Optimal. Leaf size=109 \[ \frac {\left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {2 a b (A-C) \tan (c+d x)}{d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^2}{d}+2 a A b x-\frac {b^2 (2 A-C) \tan (c+d x) \sec (c+d x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4095, 4048, 3770, 3767, 8} \[ \frac {\left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {2 a b (A-C) \tan (c+d x)}{d}+\frac {A \sin (c+d x) (a+b \sec (c+d x))^2}{d}+2 a A b x-\frac {b^2 (2 A-C) \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3767
Rule 3770
Rule 4048
Rule 4095
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A (a+b \sec (c+d x))^2 \sin (c+d x)}{d}+\int (a+b \sec (c+d x)) \left (2 A b+a C \sec (c+d x)-b (2 A-C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {A (a+b \sec (c+d x))^2 \sin (c+d x)}{d}-\frac {b^2 (2 A-C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \left (4 a A b+\left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \sec (c+d x)-4 a b (A-C) \sec ^2(c+d x)\right ) \, dx\\ &=2 a A b x+\frac {A (a+b \sec (c+d x))^2 \sin (c+d x)}{d}-\frac {b^2 (2 A-C) \sec (c+d x) \tan (c+d x)}{2 d}-(2 a b (A-C)) \int \sec ^2(c+d x) \, dx+\frac {1}{2} \left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \int \sec (c+d x) \, dx\\ &=2 a A b x+\frac {\left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {A (a+b \sec (c+d x))^2 \sin (c+d x)}{d}-\frac {b^2 (2 A-C) \sec (c+d x) \tan (c+d x)}{2 d}+\frac {(2 a b (A-C)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=2 a A b x+\frac {\left (2 A b^2+\left (2 a^2+b^2\right ) C\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {A (a+b \sec (c+d x))^2 \sin (c+d x)}{d}-\frac {2 a b (A-C) \tan (c+d x)}{d}-\frac {b^2 (2 A-C) \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.88, size = 352, normalized size = 3.23 \[ \frac {\sec ^2(c+d x) \left (\left (a^2 A+2 b^2 C\right ) \sin (c+d x)+\cos (2 (c+d x)) \left (-\left (C \left (2 a^2+b^2\right )+2 A b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+\left (2 a^2 C+2 A b^2+b^2 C\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 a A b (c+d x)\right )+a^2 A \sin (3 (c+d x))-2 a^2 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 a^2 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 a A b c+4 a A b d x+4 a b C \sin (2 (c+d x))-2 A b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 A b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-b^2 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+b^2 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 139, normalized size = 1.28 \[ \frac {8 \, A a b d x \cos \left (d x + c\right )^{2} + {\left (2 \, C a^{2} + {\left (2 \, A + C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, C a^{2} + {\left (2 \, A + C\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{2} \cos \left (d x + c\right )^{2} + 4 \, C a b \cos \left (d x + c\right ) + C b^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.27, size = 191, normalized size = 1.75 \[ \frac {4 \, {\left (d x + c\right )} A a b + \frac {4 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + {\left (2 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - {\left (2 \, C a^{2} + 2 \, A b^{2} + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (4 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.07, size = 133, normalized size = 1.22 \[ \frac {a^{2} A \sin \left (d x +c \right )}{d}+\frac {a^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+2 a A b x +\frac {2 A a b c}{d}+\frac {2 C a b \tan \left (d x +c \right )}{d}+\frac {A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {b^{2} C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {b^{2} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 140, normalized size = 1.28 \[ \frac {8 \, {\left (d x + c\right )} A a b - C b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, C a^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2 \, A b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 4 \, A a^{2} \sin \left (d x + c\right ) + 8 \, C a b \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.23, size = 188, normalized size = 1.72 \[ \frac {2\,\left (A\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+C\,a^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+\frac {C\,b^2\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+2\,A\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\right )}{d}+\frac {\frac {A\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {A\,a^2\,\sin \left (c+d\,x\right )}{4}+\frac {C\,b^2\,\sin \left (c+d\,x\right )}{2}+C\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{d\,\left (\frac {\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {1}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________