Optimal. Leaf size=282 \[ \frac {4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}-\frac {8 a^2 (4615 B+4184 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {4 a (4615 B+4184 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.57, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4157, 4103,
4101, 3888, 3885, 4086, 3877} \begin {gather*} \frac {2 a^3 (299 B+280 C) \tan (c+d x) \sec ^4(c+d x)}{1287 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^3 (4615 B+4184 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt {a \sec (c+d x)+a}}+\frac {4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 (13 B+16 C) \tan (c+d x) \sec ^4(c+d x) \sqrt {a \sec (c+d x)+a}}{143 d}-\frac {8 a^2 (4615 B+4184 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{45045 d}+\frac {4 a (4615 B+4184 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac {2 a C \tan (c+d x) \sec ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{13 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3877
Rule 3885
Rule 3888
Rule 4086
Rule 4101
Rule 4103
Rule 4157
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \sec ^4(c+d x) (a+a \sec (c+d x))^{5/2} (B+C \sec (c+d x)) \, dx\\ &=\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {2}{13} \int \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (13 B+8 C)+\frac {1}{2} a (13 B+16 C) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {4}{143} \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (247 B+216 C)+\frac {1}{4} a^2 (299 B+280 C) \sec (c+d x)\right ) \, dx\\ &=\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {\left (a^2 (4615 B+4184 C)\right ) \int \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{1287}\\ &=\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {\left (2 a^2 (4615 B+4184 C)\right ) \int \sec ^3(c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{3003}\\ &=\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {4 a (4615 B+4184 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {(4 a (4615 B+4184 C)) \int \sec (c+d x) \left (\frac {3 a}{2}-a \sec (c+d x)\right ) \sqrt {a+a \sec (c+d x)} \, dx}{15015}\\ &=\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}-\frac {8 a^2 (4615 B+4184 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {4 a (4615 B+4184 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}+\frac {\left (2 a^2 (4615 B+4184 C)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx}{6435}\\ &=\frac {4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (4615 B+4184 C) \sec ^3(c+d x) \tan (c+d x)}{9009 d \sqrt {a+a \sec (c+d x)}}+\frac {2 a^3 (299 B+280 C) \sec ^4(c+d x) \tan (c+d x)}{1287 d \sqrt {a+a \sec (c+d x)}}-\frac {8 a^2 (4615 B+4184 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{45045 d}+\frac {2 a^2 (13 B+16 C) \sec ^4(c+d x) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{143 d}+\frac {4 a (4615 B+4184 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{15015 d}+\frac {2 a C \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{13 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.55, size = 131, normalized size = 0.46 \begin {gather*} \frac {2 a^3 \left (73840 B+66944 C+8 (4615 B+4184 C) \sec (c+d x)+6 (4615 B+4184 C) \sec ^2(c+d x)+5 (4615 B+4184 C) \sec ^3(c+d x)+35 (416 B+523 C) \sec ^4(c+d x)+315 (13 B+38 C) \sec ^5(c+d x)+3465 C \sec ^6(c+d x)\right ) \tan (c+d x)}{45045 d \sqrt {a (1+\sec (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 13.88, size = 185, normalized size = 0.66
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (73840 B \left (\cos ^{6}\left (d x +c \right )\right )+66944 C \left (\cos ^{6}\left (d x +c \right )\right )+36920 B \left (\cos ^{5}\left (d x +c \right )\right )+33472 C \left (\cos ^{5}\left (d x +c \right )\right )+27690 B \left (\cos ^{4}\left (d x +c \right )\right )+25104 C \left (\cos ^{4}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+20920 C \left (\cos ^{3}\left (d x +c \right )\right )+14560 B \left (\cos ^{2}\left (d x +c \right )\right )+18305 C \left (\cos ^{2}\left (d x +c \right )\right )+4095 B \cos \left (d x +c \right )+11970 C \cos \left (d x +c \right )+3465 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{45045 d \cos \left (d x +c \right )^{6} \sin \left (d x +c \right )}\) | \(185\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.97, size = 177, normalized