*(c - I*d)^(5/2)*(b*c - a*d)*ArcTan[Tan [e + f*x]/Sqrt[c - I*d]])/f + ((a - I*b)^3*b*(c + I*d)^(5/2)*(b*c - a*d)*A rcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f))/(a^2 + b^2) + (2*(b*c - a*d)^(3/2)* (8*a^3*b*c*d - 56*a*b^3*c*d + a^4*d^2 + b^4*(8*c^2 - 15*d^2) - 6*a^2*b^2*( 4*c^2 - 3*d^2))*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d] ])/(Sqrt[b]*(a^2 + b^2)*f))/(2*(a^2 + b^2)*(b*c - a*d)) - ((b*c - a*d)*(8* a*b*c + a^2*d + 9*b^2*d)*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*T an[e + f*x])))/(4*b*(a^2 + b^2))
3.13.46.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
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[(A*b^2 - a*(b*B - a*C))*(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] + Simp[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[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[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] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*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[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(6742\) vs. \(2(317)=634\).
Time = 1.07 (sec) , antiderivative size = 6743, normalized size of antiderivative = 18.99
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(6743\) |
| default | \(\text {Expression too large to display}\) | \(6743\) |
Leaf count of result is larger than twice the leaf count of optimal. 14735 vs. \(2 (311) = 622\).
Time = 289.64 (sec) , antiderivative size = 29491, normalized size of antiderivative = 83.07 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Too large to display} \]
\[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Timed out} \]
Time = 51.84 (sec) , antiderivative size = 116010, normalized size of antiderivative = 326.79 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^3} \, dx=\text {Too large to display} \]
atan(((((20*a^16*b*d^18*f^2 + 8*a^17*c*d^17*f^2 - 4796*a^2*b^15*d^18*f^2 + 10476*a^4*b^13*d^18*f^2 + 14772*a^6*b^11*d^18*f^2 - 16644*a^8*b^9*d^18*f^ 2 - 10996*a^10*b^7*d^18*f^2 + 5892*a^12*b^5*d^18*f^2 + 764*a^14*b^3*d^18*f ^2 + 8*a^17*c^3*d^15*f^2 - 3708*b^17*c^2*d^16*f^2 + 6912*b^17*c^4*d^14*f^2 + 5820*b^17*c^6*d^12*f^2 - 4608*b^17*c^8*d^10*f^2 + 192*b^17*c^10*d^8*f^2 + 125788*a^2*b^15*c^2*d^16*f^2 - 223956*a^2*b^15*c^4*d^14*f^2 - 203692*a^ 2*b^15*c^6*d^12*f^2 + 145600*a^2*b^15*c^8*d^10*f^2 - 5248*a^2*b^15*c^10*d^ 8*f^2 + 470816*a^3*b^14*c^3*d^15*f^2 - 4848*a^3*b^14*c^5*d^13*f^2 - 482048 *a^3*b^14*c^7*d^11*f^2 + 73728*a^3*b^14*c^9*d^9*f^2 - 274940*a^4*b^13*c^2* d^16*f^2 + 324004*a^4*b^13*c^4*d^14*f^2 + 420684*a^4*b^13*c^6*d^12*f^2 - 1 83040*a^4*b^13*c^8*d^10*f^2 + 5696*a^4*b^13*c^10*d^8*f^2 + 125696*a^5*b^12 *c^3*d^15*f^2 - 188624*a^5*b^12*c^5*d^13*f^2 - 262656*a^5*b^12*c^7*d^11*f^ 2 + 51968*a^5*b^12*c^9*d^9*f^2 - 474836*a^6*b^11*c^2*d^16*f^2 + 859132*a^6 *b^11*c^4*d^14*f^2 + 822084*a^6*b^11*c^6*d^12*f^2 - 508992*a^6*b^11*c^8*d^ 10*f^2 + 17664*a^6*b^11*c^10*d^8*f^2 - 1071584*a^7*b^10*c^3*d^15*f^2 - 235 184*a^7*b^10*c^5*d^13*f^2 + 891392*a^7*b^10*c^7*d^11*f^2 - 133120*a^7*b^10 *c^9*d^9*f^2 + 325404*a^8*b^9*c^2*d^16*f^2 - 108972*a^8*b^9*c^4*d^14*f^2 - 350476*a^8*b^9*c^6*d^12*f^2 + 96768*a^8*b^9*c^8*d^10*f^2 - 3776*a^8*b^9*c ^10*d^8*f^2 - 346352*a^9*b^8*c^3*d^15*f^2 + 31056*a^9*b^8*c^5*d^13*f^2 + 3 27936*a^9*b^8*c^7*d^11*f^2 - 64256*a^9*b^8*c^9*d^9*f^2 + 323668*a^10*b^...