Optimal. Leaf size=333 \[ \frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{480 d}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{512 d}+\frac{a (12 B+5 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.949499, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4088, 4018, 4016, 3803, 3801, 215} \[ \frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{480 d}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{5/2} (1304 A+1132 B+1015 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{512 d}+\frac{a (12 B+5 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4088
Rule 4018
Rule 4016
Rule 3803
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac{1}{2} a (12 A+5 C)+\frac{1}{2} a (12 B+5 C) \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac{a (12 B+5 C) \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac{15}{4} a^2 (8 A+4 B+5 C)+\frac{1}{4} a^2 (120 A+156 B+115 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac{a^2 (120 A+156 B+115 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left (\frac{5}{8} a^3 (312 A+252 B+235 C)+\frac{3}{8} a^3 (680 A+628 B+545 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac{a^3 (680 A+628 B+545 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{1}{384} \left (a^2 (1304 A+1132 B+1015 C)\right ) \int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (1304 A+1132 B+1015 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (680 A+628 B+545 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{1}{512} \left (a^2 (1304 A+1132 B+1015 C)\right ) \int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{a^3 (1304 A+1132 B+1015 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{512 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1132 B+1015 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (680 A+628 B+545 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}+\frac{\left (a^2 (1304 A+1132 B+1015 C)\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx}{1024}\\ &=\frac{a^3 (1304 A+1132 B+1015 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{512 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1132 B+1015 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (680 A+628 B+545 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}-\frac{\left (a^2 (1304 A+1132 B+1015 C)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{512 d}\\ &=\frac{a^{5/2} (1304 A+1132 B+1015 C) \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{512 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (1304 A+1132 B+1015 C) \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{768 d \sqrt{a+a \sec (c+d x)}}+\frac{a^3 (680 A+628 B+545 C) \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{960 d \sqrt{a+a \sec (c+d x)}}+\frac{a^2 (120 A+156 B+115 C) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac{a (12 B+5 C) \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac{C \sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d}\\ \end{align*}
Mathematica [A] time = 4.68739, size = 245, normalized size = 0.74 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left (4 \sin \left (\frac{1}{2} (c+d x)\right ) ((283920 A+303048 B+321370 C) \cos (c+d x)+16 (7480 A+8444 B+8555 C) \cos (2 (c+d x))+127240 A \cos (3 (c+d x))+26080 A \cos (4 (c+d x))+19560 A \cos (5 (c+d x))+93600 A+121124 B \cos (3 (c+d x))+22640 B \cos (4 (c+d x))+16980 B \cos (5 (c+d x))+112464 B+108605 C \cos (3 (c+d x))+20300 C \cos (4 (c+d x))+15225 C \cos (5 (c+d x))+137060 C)+480 \sqrt{2} (1304 A+1132 B+1015 C) \cos ^6(c+d x) \tanh ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{491520 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.451, size = 827, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.40478, size = 1756, normalized size = 5.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]