Optimal. Leaf size=210 \[ \frac{2 a^3 (32 A+49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a^{5/2} C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.640494, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {4087, 4017, 4015, 3801, 215} \[ \frac{2 a^3 (32 A+49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a^{5/2} C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{d}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4087
Rule 4017
Rule 4015
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \int \frac{(a+a \sec (c+d x))^{5/2} \left (\frac{5 a A}{2}+\frac{7}{2} a C \sec (c+d x)\right )}{\sec ^{\frac{5}{2}}(c+d x)} \, dx}{7 a}\\ &=\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 \int \frac{(a+a \sec (c+d x))^{3/2} \left (\frac{5}{4} a^2 (8 A+7 C)+\frac{35}{4} a^2 C \sec (c+d x)\right )}{\sec ^{\frac{3}{2}}(c+d x)} \, dx}{35 a}\\ &=\frac{2 a^2 (8 A+7 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{5}{8} a^3 (32 A+49 C)+\frac{105}{8} a^3 C \sec (c+d x)\right )}{\sqrt{\sec (c+d x)}} \, dx}{105 a}\\ &=\frac{2 a^3 (32 A+49 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{21 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (8 A+7 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\left (a^2 C\right ) \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{2 a^3 (32 A+49 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{21 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (8 A+7 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (2 a^2 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 )}{d}\\ &=\frac{2 a^{5/2} C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}+\frac{2 a^3 (32 A+49 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{21 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 (8 A+7 C) \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a A (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}\\ \end{align*}
Mathematica [B] time = 6.41719, size = 474, normalized size = 2.26 \[ \frac{4 C \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left (\log \left (\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right )-\log (\sec (c+d x)+1)\right ) \left (A+C \sec ^2(c+d x)\right )}{d \left (1-\cos ^2(c+d x)\right ) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{5/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (\frac{(137 A+196 C) \sin (c) \cos (d x)}{21 d}+\frac{(31 A+14 C) \sin (2 c) \cos (2 d x)}{21 d}+\frac{(137 A+196 C) \cos (c) \sin (d x)}{21 d}+\frac{(31 A+14 C) \cos (2 c) \sin (2 d x)}{21 d}-\frac{4 \sec \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (32 A \sin \left (\frac{d x}{2}\right )+49 C \sin \left (\frac{d x}{2}\right )\right )}{21 d}-\frac{4 (32 A+49 C) \tan \left (\frac{c}{2}\right )}{21 d}+\frac{3 A \sin (3 c) \cos (3 d x)}{7 d}+\frac{A \sin (4 c) \cos (4 d x)}{14 d}+\frac{3 A \cos (3 c) \sin (3 d x)}{7 d}+\frac{A \cos (4 c) \sin (4 d x)}{14 d}\right )}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.409, size = 246, normalized size = 1.2 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{42\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 12\,A \left ( \cos \left ( dx+c \right ) \right ) ^{4}+21\,C\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{2}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) \right ) \sin \left ( dx+c \right ) -21\,C\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sqrt{2}\arctan \left ( 1/4\,\sqrt{2}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \left ( \cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) \right ) \right ) \sin \left ( dx+c \right ) +36\,A \left ( \cos \left ( dx+c \right ) \right ) ^{3}+44\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}+28\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}+92\,A\cos \left ( dx+c \right ) +196\,C\cos \left ( dx+c \right ) -184\,A-224\,C \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.24109, size = 1238, normalized size = 5.9 \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 = 0.631439, size = 1176, normalized size = 5.6 \begin{align*} \left [\frac{21 \,{\left (C a^{2} \cos \left (d x + c\right ) + C a^{2}\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - \frac{4 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac{4 \,{\left (3 \, A a^{2} \cos \left (d x + c\right )^{4} + 12 \, A a^{2} \cos \left (d x + c\right )^{3} +{\left (23 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (23 \, A + 28 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{42 \,{\left (d \cos \left (d x + c\right ) + d\right )}}, \frac{21 \,{\left (C a^{2} \cos \left (d x + c\right ) + C a^{2}\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right ) + \frac{2 \,{\left (3 \, A a^{2} \cos \left (d x + c\right )^{4} + 12 \, A a^{2} \cos \left (d x + c\right )^{3} +{\left (23 \, A + 7 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (23 \, A + 28 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{21 \,{\left (d \cos \left (d x + c\right ) + d\right )}}\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 \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )}{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]