Integrand size = 18, antiderivative size = 609 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {16 b c^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 d^2 e \sqrt {d+e x}} \]
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Time = 0.43 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {6423, 972, 759, 849, 858, 733, 435, 430, 21, 946, 174, 552, 551} \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {4 b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {4 b c \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {16 b c^3 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 d^2 e \sqrt {d+e x}}+\frac {16 b c^2 e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}} \]
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Rule 21
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 849
Rule 858
Rule 946
Rule 972
Rule 6423
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x (d+e x)^{5/2} \sqrt {1-c^2 x^2}} \, dx}{5 e} \\ & = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \left (-\frac {e}{d (d+e x)^{5/2} \sqrt {1-c^2 x^2}}-\frac {e}{d^2 (d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {1}{d^2 x \sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{5 e} \\ & = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{5 d^2}+\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {1-c^2 x^2}} \, dx}{5 d}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 d^2 e} \\ & = \frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{5 d^2 e}-\frac {\left (4 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{5 d^2 \left (c^2 d^2-e^2\right )}-\frac {\left (4 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-\frac {3 d}{2}+\frac {e x}{2}}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{15 d \left (c^2 d^2-e^2\right )} \\ & = \frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{5 d^2 e}-\frac {\left (8 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\frac {1}{4} \left (-3 c^2 d^2-e^2\right )-c^2 d e x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 d \left (c^2 d^2-e^2\right )^2}+\frac {\left (2 b c^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{5 d^2 \left (c^2 d^2-e^2\right )} \\ & = \frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (8 b c^4 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{15 \left (c^2 d^2-e^2\right )^2}-\frac {\left (2 b c^2 (c d-e) (c d+e) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{15 d \left (c^2 d^2-e^2\right )^2}+\frac {\left (4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{5 d^2 e \sqrt {d+e x}}-\frac {\left (4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}} \\ & = \frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 d^2 e \sqrt {d+e x}}-\frac {\left (16 b c^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (4 b c (c d-e) (c d+e) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 d \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}} \\ & = \frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 d \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {16 b c^2 e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {16 b c^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 d^2 e \sqrt {d+e x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 26.78 (sec) , antiderivative size = 8675, normalized size of antiderivative = 14.24 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1611\) vs. \(2(539)=1078\).
Time = 14.25 (sec) , antiderivative size = 1612, normalized size of antiderivative = 2.65
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1612\) |
default | \(\text {Expression too large to display}\) | \(1612\) |
parts | \(\text {Expression too large to display}\) | \(1634\) |
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\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
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