Integrand size = 47, antiderivative size = 407 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {(i a+b)^3 (A-i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(i a-b)^3 (A+i B-C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )-b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{105 d^4 f}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{105 d^3 f}-\frac {2 (6 b c C-7 b B d-6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{35 d^2 f}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f} \] Output:
(I*a+b)^3*(A-I*B-C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^ (1/2)/f-(I*a-b)^3*(A+I*B-C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/ (c+I*d)^(1/2)/f+2/105*(72*a^3*C*d^3-6*a^2*b*d^2*(-49*B*d+32*C*c)+21*a*b^2* d*(8*c^2*C-10*B*c*d+15*(A-C)*d^2)-b^3*(48*c^3*C-56*B*c^2*d+70*c*(A-C)*d^2+ 105*B*d^3))*(c+d*tan(f*x+e))^(1/2)/d^4/f+2/105*b*(35*b*(A*b+B*a-C*b)*d^2+4 *(-a*d+b*c)*(-7*B*b*d-6*C*a*d+6*C*b*c))*tan(f*x+e)*(c+d*tan(f*x+e))^(1/2)/ d^3/f-2/35*(-7*B*b*d-6*C*a*d+6*C*b*c)*(a+b*tan(f*x+e))^2*(c+d*tan(f*x+e))^ (1/2)/d^2/f+2/7*C*(a+b*tan(f*x+e))^3*(c+d*tan(f*x+e))^(1/2)/d/f
Time = 6.03 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {-\frac {105 (a-i b)^3 (i A+B-i C) d \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {105 i (a+i b)^3 (A+i B-C) d \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}-\frac {2 \left (-72 a^3 C d^3+6 a^2 b d^2 (32 c C-49 B d)-21 a b^2 d \left (8 c^2 C-10 B c d+15 (A-C) d^2\right )+b^3 \left (48 c^3 C-56 B c^2 d+70 c (A-C) d^2+105 B d^3\right )\right ) \sqrt {c+d \tan (e+f x)}}{d^3}+\frac {2 b \left (35 b (A b+a B-b C) d^2+4 (b c-a d) (6 b c C-7 b B d-6 a C d)\right ) \tan (e+f x) \sqrt {c+d \tan (e+f x)}}{d^2}+\frac {6 (-6 b c C+7 b B d+6 a C d) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{d}+30 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{105 d f} \] Input:
Integrate[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)) /Sqrt[c + d*Tan[e + f*x]],x]
Output:
((-105*(a - I*b)^3*(I*A + B - I*C)*d*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt [c - I*d]])/Sqrt[c - I*d] + ((105*I)*(a + I*b)^3*(A + I*B - C)*d*ArcTanh[S qrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] - (2*(-72*a^3*C*d^3 + 6*a^2*b*d^2*(32*c*C - 49*B*d) - 21*a*b^2*d*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2) + b^3*(48*c^3*C - 56*B*c^2*d + 70*c*(A - C)*d^2 + 105*B*d^3))*Sqr t[c + d*Tan[e + f*x]])/d^3 + (2*b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a *d)*(6*b*c*C - 7*b*B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/ d^2 + (6*(-6*b*c*C + 7*b*B*d + 6*a*C*d)*(a + b*Tan[e + f*x])^2*Sqrt[c + d* Tan[e + f*x]])/d + 30*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]])/( 105*d*f)
Time = 4.74 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.03, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.383, Rules used = {3042, 4130, 27, 3042, 4130, 27, 3042, 4120, 27, 3042, 4113, 3042, 4022, 3042, 4020, 25, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{\sqrt {c+d \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 \int -\frac {(a+b \tan (e+f x))^2 \left ((6 b c C-6 a d C-7 b B d) \tan ^2(e+f x)-7 (A b-C b+a B) d \tan (e+f x)+6 b c C-a (7 A-C) d\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{7 d}+\frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\int \frac {(a+b \tan (e+f x))^2 \left ((6 b c C-6 a d C-7 b B d) \tan ^2(e+f x)-7 (A b-C b+a B) d \tan (e+f x)+6 b c C-a (7 A-C) d\right )}{\sqrt {c+d \tan (e+f x)}}dx}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\int \frac {(a+b \tan (e+f x))^2 \left ((6 b c C-6 a d C-7 b B d) \tan (e+f x)^2-7 (A b-C b+a B) d \tan (e+f x)+6 b c C-a (7 A-C) d\right )}{\sqrt {c+d \tan (e+f x)}}dx}{7 d}\) |
