Optimal. Leaf size=169 \[ \frac {64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (21 A+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.21, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {4168, 4086,
3878, 3877} \begin {gather*} \frac {64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (21 A+13 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{315 d}+\frac {2 a (21 A+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac {4 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3877
Rule 3878
Rule 4086
Rule 4168
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {2 \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (9 A+7 C)-a C \sec (c+d x)\right ) \, dx}{9 a}\\ &=-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{21} (21 A+13 C) \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{105} (8 a (21 A+13 C)) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 (21 A+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}+\frac {1}{315} \left (32 a^2 (21 A+13 C)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (21 A+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.47, size = 125, normalized size = 0.74 \begin {gather*} \frac {a^2 (2961 A+2908 C+4 (441 A+698 C) \cos (c+d x)+4 (966 A+803 C) \cos (2 (c+d x))+588 A \cos (3 (c+d x))+584 C \cos (3 (c+d x))+903 A \cos (4 (c+d x))+584 C \cos (4 (c+d x))) \sec ^4(c+d x) \sqrt {a (1+\sec (c+d x))} \tan \left (\frac {1}{2} (c+d x)\right )}{1260 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 11.71, size = 132, normalized size = 0.78
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (903 A \left (\cos ^{4}\left (d x +c \right )\right )+584 C \left (\cos ^{4}\left (d x +c \right )\right )+294 A \left (\cos ^{3}\left (d x +c \right )\right )+292 C \left (\cos ^{3}\left (d x +c \right )\right )+63 A \left (\cos ^{2}\left (d x +c \right )\right )+219 C \left (\cos ^{2}\left (d x +c \right )\right )+130 C \cos \left (d x +c \right )+35 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{315 d \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.47, size = 130, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left ({\left (903 \, A + 584 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 2 \, {\left (147 \, A + 146 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 130 \, C a^{2} \cos \left (d x + c\right ) + 35 \, 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 )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.59, size = 261, normalized size = 1.54 \begin {gather*} \frac {8 \, {\left ({\left ({\left (4 \, {\left (2 \, \sqrt {2} {\left (21 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, \sqrt {2} {\left (21 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, C a^{7} \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} + 63 \, \sqrt {2} {\left (21 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 13 \, C a^{7} \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} - 210 \, \sqrt {2} {\left (5 \, A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 3 \, C a^{7} \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} + 315 \, \sqrt {2} {\left (A a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{315 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{4} \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 = 11.20, size = 766, normalized size = 4.53 \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 (\frac {A\,a^2\,2{}\mathrm {i}}{d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (903\,A+584\,C\right )\,2{}\mathrm {i}}{315\,d}\right )}{{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1}-\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 {A\,a^2\,2{}\mathrm {i}}{d}-\frac {a^2\,\left (A+2\,C\right )\,4{}\mathrm {i}}{d}+\frac {a^2\,\left (21\,A-32\,C\right )\,2{}\mathrm {i}}{105\,d}\right )-\frac {A\,a^2\,2{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (5\,A+2\,C\right )\,4{}\mathrm {i}}{5\,d}-\frac {a^2\,\left (5\,A+32\,C\right )\,2{}\mathrm {i}}{5\,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 )}^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 {A\,a^2\,4{}\mathrm {i}}{3\,d}-\frac {a^2\,\left (3\,A+C\right )\,8{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (13\,A+20\,C\right )\,4{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (A+C\right )\,40{}\mathrm {i}}{9\,d}\right )-\frac {A\,a^2\,4{}\mathrm {i}}{3\,d}+\frac {a^2\,\left (3\,A+C\right )\,8{}\mathrm {i}}{9\,d}-\frac {a^2\,\left (13\,A+20\,C\right )\,4{}\mathrm {i}}{9\,d}+\frac {a^2\,\left (A+C\right )\,40{}\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 {A\,a^2\,10{}\mathrm {i}}{3\,d}-\frac {a^2\,\left (189\,A+292\,C\right )\,2{}\mathrm {i}}{315\,d}\right )-\frac {A\,a^2\,2{}\mathrm {i}}{3\,d}+\frac {a^2\,\left (9\,A+4\,C\right )\,2{}\mathrm {i}}{3\,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 )}+\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 {A\,a^2\,10{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (3\,A+4\,C\right )\,10{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (9\,A-16\,C\right )\,2{}\mathrm {i}}{63\,d}+\frac {a^2\,\left (11\,A+20\,C\right )\,2{}\mathrm {i}}{7\,d}\right )-\frac {A\,a^2\,2{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (11\,A+4\,C\right )\,2{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (5\,A+12\,C\right )\,6{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (5\,A-16\,C\right )\,2{}\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________