Optimal. Leaf size=116 \[ \frac {2 \left (a^2 C+3 a b B+b^2 C\right ) \tan (c+d x)}{3 d}+\frac {\left (2 a^2 B+2 a b C+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b (2 a C+3 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {4056, 4048, 3770, 3767, 8} \[ \frac {2 \left (a^2 C+3 a b B+b^2 C\right ) \tan (c+d x)}{3 d}+\frac {\left (2 a^2 B+2 a b C+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b (2 a C+3 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac {C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rule 3770
Rule 4048
Rule 4056
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^2 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+b \sec (c+d x)) \left ((3 a B+2 b C) \sec (c+d x)+(3 b B+2 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {b (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{6} \int \left (3 \left (2 a^2 B+b^2 B+2 a b C\right ) \sec (c+d x)+4 \left (3 a b B+a^2 C+b^2 C\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {b (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}+\frac {1}{2} \left (2 a^2 B+b^2 B+2 a b C\right ) \int \sec (c+d x) \, dx+\frac {1}{3} \left (2 \left (3 a b B+a^2 C+b^2 C\right )\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {\left (2 a^2 B+b^2 B+2 a b C\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {b (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}-\frac {\left (2 \left (3 a b B+a^2 C+b^2 C\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {\left (2 a^2 B+b^2 B+2 a b C\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {2 \left (3 a b B+a^2 C+b^2 C\right ) \tan (c+d x)}{3 d}+\frac {b (3 b B+2 a C) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {C (a+b \sec (c+d x))^2 \tan (c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.48, size = 92, normalized size = 0.79 \[ \frac {3 \left (2 a^2 B+2 a b C+b^2 B\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (2 \left (3 a^2 C+6 a b B+b^2 C \tan ^2(c+d x)+3 b^2 C\right )+3 b (2 a C+b B) \sec (c+d x)\right )}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 150, normalized size = 1.29 \[ \frac {3 \, {\left (2 \, B a^{2} + 2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (2 \, B a^{2} + 2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, C b^{2} + 2 \, {\left (3 \, C a^{2} + 6 \, B a b + 2 \, C b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.33, size = 294, normalized size = 2.53 \[ \frac {3 \, {\left (2 \, B a^{2} + 2 \, C a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (2 \, B a^{2} + 2 \, C a b + B b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.22, size = 174, normalized size = 1.50 \[ \frac {B \,a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{2} C \tan \left (d x +c \right )}{d}+\frac {2 B a b \tan \left (d x +c \right )}{d}+\frac {a b C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {C a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{2} B \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {b^{2} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {2 b^{2} C \tan \left (d x +c \right )}{3 d}+\frac {b^{2} C \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.33, size = 165, normalized size = 1.42 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C b^{2} - 6 \, C a b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 3 \, B b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, B a^{2} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 12 \, C a^{2} \tan \left (d x + c\right ) + 24 \, B a b \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.38, size = 227, normalized size = 1.96 \[ \frac {\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (B\,a^2+C\,a\,b+\frac {B\,b^2}{2}\right )}{4\,B\,a^2+4\,C\,a\,b+2\,B\,b^2}\right )\,\left (2\,B\,a^2+2\,C\,a\,b+B\,b^2\right )}{d}-\frac {\left (2\,C\,a^2-B\,b^2+2\,C\,b^2+4\,B\,a\,b-2\,C\,a\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-4\,C\,a^2-8\,B\,a\,b-\frac {4\,C\,b^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (B\,b^2+2\,C\,a^2+2\,C\,b^2+4\,B\,a\,b+2\,C\,a\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________