Optimal. Leaf size=303 \[ \frac {2 \left (25 a^4-31 a^2 b^2+6 b^4\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{105 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {4 b \left (41 a^2-3 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2+3 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.59, antiderivative size = 303, 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 = {3949, 4189,
4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \left (25 a^2+3 b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{105 a d \sqrt {\sec (c+d x)}}+\frac {4 b \left (41 a^2-3 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^2 d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 \left (25 a^4-31 a^2 b^2+6 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^2 d \sqrt {a+b \sec (c+d x)}}+\frac {16 b \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \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 3943
Rule 3949
Rule 4120
Rule 4189
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac {7}{2}}(c+d x)} \, dx &=\frac {2 a \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 {-8 a b-\left (5 a^2+7 b^2\right ) \sec (c+d x)-4 a b \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 a \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \int \frac {\frac {1}{2} a \left (25 a^2+3 b^2\right )+22 a^2 b \sec (c+d x)+8 a 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 a \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2+3 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}-\frac {4 \int \frac {-\frac {1}{2} a b \left (41 a^2-3 b^2\right )-\frac {1}{4} a^2 \left (25 a^2+51 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 a \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2+3 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}+\frac {1}{105} \left (2 b \left (41-\frac {3 b^2}{a^2}\right )\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx+\frac {\left (25 a^4-31 a^2 b^2+6 b^4\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{105 a^2}\\ &=\frac {2 a \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2+3 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}+\frac {\left (\left (25 a^4-31 a^2 b^2+6 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^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (2 b \left (41-\frac {3 b^2}{a^2}\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{105 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 a \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2+3 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}+\frac {\left (\left (25 a^4-31 a^2 b^2+6 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^2 \sqrt {a+b \sec (c+d x)}}+\frac {\left (2 b \left (41-\frac {3 b^2}{a^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 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=\frac {2 \left (25 a^4-31 a^2 b^2+6 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^2 d \sqrt {a+b \sec (c+d x)}}+\frac {4 b \left (41-\frac {3 b^2}{a^2}\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{105 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 a \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {16 b \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (25 a^2+3 b^2\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{105 a d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.15, size = 237, normalized size = 0.78 \begin {gather*} \frac {(a+b \sec (c+d x))^{3/2} \left (16 b \left (41 a^3+41 a^2 b-3 a b^2-3 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^4-31 a^2 b^2+6 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 (178 a^2 b+12 b^3+a \left (145 a^2+108 b^2\right ) \cos (c+d x)+78 a^2 b \cos (2 (c+d x))+15 a^3 \cos (3 (c+d x))\right ) \sin (c+d x)\right )}{420 a^2 d (b+a \cos (c+d x))^2 \sec ^{\frac {3}{2}}(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.
\(2(327)=654\).
time = 0.24, size = 2050, normalized size = 6.77
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 = 0.91, size = 501, normalized size = 1.65 \begin {gather*} \frac {\sqrt {2} {\left (-75 i \, a^{4} + 11 i \, a^{2} b^{2} - 12 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} - 11 i \, a^{2} b^{2} + 12 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 ) - 6 \, \sqrt {2} {\left (-41 i \, a^{3} b + 3 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 ) - 6 \, \sqrt {2} {\left (41 i \, a^{3} b - 3 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} + 24 \, a^{3} b \cos \left (d x + c\right )^{2} + {\left (25 \, a^{4} + 3 \, 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^{3} 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 {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\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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