Optimal. Leaf size=334 \[ \frac{\left (224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right ) \tan (c+d x)}{60 b d}+\frac{\left (32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}+\frac{\left (24 a^2 A b-4 a^3 B+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}+\frac{\left (48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.711026, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4010, 4002, 3997, 3787, 3770, 3767, 8} \[ \frac{\left (224 a^2 A b^3+24 a^4 A b+121 a^3 b^2 B-4 a^5 B+128 a b^4 B+32 A b^5\right ) \tan (c+d x)}{60 b d}+\frac{\left (32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (-4 a^2 B+24 a A b+25 b^2 B\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}+\frac{\left (24 a^2 A b-4 a^3 B+53 a b^2 B+32 A b^3\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}+\frac{\left (48 a^3 A b+178 a^2 b^2 B-8 a^4 B+232 a A b^3+75 b^4 B\right ) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4010
Rule 4002
Rule 3997
Rule 3787
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^4 (5 b B+(6 A b-a B) \sec (c+d x)) \, dx}{6 b}\\ &=\frac{(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (3 b (8 A b+7 a B)+\left (24 a A b-4 a^2 B+25 b^2 B\right ) \sec (c+d x)\right ) \, dx}{30 b}\\ &=\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (3 b \left (56 a A b+24 a^2 B+25 b^2 B\right )+3 \left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) \sec (c+d x)\right ) \, dx}{120 b}\\ &=\frac{\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) (a+b \sec (c+d x)) \left (3 b \left (216 a^2 A b+64 A b^3+64 a^3 B+181 a b^2 B\right )+3 \left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \sec (c+d x)\right ) \, dx}{360 b}\\ &=\frac{\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{\int \sec (c+d x) \left (45 b \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right )+12 \left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \sec (c+d x)\right ) \, dx}{720 b}\\ &=\frac{\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}+\frac{1}{16} \left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \int \sec (c+d x) \, dx+\frac{\left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \int \sec ^2(c+d x) \, dx}{60 b}\\ &=\frac{\left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}-\frac{\left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{60 b d}\\ &=\frac{\left (32 a^3 A b+24 a A b^3+8 a^4 B+36 a^2 b^2 B+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left (24 a^4 A b+224 a^2 A b^3+32 A b^5-4 a^5 B+121 a^3 b^2 B+128 a b^4 B\right ) \tan (c+d x)}{60 b d}+\frac{\left (48 a^3 A b+232 a A b^3-8 a^4 B+178 a^2 b^2 B+75 b^4 B\right ) \sec (c+d x) \tan (c+d x)}{240 d}+\frac{\left (24 a^2 A b+32 A b^3-4 a^3 B+53 a b^2 B\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{120 b d}+\frac{\left (24 a A b-4 a^2 B+25 b^2 B\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{120 b d}+\frac{(6 A b-a B) (a+b \sec (c+d x))^4 \tan (c+d x)}{30 b d}+\frac{B (a+b \sec (c+d x))^5 \tan (c+d x)}{6 b d}\\ \end{align*}
Mathematica [A] time = 2.86417, size = 244, normalized size = 0.73 \[ \frac{15 \left (32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (160 b \left (3 a^2 A b+2 a^3 B+4 a b^2 B+A b^3\right ) \tan ^2(c+d x)+10 b^2 \left (36 a^2 B+24 a A b+5 b^2 B\right ) \sec ^3(c+d x)+15 \left (32 a^3 A b+36 a^2 b^2 B+8 a^4 B+24 a A b^3+5 b^4 B\right ) \sec (c+d x)+240 \left (6 a^2 A b^2+a^4 A+4 a^3 b B+4 a b^3 B+A b^4\right )+48 b^3 (4 a B+A b) \tan ^4(c+d x)+40 b^4 B \sec ^5(c+d x)\right )}{240 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.047, size = 550, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.996663, size = 640, normalized size = 1.92 \begin{align*} \frac{640 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} b + 960 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b^{2} + 128 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} B a b^{3} + 32 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A b^{4} - 5 \, B b^{4}{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, B a^{2} b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, A a b^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, B a^{4}{\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 )} - 480 \, A a^{3} 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 )} + 480 \, A a^{4} \tan \left (d x + c\right )}{480 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.702187, size = 797, normalized size = 2.39 \begin{align*} \frac{15 \,{\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \,{\left (15 \, A a^{4} + 40 \, B a^{3} b + 60 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{5} + 40 \, B b^{4} + 15 \,{\left (8 \, B a^{4} + 32 \, A a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{4} + 32 \,{\left (10 \, B a^{3} b + 15 \, A a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4}\right )} \cos \left (d x + c\right )^{3} + 10 \,{\left (36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 48 \,{\left (4 \, B a b^{3} + A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B \sec{\left (c + d x \right )}\right ) \left (a + b \sec{\left (c + d x \right )}\right )^{4} \sec ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.28367, size = 1601, normalized size = 4.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]