size = 0.63 \begin {gather*} \frac {2 \, {\left (16 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{6} + 8 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 6 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (4615 \, B + 4184 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (416 \, B + 523 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 315 \, {\left (13 \, B + 38 \, C\right )} a^{2} \cos \left (d x + c\right ) + 3465 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.43, size = 351, normalized size = 1.24 \begin {gather*} \frac {8 \, {\left ({\left ({\left ({\left ({\left (4 \, {\left (2 \, \sqrt {2} {\left (1625 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13 \, \sqrt {2} {\left (1625 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 143 \, \sqrt {2} {\left (1625 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 1483 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 858 \, \sqrt {2} {\left (415 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 362 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6006 \, \sqrt {2} {\left (50 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 49 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30030 \, \sqrt {2} {\left (5 \, B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 4 \, C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45045 \, \sqrt {2} {\left (B a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{9} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{45045 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{6} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 14.05, size = 988, normalized size = 3.50 \begin {gather*} \frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (-\frac {B\,a^2\,16{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (4\,B+5\,C\right )\,32{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (650\,B+811\,C\right )\,32{}\mathrm {i}}{9009\,d}\right )+\frac {C\,a^2\,32{}\mathrm {i}}{d}-\frac {a^2\,\left (5\,B+2\,C\right )\,16{}\mathrm {i}}{7\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^3}+\frac {\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {B\,a^2\,16{}\mathrm {i}}{5\,d}-\frac {a^2\,\left (403\,B+1046\,C\right )\,16{}\mathrm {i}}{15015\,d}\right )+\frac {a^2\,\left (5\,B+2\,C\right )\,16{}\mathrm {i}}{5\,d}\right )\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^2}-\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {B\,a^2\,16{}\mathrm {i}}{3\,d}+\frac {a^2\,\left (9\,B+10\,C\right )\,16{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (13\,B-6\,C\right )\,64{}\mathrm {i}}{1287\,d}\right )+\frac {a^2\,\left (B+6\,C\right )\,64{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (5\,B+2\,C\right )\,16{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (5\,B+16\,C\right )\,16{}\mathrm {i}}{9\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {B\,a^2\,16{}\mathrm {i}}{13\,d}+\frac {a^2\,\left (3\,B+4\,C\right )\,80{}\mathrm {i}}{13\,d}-\frac {a^2\,\left (5\,B+2\,C\right )\,16{}\mathrm {i}}{13\,d}-\frac {a^2\,\left (11\,B+10\,C\right )\,16{}\mathrm {i}}{13\,d}\right )-\frac {B\,a^2\,16{}\mathrm {i}}{13\,d}-\frac {a^2\,\left (3\,B+4\,C\right )\,80{}\mathrm {i}}{13\,d}+\frac {a^2\,\left (5\,B+2\,C\right )\,16{}\mathrm {i}}{13\,d}+\frac {a^2\,\left (11\,B+10\,C\right )\,16{}\mathrm {i}}{13\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^6}-\frac {\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {B\,a^2\,16{}\mathrm {i}}{11\,d}+\frac {C\,a^2\,1792{}\mathrm {i}}{143\,d}+\frac {a^2\,\left (B+2\,C\right )\,80{}\mathrm {i}}{11\,d}-\frac {a^2\,\left (B+C\right )\,160{}\mathrm {i}}{11\,d}\right )-\frac {C\,a^2\,128{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (B-8\,C\right )\,16{}\mathrm {i}}{11\,d}+\frac {a^2\,\left (5\,B+2\,C\right )\,16{}\mathrm {i}}{11\,d}-\frac {a^2\,\left (5\,B+9\,C\right )\,32{}\mathrm {i}}{11\,d}\right )}{\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,{\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )}^5}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (4615\,B+4184\,C\right )\,32{}\mathrm {i}}{45045\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\sqrt {a+\frac {a}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (4615\,B+4184\,C\right )\,16{}\mathrm {i}}{45045\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________