| \(\Big \downarrow \) 4130 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 \int -\frac {(a+b \tan (e+f x)) \left (4 c (6 c C-7 B d) b^2-a d (48 c C+7 B d) b+a^2 (35 A-11 C) d^2+\left (35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right ) \tan ^2(e+f x)+35 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{5 d}+\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\int \frac {(a+b \tan (e+f x)) \left (4 c (6 c C-7 B d) b^2-a d (48 c C+7 B d) b+a^2 (35 A-11 C) d^2+\left (35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right ) \tan ^2(e+f x)+35 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\int \frac {(a+b \tan (e+f x)) \left (4 c (6 c C-7 B d) b^2-a d (48 c C+7 B d) b+a^2 (35 A-11 C) d^2+\left (35 b (A b-C b+a B) d^2+4 (b c-a d) (6 b c C-6 a d C-7 b B d)\right ) \tan (e+f x)^2+35 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 4120 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}-\frac {2 \int -\frac {-2 c \left (24 C c^2-28 B d c+35 (A-C) d^2\right ) b^3+42 a c d (4 c C-5 B d) b^2-3 a^2 d^2 (64 c C+7 B d) b+3 a^3 (35 A-11 C) d^3+\left (-\left (\left (48 C c^3-56 B d c^2+70 (A-C) d^2 c+105 B d^3\right ) b^3\right )+21 a d \left (8 C c^2-10 B d c+15 (A-C) d^2\right ) b^2-6 a^2 d^2 (32 c C-49 B d) b+72 a^3 C d^3\right ) \tan ^2(e+f x)+105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)}{2 \sqrt {c+d \tan (e+f x)}}dx}{3 d}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {\int \frac {-2 c \left (24 C c^2-28 B d c+35 (A-C) d^2\right ) b^3+42 a c d (4 c C-5 B d) b^2-3 a^2 d^2 (64 c C+7 B d) b+3 a^3 (35 A-11 C) d^3+\left (-\left (\left (48 C c^3-56 B d c^2+70 (A-C) d^2 c+105 B d^3\right ) b^3\right )+21 a d \left (8 C c^2-10 B d c+15 (A-C) d^2\right ) b^2-6 a^2 d^2 (32 c C-49 B d) b+72 a^3 C d^3\right ) \tan ^2(e+f x)+105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}+\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {\int \frac {-2 c \left (24 C c^2-28 B d c+35 (A-C) d^2\right ) b^3+42 a c d (4 c C-5 B d) b^2-3 a^2 d^2 (64 c C+7 B d) b+3 a^3 (35 A-11 C) d^3+\left (-\left (\left (48 C c^3-56 B d c^2+70 (A-C) d^2 c+105 B d^3\right ) b^3\right )+21 a d \left (8 C c^2-10 B d c+15 (A-C) d^2\right ) b^2-6 a^2 d^2 (32 c C-49 B d) b+72 a^3 C d^3\right ) \tan (e+f x)^2+105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{3 d}+\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {\int \frac {105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)-105 \left (-\left ((A-C) a^3\right )+3 b B a^2+3 b^2 (A-C) a-b^3 B\right ) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}}{3 d}+\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {\int \frac {105 \left (B a^3+3 b (A-C) a^2-3 b^2 B a-b^3 (A-C)\right ) d^3 \tan (e+f x)-105 \left (-\left ((A-C) a^3\right )+3 b B a^2+3 b^2 (A-C) a-b^3 B\right ) d^3}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}}{3 d}+\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}+\frac {\frac {105}{2} d^3 (a+i b)^3 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {105}{2} d^3 (a-i b)^3 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}}{3 d}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}+\frac {\frac {105}{2} d^3 (a+i b)^3 (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {105}{2} d^3 (a-i b)^3 (A-i B-C) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}}{3 