Optimal. Leaf size=313 \[ -\frac{a \left (9 a^2+11 b^2\right ) \tan (c+d x) \sqrt [3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{10 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}+\frac{\left (18 a^2+25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{20 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.488216, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3865, 4082, 4007, 3834, 139, 138} \[ -\frac{a \left (9 a^2+11 b^2\right ) \tan (c+d x) \sqrt [3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{10 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt [3]{a+b \sec (c+d x)}}+\frac{\left (18 a^2+25 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{20 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3865
Rule 4082
Rule 4007
Rule 3834
Rule 139
Rule 138
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx &=\frac{3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac{3 \int \frac{\sec (c+d x) \left (a+\frac{5}{3} b \sec (c+d x)-2 a \sec ^2(c+d x)\right )}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{8 b}\\ &=-\frac{9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac{3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac{9 \int \frac{\sec (c+d x) \left (\frac{a b}{3}+\frac{1}{9} \left (18 a^2+25 b^2\right ) \sec (c+d x)\right )}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{40 b^2}\\ &=-\frac{9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac{3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}-\frac{\left (a \left (9 a^2+11 b^2\right )\right ) \int \frac{\sec (c+d x)}{\sqrt [3]{a+b \sec (c+d x)}} \, dx}{20 b^3}+\frac{\left (18 a^2+25 b^2\right ) \int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \, dx}{40 b^3}\\ &=-\frac{9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac{3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac{\left (a \left (9 a^2+11 b^2\right ) \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{1+x} \sqrt [3]{a+b x}} \, dx,x,\sec (c+d x)\right )}{20 b^3 d \sqrt{1-\sec (c+d x)} \sqrt{1+\sec (c+d x)}}-\frac{\left (\left (18 a^2+25 b^2\right ) \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\sec (c+d x)\right )}{40 b^3 d \sqrt{1-\sec (c+d x)} \sqrt{1+\sec (c+d x)}}\\ &=-\frac{9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac{3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}-\frac{\left (\left (18 a^2+25 b^2\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{-a-b}-\frac{b x}{-a-b}\right )^{2/3}}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\sec (c+d x)\right )}{40 b^3 d \sqrt{1-\sec (c+d x)} \sqrt{1+\sec (c+d x)} \left (-\frac{a+b \sec (c+d x)}{-a-b}\right )^{2/3}}+\frac{\left (a \left (9 a^2+11 b^2\right ) \sqrt [3]{-\frac{a+b \sec (c+d x)}{-a-b}} \tan (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} \sqrt{1+x} \sqrt [3]{-\frac{a}{-a-b}-\frac{b x}{-a-b}}} \, dx,x,\sec (c+d x)\right )}{20 b^3 d \sqrt{1-\sec (c+d x)} \sqrt{1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}}\\ &=-\frac{9 a (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 b^2 d}+\frac{3 \sec (c+d x) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{8 b d}+\frac{\left (18 a^2+25 b^2\right ) F_1\left (\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^{2/3} \tan (c+d x)}{20 \sqrt{2} b^3 d \sqrt{1+\sec (c+d x)} \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{2/3}}-\frac{a \left (9 a^2+11 b^2\right ) F_1\left (\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right ) \sqrt [3]{\frac{a+b \sec (c+d x)}{a+b}} \tan (c+d x)}{10 \sqrt{2} b^3 d \sqrt{1+\sec (c+d x)} \sqrt [3]{a+b \sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 26.4094, size = 19015, normalized size = 60.75 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.121, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}{\frac{1}{\sqrt [3]{a+b\sec \left ( dx+c \right ) }}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}, x\right ) \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 \frac{\sec ^{4}{\left (c + d x \right )}}{\sqrt [3]{a + b \sec{\left (c + d x \right )}}}\, dx \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{\sec \left (d x + c\right )^{4}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]