Optimal. Leaf size=731 \[ \frac {15 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac {15 \left (1+\sqrt {3}\right ) \tan (c+d x) \sqrt [3]{\sec (c+d x)+1}}{7 a d \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right ) (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}-\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) \tan (c+d x) \sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7\ 2^{2/3} a d (1-\sec (c+d x)) \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} (a \sec (c+d x)+a)^{2/3}}-\frac {15 \sqrt [3]{2} \sqrt [4]{3} \tan (c+d x) \sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a d (1-\sec (c+d x)) \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} (a \sec (c+d x)+a)^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.67, antiderivative size = 731, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3797, 3828, 3827, 51, 63, 308, 225, 1881} \[ \frac {15 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac {15 \left (1+\sqrt {3}\right ) \tan (c+d x) \sqrt [3]{\sec (c+d x)+1}}{7 a d \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right ) (a \sec (c+d x)+a)^{2/3}}-\frac {3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}-\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) \tan (c+d x) \sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7\ 2^{2/3} a d (1-\sec (c+d x)) \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} (a \sec (c+d x)+a)^{2/3}}-\frac {15 \sqrt [3]{2} \sqrt [4]{3} \tan (c+d x) \sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right ) \sqrt {\frac {(\sec (c+d x)+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{\sec (c+d x)+1}+2^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{7 a d (1-\sec (c+d x)) \sqrt {-\frac {\sqrt [3]{\sec (c+d x)+1} \left (\sqrt [3]{2}-\sqrt [3]{\sec (c+d x)+1}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{\sec (c+d x)+1}\right )^2}} (a \sec (c+d x)+a)^{2/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 225
Rule 308
Rule 1881
Rule 3797
Rule 3827
Rule 3828
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx &=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {5 \int \frac {\sec (c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx}{7 a}\\ &=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {\left (5 (1+\sec (c+d x))^{2/3}\right ) \int \frac {\sec (c+d x)}{(1+\sec (c+d x))^{2/3}} \, dx}{7 a (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}-\frac {\left (5 \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} (1+x)^{7/6}} \, dx,x,\sec (c+d x)\right )}{7 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {15 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}+\frac {\left (5 \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt [6]{1+x}} \, dx,x,\sec (c+d x)\right )}{7 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {15 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}+\frac {\left (30 \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {15 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}-\frac {\left (15 \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {2^{2/3} \left (-1+\sqrt {3}\right )-2 x^4}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}-\frac {\left (15\ 2^{2/3} \left (1-\sqrt {3}\right ) \sqrt [6]{1+\sec (c+d x)} \tan (c+d x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-x^6}} \, dx,x,\sqrt [6]{1+\sec (c+d x)}\right )}{7 a d \sqrt {1-\sec (c+d x)} (a+a \sec (c+d x))^{2/3}}\\ &=-\frac {3 \tan (c+d x)}{7 d (a+a \sec (c+d x))^{5/3}}+\frac {15 \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3}}+\frac {15 \left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)} \tan (c+d x)}{7 a d (a+a \sec (c+d x))^{2/3} \left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )}-\frac {15 \sqrt [3]{2} \sqrt [4]{3} E\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7 a d (1-\sec (c+d x)) (a+a \sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}-\frac {5\ 3^{3/4} \left (1-\sqrt {3}\right ) F\left (\cos ^{-1}\left (\frac {\sqrt [3]{2}-\left (1-\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}{\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right ) \sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right ) \sqrt {\frac {2^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\sec (c+d x)}+(1+\sec (c+d x))^{2/3}}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}} \tan (c+d x)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (a+a \sec (c+d x))^{2/3} \sqrt {-\frac {\sqrt [3]{1+\sec (c+d x)} \left (\sqrt [3]{2}-\sqrt [3]{1+\sec (c+d x)}\right )}{\left (\sqrt [3]{2}-\left (1+\sqrt {3}\right ) \sqrt [3]{1+\sec (c+d x)}\right )^2}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.36, size = 90, normalized size = 0.12 \[ \frac {\tan (c+d x) \left (5\ 2^{5/6} \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sqrt [6]{\sec (c+d x)+1} \, _2F_1\left (\frac {1}{2},\frac {7}{6};\frac {3}{2};\frac {1}{2} (1-\sec (c+d x))\right )-3\right )}{7 d (a (\sec (c+d x)+1))^{5/3}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \sec \left (d x + c\right )^{2}}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}\left (d x +c \right )}{\left (a +a \sec \left (d x +c \right )\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________