Optimal. Leaf size=172 \[ \frac{2 a^2 (4 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{6 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.654635, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4221, 3044, 2975, 2980, 2771} \[ \frac{2 a^2 (4 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{6 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4221
Rule 3044
Rule 2975
Rule 2980
Rule 2771
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{9}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{9}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{3 a A}{2}+\frac{1}{2} a (2 A+7 C) \cos (c+d x)\right )}{\cos ^{\frac{7}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{6 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{7}{4} a^2 (4 A+5 C)+\frac{1}{4} a^2 (16 A+35 C) \cos (c+d x)\right )}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{2 a^2 (4 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{6 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{105} \left (a (104 A+175 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 (104 A+175 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \cos (c+d x)}}+\frac{2 a^2 (4 A+5 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{15 d \sqrt{a+a \cos (c+d x)}}+\frac{6 a A \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac{2 A (a+a \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.55075, size = 102, normalized size = 0.59 \[ \frac{a \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((468 A+525 C) \cos (c+d x)+2 (52 A+35 C) \cos (2 (c+d x))+104 A \cos (3 (c+d x))+164 A+175 C \cos (3 (c+d x))+70 C)}{210 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.168, size = 108, normalized size = 0.6 \begin{align*} -{\frac{2\,a \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 104\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+175\,C \left ( \cos \left ( dx+c \right ) \right ) ^{3}+52\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+35\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+39\,A\cos \left ( dx+c \right ) +15\,A \right ) \cos \left ( dx+c \right ) }{105\,d\sin \left ( dx+c \right ) } \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{9}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.76623, size = 711, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.44365, size = 274, normalized size = 1.59 \begin{align*} \frac{2 \,{\left ({\left (104 \, A + 175 \, C\right )} a \cos \left (d x + c\right )^{3} +{\left (52 \, A + 35 \, C\right )} a \cos \left (d x + c\right )^{2} + 39 \, A a \cos \left (d x + c\right ) + 15 \, A a\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )} \sqrt{\cos \left (d x + c\right )}} \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 \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \left (d x + c\right )^{\frac{9}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]