d}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}+\frac {\frac {105 i d^3 (a-i b)^3 (A-i B-C) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {105 i d^3 (a+i b)^3 (A+i B-C) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}}{3 d}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}+\frac {-\frac {105 i d^3 (a-i b)^3 (A-i B-C) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}+\frac {105 i d^3 (a+i b)^3 (A+i B-C) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}+\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}}{3 d}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}+\frac {\frac {105 d^2 (a-i b)^3 (A-i B-C) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {105 d^2 (a+i b)^3 (A+i B-C) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{f}+\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}}{3 d}}{5 d}}{7 d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 C (a+b \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}{7 d f}-\frac {\frac {2 (-6 a C d-7 b B d+6 b c C) (a+b \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}{5 d f}-\frac {\frac {2 b \tan (e+f x) \sqrt {c+d \tan (e+f x)} \left (35 b d^2 (a B+A b-b C)+4 (b c-a d) (-6 a C d-7 b B d+6 b c C)\right )}{3 d f}+\frac {\frac {2 \sqrt {c+d \tan (e+f x)} \left (72 a^3 C d^3-6 a^2 b d^2 (32 c C-49 B d)+21 a b^2 d \left (15 d^2 (A-C)-10 B c d+8 c^2 C\right )-\left (b^3 \left (70 c d^2 (A-C)-56 B c^2 d+105 B d^3+48 c^3 C\right )\right )\right )}{d f}+\frac {105 d^3 (a-i b)^3 (A-i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {105 d^3 (a+i b)^3 (A+i B-C) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{3 d}}{5 d}}{7 d}\) |
Input:
Int[((a + b*Tan[e + f*x])^3*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/Sqrt[ c + d*Tan[e + f*x]],x]
Output:
(2*C*(a + b*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]])/(7*d*f) - ((2*(6*b*c *C - 7*b*B*d - 6*a*C*d)*(a + b*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]])/( 5*d*f) - ((2*b*(35*b*(A*b + a*B - b*C)*d^2 + 4*(b*c - a*d)*(6*b*c*C - 7*b* B*d - 6*a*C*d))*Tan[e + f*x]*Sqrt[c + d*Tan[e + f*x]])/(3*d*f) + ((105*(a - I*b)^3*(A - I*B - C)*d^3*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I *d]*f) + (105*(a + I*b)^3*(A + I*B - C)*d^3*ArcTan[Tan[e + f*x]/Sqrt[c + I *d]])/(Sqrt[c + I*d]*f) + (2*(72*a^3*C*d^3 - 6*a^2*b*d^2*(32*c*C - 49*B*d) + 21*a*b^2*d*(8*c^2*C - 10*B*c*d + 15*(A - C)*d^2) - b^3*(48*c^3*C - 56*B *c^2*d + 70*c*(A - C)*d^2 + 105*B*d^3))*Sqrt[c + d*Tan[e + f*x]])/(d*f))/( 3*d))/(5*d))/(7*d)
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_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[b*C*Tan[e + f*x]*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 2))), x] - Simp[1/(d*(n + 2)) Int[(c + d*Tan[e + f*x])^n*Si mp[b*c*C - a*A*d*(n + 2) - (A*b + a*B - b*C)*d*(n + 2)*Tan[e + f*x] - (a*C* d*(n + 2) - b*(c*C - B*d*(n + 2)))*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[c^2 + d^2, 0] && !LtQ[n, -1]
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])))
Leaf count of result is larger than twice the leaf count of optimal. \(5977\) vs. \(2(371)=742\).
Time = 0.42 (sec) , antiderivative size = 5978, normalized size of antiderivative = 14.69
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(5978\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(25426\) |
| default | \(\text {Expression too large to display}\) | \(25426\) |
Input:
int((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1 /2),x,method=_RETURNVERBOSE)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 37247 vs. \(2 (363) = 726\).
