Optimal. Leaf size=138 \[ \frac {64 a^3 (7 B+5 C) \tan (c+d x)}{105 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (7 B+5 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{105 d}+\frac {2 a (7 B+5 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4054, 12, 3793, 3792} \[ \frac {64 a^3 (7 B+5 C) \tan (c+d x)}{105 d \sqrt {a \sec (c+d x)+a}}+\frac {16 a^2 (7 B+5 C) \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{105 d}+\frac {2 a (7 B+5 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3792
Rule 3793
Rule 4054
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {2 \int \frac {1}{2} a (7 B+5 C) \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx}{7 a}\\ &=\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{7} (7 B+5 C) \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx\\ &=\frac {2 a (7 B+5 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{35} (8 a (7 B+5 C)) \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac {16 a^2 (7 B+5 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac {2 a (7 B+5 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}+\frac {1}{105} \left (32 a^2 (7 B+5 C)\right ) \int \sec (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {64 a^3 (7 B+5 C) \tan (c+d x)}{105 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (7 B+5 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{105 d}+\frac {2 a (7 B+5 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac {2 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.37, size = 79, normalized size = 0.57 \[ \frac {2 a^3 \tan (c+d x) \left (3 (7 B+20 C) \sec ^2(c+d x)+(98 B+115 C) \sec (c+d x)+301 B+15 C \sec ^3(c+d x)+230 C\right )}{105 d \sqrt {a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 115, normalized size = 0.83 \[ \frac {2 \, {\left ({\left (301 \, B + 230 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (98 \, B + 115 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (7 \, B + 20 \, C\right )} a^{2} \cos \left (d x + c\right ) + 15 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.86, size = 216, normalized size = 1.57 \[ \frac {8 \, {\left ({\left (4 \, {\left (2 \, \sqrt {2} {\left (7 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5 \, C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, \sqrt {2} {\left (7 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5 \, C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, \sqrt {2} {\left (7 \, B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 5 \, C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 105 \, \sqrt {2} {\left (B a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + C a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{105 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a\right )}^{3} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.44, size = 119, normalized size = 0.86 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (301 B \left (\cos ^{3}\left (d x +c \right )\right )+230 C \left (\cos ^{3}\left (d x +c \right )\right )+98 B \left (\cos ^{2}\left (d x +c \right )\right )+115 C \left (\cos ^{2}\left (d x +c \right )\right )+21 B \cos \left (d x +c \right )+60 C \cos \left (d x +c \right )+15 C \right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{105 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.43, size = 590, normalized size = 4.28 \[ \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 {B\,a^2\,2{}\mathrm {i}}{d}-\frac {a^2\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (301\,B+230\,C\right )\,2{}\mathrm {i}}{105\,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 {B\,a^2\,2{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (3\,B+4\,C\right )\,10{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (5\,B+2\,C\right )\,2{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (11\,B+10\,C\right )\,2{}\mathrm {i}}{7\,d}\right )+\frac {B\,a^2\,2{}\mathrm {i}}{7\,d}+\frac {a^2\,\left (3\,B+4\,C\right )\,10{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (5\,B+2\,C\right )\,2{}\mathrm {i}}{7\,d}-\frac {a^2\,\left (11\,B+10\,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}-\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^2\,\left (5\,B+2\,C\right )\,2{}\mathrm {i}}{5\,d}-\frac {a^2\,\left (5\,B+9\,C\right )\,4{}\mathrm {i}}{5\,d}+\frac {a^2\,\left (7\,B-8\,C\right )\,2{}\mathrm {i}}{35\,d}\right )-\frac {B\,a^2\,2{}\mathrm {i}}{5\,d}-\frac {a^2\,\left (B+2\,C\right )\,2{}\mathrm {i}}{d}+\frac {a^2\,\left (B+C\right )\,4{}\mathrm {i}}{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^2\,\left (5\,B+2\,C\right )\,2{}\mathrm {i}}{3\,d}-\frac {a^2\,\left (63\,B+80\,C\right )\,2{}\mathrm {i}}{105\,d}\right )-\frac {B\,a^2\,2{}\mathrm {i}}{3\,d}+\frac {a^2\,\left (9\,B+10\,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 )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \left (B + C \sec {\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________