Optimal. Leaf size=120 \[ \frac {(A+3 B-7 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A-B+C) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac {2 C \tan (c+d x)}{a d \sqrt {a \sec (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 135, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {4078, 4001, 3795, 203} \[ \frac {(A+3 B-7 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(A-B+5 C) \tan (c+d x)}{2 a d \sqrt {a \sec (c+d x)+a}}-\frac {(A-B+C) \tan (c+d x) \sec (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 3795
Rule 4001
Rule 4078
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {\int \frac {\sec (c+d x) \left (a (A+B-C)+\frac {1}{2} a (A-B+5 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(A-B+5 C) \tan (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}+\frac {(A+3 B-7 C) \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a}\\ &=-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(A-B+5 C) \tan (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}-\frac {(A+3 B-7 C) \operatorname {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{2 a d}\\ &=\frac {(A+3 B-7 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B+C) \sec (c+d x) \tan (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(A-B+5 C) \tan (c+d x)}{2 a d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.49, size = 748, normalized size = 6.23 \[ \frac {4 \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}} \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )} \cos ^3\left (\frac {1}{2} (c+d x)\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {(7 A-3 B-C) \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {2 \sin ^2\left (\frac {1}{2} (c+d x)\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};-\frac {\sin ^2\left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}+5 \sqrt {-\frac {\sin ^2\left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}} \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^2 \left (3-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {-\frac {\sin ^2\left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}-\tanh ^{-1}\left (\sqrt {-\frac {\sin ^2\left (\frac {1}{2} (c+d x)\right )}{1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )\right )\right )}{10 \left (1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )\right )^{3/2}}+\frac {(A-B+C) \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}{1-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {(A-B+C) \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}{\sin \left (\frac {1}{2} (c+d x)\right )+1}+\frac {(A-B+C) \left (2 \sin \left (\frac {1}{2} (c+d x)\right )+1\right )}{4 \left (1-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}-\frac {(A-B+C) \left (1-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{4 \left (\sin \left (\frac {1}{2} (c+d x)\right )+1\right ) \sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}+\frac {3}{2} (A-B+C) \tan ^{-1}\left (\frac {1-2 \sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )-\frac {3}{2} (A-B+C) \tan ^{-1}\left (\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )+1}{\sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )+\frac {4 A \sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {1-2 \sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )}{d \sqrt {\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.50, size = 398, normalized size = 3.32 \[ \left [\frac {\sqrt {2} {\left ({\left (A + 3 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (A + 3 \, B - 7 \, C\right )} \cos \left (d x + c\right ) + A + 3 \, B - 7 \, C\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, {\left ({\left (A - B + 5 \, C\right )} \cos \left (d x + c\right ) + 4 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{8 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, -\frac {\sqrt {2} {\left ({\left (A + 3 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (A + 3 \, B - 7 \, C\right )} \cos \left (d x + c\right ) + A + 3 \, B - 7 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2 \, {\left ({\left (A - B + 5 \, C\right )} \cos \left (d x + c\right ) + 4 \, C\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{4 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 2.26, size = 201, normalized size = 1.68 \[ \frac {\frac {{\left (\frac {\sqrt {2} {\left (A a^{2} - B a^{2} + C a^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} - \frac {\sqrt {2} {\left (A a^{2} - B a^{2} + 9 \, C a^{2}\right )}}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}} + \frac {\sqrt {2} {\left (A + 3 \, B - 7 \, C\right )} \log \left ({\left | -\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.78, size = 581, normalized size = 4.84 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right ) \left (A \sin \left (d x +c \right ) \ln \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+3 B \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sin \left (d x +c \right )}\right ) \cos \left (d x +c \right )-7 C \sin \left (d x +c \right ) \ln \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+A \ln \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+3 B \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \ln \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right )-7 C \ln \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sin \left (d x +c \right )}\right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-2 A \left (\cos ^{2}\left (d x +c \right )\right )+2 B \left (\cos ^{2}\left (d x +c \right )\right )-10 C \left (\cos ^{2}\left (d x +c \right )\right )+2 A \cos \left (d x +c \right )-2 B \cos \left (d x +c \right )+2 C \cos \left (d x +c \right )+8 C \right )}{4 d \sin \left (d x +c \right )^{3} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\cos \left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________