Time = 11.45 (sec) , antiderivative size = 37247, normalized size of antiderivative = 91.52 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(1/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \] Input:
integrate((a+b*tan(f*x+e))**3*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f* x+e))**(1/2),x)
Output:
Integral((a + b*tan(e + f*x))**3*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/ sqrt(c + d*tan(e + f*x)), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Timed out} \] Input:
integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(1/2),x, algorithm="maxima")
Output:
Timed out
Exception generated. \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+ e))^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[0,24,9]%%%}+%%%{10,[0,22,9]%%%}+%%%{45,[0,20,9]%%%} +%%%{120,
Time = 104.88 (sec) , antiderivative size = 28858, normalized size of antiderivative = 70.90 \[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \] Input:
int(((a + b*tan(e + f*x))^3*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(1/2),x)
Output:
atan(((((8*(4*A*a^3*d^3*f^2 - 12*A*a*b^2*d^3*f^2 + 4*A*b^3*c*d^2*f^2 - 12* A*a^2*b*c*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^ 6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120* A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - ( 16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^ 4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a ^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a*b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60* A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2* f^4 + d^2*f^4)))^(1/2))*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b ^5*d*f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c* f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A ^4*b^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^ 4 + 6*A^4*a^10*b^2))^(1/2) - 4*A^2*a^6*c*f^2 + 4*A^2*b^6*c*f^2 - 24*A^2*a* b^5*d*f^2 - 24*A^2*a^5*b*d*f^2 - 60*A^2*a^2*b^4*c*f^2 + 60*A^2*a^4*b^2*c*f ^2 + 80*A^2*a^3*b^3*d*f^2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - (16*(c + d*ta n(e + f*x))^(1/2)*(A^2*a^6*d^2 - A^2*b^6*d^2 + 15*A^2*a^2*b^4*d^2 - 15*A^2 *a^4*b^2*d^2))/f^2)*((((8*A^2*a^6*c*f^2 - 8*A^2*b^6*c*f^2 + 48*A^2*a*b^5*d *f^2 + 48*A^2*a^5*b*d*f^2 + 120*A^2*a^2*b^4*c*f^2 - 120*A^2*a^4*b^2*c*f^2 - 160*A^2*a^3*b^3*d*f^2)^2/4 - (16*c^2*f^4 + 16*d^2*f^4)*(A^4*a^12 + A^4*b ^12 + 6*A^4*a^2*b^10 + 15*A^4*a^4*b^8 + 20*A^4*a^6*b^6 + 15*A^4*a^8*b^4...
\[ \int \frac {(a+b \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {2 \sqrt {d \tan \left (f x +e \right )+c}\, a^{4}+\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{5}}{d \tan \left (f x +e \right )+c}d x \right ) b^{3} c d f +3 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{4}}{d \tan \left (f x +e \right )+c}d x \right ) a \,b^{2} c d f +\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{4}}{d \tan \left (f x +e \right )+c}d x \right ) b^{4} d f +3 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}}{d \tan \left (f x +e \right )+c}d x \right ) a^{2} b c d f +4 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}}{d \tan \left (f x +e \right )+c}d x \right ) a \,b^{3} d f -\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{d \tan \left (f x +e \right )+c}d x \right ) a^{4} d f +\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{d \tan \left (f x +e \right )+c}d x \right ) a^{3} c d f +6 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{d \tan \left (f x +e \right )+c}d x \right ) a^{2} b^{2} d f +4 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )}{d \tan \left (f x +e \right )+c}d x \right ) a^{3} b d f}{d f} \] Input:
int((a+b*tan(f*x+e))^3*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1 /2),x)
Output:
(2*sqrt(tan(e + f*x)*d + c)*a**4 + int((sqrt(tan(e + f*x)*d + c)*tan(e + f *x)**5)/(tan(e + f*x)*d + c),x)*b**3*c*d*f + 3*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)*d + c),x)*a*b**2*c*d*f + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**4)/(tan(e + f*x)*d + c),x)*b**4*d*f + 3*int(( sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)*d + c),x)*a**2*b*c *d*f + 4*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**3)/(tan(e + f*x)*d + c),x)*a*b**3*d*f - int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)*d + c),x)*a**4*d*f + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/ (tan(e + f*x)*d + c),x)*a**3*c*d*f + 6*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)*d + c),x)*a**2*b**2*d*f + 4*int((sqrt(tan(e + f* x)*d + c)*tan(e + f*x))/(tan(e + f*x)*d + c),x)*a**3*b*d*f)/(d*f)