Optimal. Leaf size=136 \[ -\frac {4 d^2 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{77 b \sqrt {d \tan (a+b x)}}-\frac {4 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{77 b}-\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{77 b}+\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2691, 2693,
2694, 2653, 2720} \begin {gather*} -\frac {4 d^2 \sqrt {\sin (2 a+2 b x)} \sec (a+b x) F\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{77 b \sqrt {d \tan (a+b x)}}+\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b}-\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{77 b}-\frac {4 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{77 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2653
Rule 2691
Rule 2693
Rule 2694
Rule 2720
Rubi steps
\begin {align*} \int \sec ^5(a+b x) (d \tan (a+b x))^{3/2} \, dx &=\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b}-\frac {1}{11} d^2 \int \frac {\sec ^5(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=-\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{77 b}+\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b}-\frac {1}{77} \left (6 d^2\right ) \int \frac {\sec ^3(a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=-\frac {4 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{77 b}-\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{77 b}+\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b}-\frac {1}{77} \left (4 d^2\right ) \int \frac {\sec (a+b x)}{\sqrt {d \tan (a+b x)}} \, dx\\ &=-\frac {4 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{77 b}-\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{77 b}+\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b}-\frac {\left (4 d^2 \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)} \sqrt {\sin (a+b x)}} \, dx}{77 \sqrt {\cos (a+b x)} \sqrt {d \tan (a+b x)}}\\ &=-\frac {4 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{77 b}-\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{77 b}+\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b}-\frac {\left (4 d^2 \sec (a+b x) \sqrt {\sin (2 a+2 b x)}\right ) \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx}{77 \sqrt {d \tan (a+b x)}}\\ &=-\frac {4 d^2 F\left (\left .a-\frac {\pi }{4}+b x\right |2\right ) \sec (a+b x) \sqrt {\sin (2 a+2 b x)}}{77 b \sqrt {d \tan (a+b x)}}-\frac {4 d \sec (a+b x) \sqrt {d \tan (a+b x)}}{77 b}-\frac {2 d \sec ^3(a+b x) \sqrt {d \tan (a+b x)}}{77 b}+\frac {2 d \sec ^5(a+b x) \sqrt {d \tan (a+b x)}}{11 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.91, size = 90, normalized size = 0.66 \begin {gather*} -\frac {d \sec ^5(a+b x) \left (-23+6 \cos (2 (a+b x))+\cos (4 (a+b x))+16 \cos ^6(a+b x) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\tan ^2(a+b x)\right ) \sqrt {\sec ^2(a+b x)}\right ) \sqrt {d \tan (a+b x)}}{154 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.54, size = 251, normalized size = 1.85
method | result | size |
default | \(-\frac {\left (-1+\cos \left (b x +a \right )\right ) \left (-4 \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {\cos \left (b x +a \right )-1+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \left (\cos ^{5}\left (b x +a \right )\right ) \sin \left (b x +a \right ) \EllipticF \left (\sqrt {-\frac {\cos \left (b x +a \right )-1-\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}\, \left (\cos ^{5}\left (b x +a \right )\right )-2 \left (\cos ^{4}\left (b x +a \right )\right ) \sqrt {2}+\left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}-\left (\cos ^{2}\left (b x +a \right )\right ) \sqrt {2}-7 \cos \left (b x +a \right ) \sqrt {2}+7 \sqrt {2}\right ) \left (\cos \left (b x +a \right )+1\right )^{2} \left (\frac {d \sin \left (b x +a \right )}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sqrt {2}}{77 b \sin \left (b x +a \right )^{5} \cos \left (b x +a \right )^{4}}\) | \(251\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains complex when optimal does not.
time = 0.11, size = 125, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {i \, d} d \cos \left (b x + a\right )^{5} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + 2 \, \sqrt {-i \, d} d \cos \left (b x + a\right )^{5} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - {\left (2 \, d \cos \left (b x + a\right )^{4} + d \cos \left (b x + a\right )^{2} - 7 \, d\right )} \sqrt {\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}\right )}}{77 \, b \cos \left (b x + a\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}} \sec ^{5}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{3/2}}{{\cos \left (a+b\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________