Optimal. Leaf size=305 \[ \frac {2 \left (25 a^4-17 a^2 b^2-8 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{105 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (19 a^2+8 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2-4 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.56, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3942, 4189,
4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \left (25 a^2-4 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{105 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 b \left (19 a^2+8 b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^3 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 \left (25 a^4-17 a^2 b^2-8 b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{105 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 b \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 3941
Rule 3942
Rule 3943
Rule 4120
Rule 4189
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{2}}(c+d x)} \, dx &=\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \int \frac {b+5 a \sec (c+d x)+4 b \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {\frac {1}{2} \left (-25 a^2+4 b^2\right )-\frac {23}{2} a b \sec (c+d x)-b^2 \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{35 a}\\ &=\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2-4 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt {\sec (c+d x)}}+\frac {4 \int \frac {\frac {1}{4} b \left (19 a^2+8 b^2\right )+\frac {1}{4} a \left (25 a^2+2 b^2\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{105 a^2}\\ &=\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2-4 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt {\sec (c+d x)}}+\frac {\left (b \left (19 a^2+8 b^2\right )\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{105 a^3}+\frac {\left (25 a^4-17 a^2 b^2-8 b^4\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{105 a^3}\\ &=\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2-4 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (25 a^4-17 a^2 b^2-8 b^4\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{105 a^3 \sqrt {a+b \sec (c+d x)}}+\frac {\left (b \left (19 a^2+8 b^2\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{105 a^3 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2-4 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (25 a^4-17 a^2 b^2-8 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{105 a^3 \sqrt {a+b \sec (c+d x)}}+\frac {\left (b \left (19 a^2+8 b^2\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{105 a^3 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=\frac {2 \left (25 a^4-17 a^2 b^2-8 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{105 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 b \left (19 a^2+8 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 a d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2-4 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a^2 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.82, size = 237, normalized size = 0.78 \begin {gather*} \frac {\sqrt {a+b \sec (c+d x)} \left (8 b \left (19 a^3+19 a^2 b+8 a b^2+8 b^3\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+8 \left (25 a^4-17 a^2 b^2-8 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+2 a \left (136 a^2 b-16 b^3+a \left (145 a^2-4 b^2\right ) \cos (c+d x)+36 a^2 b \cos (2 (c+d x))+15 a^3 \cos (3 (c+d x))\right ) \sin (c+d x)\right )}{420 a^3 d (b+a \cos (c+d x)) \sqrt {\sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2049\) vs.
\(2(329)=658\).
time = 0.26, size = 2050, normalized size = 6.72
method | result | size |
default | \(\text {Expression too large to display}\) | \(2050\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.88, size = 501, normalized size = 1.64 \begin {gather*} \frac {\sqrt {2} {\left (-75 i \, a^{4} + 32 i \, a^{2} b^{2} + 16 i \, b^{4}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (75 i \, a^{4} - 32 i \, a^{2} b^{2} - 16 i \, b^{4}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-19 i \, a^{3} b - 8 i \, a b^{3}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (19 i \, a^{3} b + 8 i \, a b^{3}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) + \frac {6 \, {\left (15 \, a^{4} \cos \left (d x + c\right )^{3} + 3 \, a^{3} b \cos \left (d x + c\right )^{2} + {\left (25 \, a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, a^{4} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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